text stringlengths 1 2.12k | source dict |
|---|---|
Below is the comprehensive list of Greek symbols/characters/alphabets generally used for preparing a document using LaTex. Note that The following table provides a comprehensive list/guide of mathematical Integral symbols using an appropriate example. command to add a little whitespace between them: The choices available with all fonts are: When using the STIX fonts, you also have the This is a small change from regular TeX, where the dollar sign in Note that you can set the integral boundaries by using the underscore _ and circumflex ^ symbol as seen below. The subsequent table provides a list/guide of commonly used Binary Symbols/Operators using an appropriate example. to blend well with Times), or a Unicode font that you provide. width of the symbols below: Care should be taken when putting accents on lower-case i's and j's.
Online LaTeX equation editor, generate your mathematical expressions using LaTeX with a simple way. If you think I forgot some very important basic function or symbol here, please let me know. You can try testing these Trigonometric functions commands directly on our online LaTeX compilerfor a better understanding. | {
"domain": "com.ng",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9648551566309688,
"lm_q1q2_score": 0.8839130902356964,
"lm_q2_score": 0.9161096193153989,
"openwebmath_perplexity": 13979.083181375348,
"openwebmath_score": 0.7525127530097961,
"tags": null,
"url": "https://nndcgroup.com.ng/blog/1k4v2s2.php?c0ec0e=latex-math-symbols"
} |
The following table provides a comprehensive list/guide of Trigonometric functions along with their LaTeX command. While the syntax inside the pair of dollar signs ($) aims to be TeX-like, Even though commands follow a logical naming scheme, you will probably need a table for the most common math symbols at some point. Additionally, you can use \mathdefault{...} or its alias Therefore, these Any text element can use math text. in TeX. To test these binary symbols/operators commands directly on our online LaTeX compiler for a better understanding. You can try testing these Integral symbol commands directly on our online LaTeX compiler for a better understanding. Columns are separated with ampersand & and rows with a double backslash \\ (the linebreak command). \leftarrow, \sum, \int. inside a pair of dollar signs ($). character can not be found in the custom font. The fonts used should have a Unicode mapping in order to find any fairly tricky to use, and should be considered an experimental feature for Of course LaTeX is able to typeset matrices as well. LaTeX offers math symbols for various kinds of integrals out of the box. mathtext also provides a way to use custom fonts for math. Note that you can set the integral boundaries by using the underscore _ and circumflex ^ symbol as seen below.
in the following \imath is used to avoid the extra dot over the i: You can also use a large number of the TeX symbols, as in \infty,
For this purpose LaTeX offers the following environments. LaTeX offers math symbols for various kinds of integrals out of the box. These are generally used for preparing any document using LaTex. you can then set the following parameters, which control which font file to use Example: $\begin{bmatrix}1 & 0 & \cdots & 0\\1 & 0 & \cdots & 0\\\vdots & \vdots & \ddots & \vdots \\1 & 0 & 0 & 0\end{bmatrix}$. You can use a subset TeX markup in any matplotlib text string by placing it [3] | {
"domain": "com.ng",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9648551566309688,
"lm_q1q2_score": 0.8839130902356964,
"lm_q2_score": 0.9161096193153989,
"openwebmath_perplexity": 13979.083181375348,
"openwebmath_score": 0.7525127530097961,
"tags": null,
"url": "https://nndcgroup.com.ng/blog/1k4v2s2.php?c0ec0e=latex-math-symbols"
} |
for a particular set of math characters. . a Roman font have shortcuts. expressions blend well with other text in the plot. Lower and Upper integral boundaries can be set using the symbol underscore character "_" and "^", respectively. information. Mathtext should be placed between a pair of dollar signs ($). follows: Here "s" and "t" are variable in italics font (default), "sin" is in Roman Doing things the obvious way produces brackets that are too You can try all the LaTeX symbols in the sandbox below. . . The following table provides a comprehensive list/guide of Trigonometric functions along with their LaTeX command. Customizing Matplotlib with style sheets and rcParams, Fractions, binomials, and stacked numbers. (from (La)TeX), STIX fonts (with are designed Table 258: undertilde Extensible Accents . Here are some more basic functions which don't fit in the categories mentioned above. Jump to navigation Jump to search This is an information page. This default can be changed using the mathtext.default rcParam. to use characters from the default Computer Modern fonts whenever a particular Wikipedia:LaTeX symbols. . fractions or sub/superscripts: The default font is italics for mathematical symbols. Text rendering With LaTeX). easy to display monetary values, e.g., "$100.00", if a single dollar sign . around fractions. the calligraphy A is squished into the sin. sign. . Play around with the commands, make your own formulas and copy the code to your document. All the Greek symbols/characters/alphabets are showed along with the LaTex rendered output. | {
"domain": "com.ng",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9648551566309688,
"lm_q1q2_score": 0.8839130902356964,
"lm_q2_score": 0.9161096193153989,
"openwebmath_perplexity": 13979.083181375348,
"openwebmath_score": 0.7525127530097961,
"tags": null,
"url": "https://nndcgroup.com.ng/blog/1k4v2s2.php?c0ec0e=latex-math-symbols"
} |
Form W7 記入例, Black Mouth Cur Puppies, How Many Chapters In The Ocean At The End Of The Lane, Dan Heder Wife, Toph Is Blind, Whirlpool Wrt311fzdm00 Manual, Xfi Pods Ethernet Port, Aylin Erçel Cause Of Death, Psalm 92 Prayer, Recetas Con Yuca Rallada, My Hero Academia Season 4 Opening Lyrics, Eddie Stanky 42, Kidy Fox Wake Up West Texas, Rockall Energy Stock, Soundcloud Playlist To Mp3, Fly Love Lyrics Portuguese, Proper Noun Of Athlete, Rappers Without Kids, Uromastyx Loricata For Sale, Choir Concert Reflection Essay, Dave Schultz Death Scene, Hippopotamus Heart Size, Myrtle Urkel Episodes, Lumber Tycoon 2 Guide, Leonberger Breeder California, My Hero Academia Crossover, Gillette Sales Plummet, Unblocked Google Drive Movies, Koolkat Truck Air Conditioning Price, Three Bears Chow Chows, Anisha Gregg Avani, Rurik Descendants Today, Hendrick Motorsports Employees, Prabal Gurung Family, Woke Wednesday Meaning, Dispatch Subscription Management, Taekwondo Lesson Plan Pdf, How Did Andy Biersack And Juliet Simms Meet, Agent Originaux R6, Dani Gabriel Physiotherapist, Annelle Dupuy Desoto, Measles Powerpoint Template, Peliculas Completas En Castellano Youtube, Southern Stingray Sting, Avatar 2 Online, Phoebe Cates Parents, Franchise Starbucks Canada Prix, Warrior In Samoan, Killing Lincoln Sparknotes, Rich Eisen Show Hiatus, Triple Cave Spider Xp Farm, The Workmanship Of Your Timbrels And Pipes, Mo Duinne Gaelic Pronunciation, Vw Trike For Sale On Craigslist, Custom Supercar Builder, Goat Fever Reducer, Anime Face Mask, English Bulldog Bite Force, Moff Gideon Costume, Julian Clary Ian Mackley Split, Used Cable Lasher For Sale, Best Electric Bike For Over 60s, 1993 Honda Goldwing Trike, Extreme Hills M Minecraft, Jt The Bigga Figga Height, Modern Dollhouse Plans, Isuzu Box Truck Vin Location, Upvc Windows Northern Ireland Prices, Karthika Masam Dates 2020, Funkoverse Jurassic Park How To Play, Iterative Preorder Traversal Without Stack, Winterizer Fertilizer Walmart, Is Stockx | {
"domain": "com.ng",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9648551566309688,
"lm_q1q2_score": 0.8839130902356964,
"lm_q2_score": 0.9161096193153989,
"openwebmath_perplexity": 13979.083181375348,
"openwebmath_score": 0.7525127530097961,
"tags": null,
"url": "https://nndcgroup.com.ng/blog/1k4v2s2.php?c0ec0e=latex-math-symbols"
} |
How To Play, Iterative Preorder Traversal Without Stack, Winterizer Fertilizer Walmart, Is Stockx Legit, The Curse Of Frankenstein Dailymotion, Pipefitter Blue Book App, Citadel Vs Citadel Securities, 三浦 春 馬 結婚, | {
"domain": "com.ng",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9648551566309688,
"lm_q1q2_score": 0.8839130902356964,
"lm_q2_score": 0.9161096193153989,
"openwebmath_perplexity": 13979.083181375348,
"openwebmath_score": 0.7525127530097961,
"tags": null,
"url": "https://nndcgroup.com.ng/blog/1k4v2s2.php?c0ec0e=latex-math-symbols"
} |
# Wrapping text in enumeration environment around a table
I would like to wrap the text in an enumerate environment around a table, this is my actual situation (sorry for including so much code, but I wanted to give a precise idea of the number of lines in the enumerate environment): | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.977022627413796,
"lm_q1q2_score": 0.883864040811854,
"lm_q2_score": 0.9046505331728752,
"openwebmath_perplexity": 5530.152522412758,
"openwebmath_score": 1.0000078678131104,
"tags": null,
"url": "http://tex.stackexchange.com/questions/53702/wrapping-text-in-enumeration-environment-around-a-table?answertab=active"
} |
\subsection{Given the following data set where “Target 2” represent the class attribute,
compute the naive Bayesian classification for the instance $<L,white>$ and $<XS,?>$.}
\begin{table}[h!t]
\centering
\begin{tabular}{ccc}
\toprule
Size & Color & Target2 \\
\midrule
XS & green & Yes \\
L & green & Yes \\
XS & white & No \\
M & black & No \\
XL & green & Yes \\
XS & white & Yes \\
L & black & No \\
M & green & Yes \\
\bottomrule
\end{tabular}
\end{table}
\begin{enumerate}
\item In this case the experience $E$ is made up of $e_{1} = L$ and $e_{2} = white$,
which in Naive Bayes are to be considered as independent, therefore we have:
\begin{align*}
P(yes|E) &= P(L|yes)\cdot P(white|yes) \cdot P(yes) \\
&= \sfrac{1}{5}\times \sfrac{1}{5} \times \sfrac{5}{8} \\
&= 0.025
\end{align*}
\begin{align*}
P(no|E) &= P(L|no) \cdot P(white|no) \cdot P(no) \\
&= \sfrac{1}{3} \times \sfrac{1}{3} times \sfrac{3}{8} \\
&= 0.041
\end{align*}
Then we normalize:
\begin{align*}
P(yes) &= \frac{0.025}{0.066}
\simeq 0.38
\end{align*}
\begin{align*}
P(no) &= \frac{0.041}{0.066}
\simeq 0.62
\end{align*}
As $P(no) > P(yes)$, we label $<L,white>$ as no''.
\item Now we have to classify a sample with a missing value. During the testing phase
we simply omit the attribute\footnote{Classification Other Methods, slide 24}:
\begin{align*}
P(yes|XS) &= P(XS|yes) \cdot P(yes) = \sfrac{2}{5} \times \sfrac{5}{8}
= 0.25
\end{align*}
\begin{align*}
P(no|XS) &= P(XS|no) \cdot P(no)
= \sfrac{1}{3} \times \sfrac{3}{8}
= 0.125
\end{align*}
Let us normalize
\begin{align*}
P(yes) &= \sfrac{0.25}{0.375} \simeq 0.7 \\
P(no) &= \sfrac{0.125}{0.375} \simeq 0.3
\end{align*}
Bottom line this sample is classified as yes''.
\end{enumerate} | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.977022627413796,
"lm_q1q2_score": 0.883864040811854,
"lm_q2_score": 0.9046505331728752,
"openwebmath_perplexity": 5530.152522412758,
"openwebmath_score": 1.0000078678131104,
"tags": null,
"url": "http://tex.stackexchange.com/questions/53702/wrapping-text-in-enumeration-environment-around-a-table?answertab=active"
} |
I have tried using the wrapfig and the floatftl package unsuccessfully, the table was moved to the end of the list in both cases. I have considered using two minipage environments, but I would like the text to actually wrap around the table.
-
It helps when you post questions to make complete documents including loading all the packages you need, I guessed
\usepackage{booktabs,xfrac,amsmath}
in this case. Also I fixed a few font issues (for multi-letter identifiers and angle brackets)
Changing margins within a LaTeX list is a bit delicate, but this is I think the layout you want
\documentclass{article}
\usepackage{booktabs,xfrac,amsmath}
\begin{document}
\subsection{Given the following data set where “Target 2” represent the class attribute,
compute the naive Bayesian classification for the instance $\langle L,white\rangle$ and $\langle \mathit{XS},?\rangle$.}
\savebox0{%
\begin{tabular}{ccc}
\toprule
Size & Color & Target2 \\
\midrule
XS & green & Yes \\
L & green & Yes \\
XS & white & No \\
M & black & No \\
XL & green & Yes \\
XS & white & Yes \\
L & black & No \\
M & green & Yes \\
\bottomrule
\end{tabular}}
\begin{enumerate}
\makeatletter
\dimen@\wd0
\parshape \@ne \@totalleftmargin \linewidth
\hbox to \textwidth{\hfill\vtop to \z@{\vskip1em \box\z@\vss}}
\item
In this case the experience $E$ is made up of $e_{1} = L$ and $e_{2} = \mathit{white}$,
which in Naive Bayes are to be considered as independent, therefore we have:
\begin{align*}
P(\mathit{yes}|E) &= P(L|\mathit{yes})\cdot P(\mathit{white}|\mathit{yes}) \cdot P(\mathit{yes}) \\
&= \sfrac{1}{5}\times \sfrac{1}{5} \times \sfrac{5}{8} \\
&= 0.025
\end{align*}
\begin{align*}
P(\mathit{no}|E) &= P(L|\mathit{no}) \cdot P(\mathit{white}|\mathit{no}) \cdot P(\mathit{no}) \\
&= \sfrac{1}{3} \times \sfrac{1}{3} \times \sfrac{3}{8} \\
&= 0.041
\end{align*} | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.977022627413796,
"lm_q1q2_score": 0.883864040811854,
"lm_q2_score": 0.9046505331728752,
"openwebmath_perplexity": 5530.152522412758,
"openwebmath_score": 1.0000078678131104,
"tags": null,
"url": "http://tex.stackexchange.com/questions/53702/wrapping-text-in-enumeration-environment-around-a-table?answertab=active"
} |
\parshape \@ne \@totalleftmargin \linewidth
Then we normalize:
\begin{align*}
P(\mathit{yes}) &= \frac{0.025}{0.066}
\simeq 0.38
\end{align*}
\begin{align*}
P(\mathit{no}) &= \frac{0.041}{0.066}
\simeq 0.62
\end{align*}
As $P(\mathit{no}) > P(\mathit{yes})$, we label $\langle L,\mathit{white}\rangle$ as no''.
\item Now we have to classify a sample with a missing value. During the testing phase
we simply omit the attribute\footnote{Classification Other Methods, slide 24}:
\begin{align*}
P(\mathit{yes}|\mathit{XS}) &= P(\mathit{XS}|\mathit{yes}) \cdot P(\mathit{yes}) = \sfrac{2}{5} \times \sfrac{5}{8}
= 0.25
\end{align*}
\begin{align*}
P(\mathit{no}|\mathit{XS}) &= P(\mathit{XS}|\mathit{no}) \cdot P(\mathit{no})
= \sfrac{1}{3} \times \sfrac{3}{8}
= 0.125
\end{align*}
Let us normalize
\begin{align*}
P(\mathit{yes}) &= \sfrac{0.25}{0.375} \simeq 0.7 \\
P(\mathit{no}) &= \sfrac{0.125}{0.375} \simeq 0.3
\end{align*}
Bottom line this sample is classified as yes''.
\end{enumerate}
\end{document}
-
this is great, thank you very much David! – Edo Apr 29 '12 at 10:26
Not quite right: note the "therefore we have:" is stretched out. That's probably fixable but a workaround is to put a blank line after that line, before the following align. – David Carlisle Apr 29 '12 at 13:27
I had noticed that, but I guess that a \hfill should do the job. Still it is a great answer, even though I hoped there was some package to manage these situations, thank you very much again. – Edo Apr 29 '12 at 13:30 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.977022627413796,
"lm_q1q2_score": 0.883864040811854,
"lm_q2_score": 0.9046505331728752,
"openwebmath_perplexity": 5530.152522412758,
"openwebmath_score": 1.0000078678131104,
"tags": null,
"url": "http://tex.stackexchange.com/questions/53702/wrapping-text-in-enumeration-environment-around-a-table?answertab=active"
} |
# piecewise
Conditionally defined expression or function
## Description
example
pw = piecewise(cond1,val1,cond2,val2,...) returns the piecewise expression or function pw whose value is val1 when condition cond1 is true, is val2 when cond2 is true, and so on. If no condition is true, the value of pw is NaN.
example
pw = piecewise(cond1,val1,cond2,val2,...,otherwiseVal) returns the piecewise expression or function pw that has the value otherwiseVal if no condition is true.
## Examples
collapse all
Define the following piecewise expression by using piecewise.
$\mathit{y}=\left\{\begin{array}{ll}-1& \mathit{x}<0\\ 1& \mathit{x}>0\end{array}$
syms x
y = piecewise(x < 0,-1,x > 0,1)
y =
Evaluate y at -2, 0, and 2 by using subs to substitute for x. Because y is undefined at x = 0, the value is NaN.
subs(y,x,[-2 0 2])
ans = $\left(\begin{array}{ccc}-1& \mathrm{NaN}& 1\end{array}\right)$
Define the following function symbolically.
$\mathit{y}\left(\mathit{x}\right)=\left\{\begin{array}{ll}-1& \mathit{x}<0\\ 1& \mathit{x}>0\end{array}$
syms y(x)
y(x) = piecewise(x < 0,-1,x > 0,1)
y(x) =
Because y(x) is a symbolic function, you can directly evaluate it for values of x. Evaluate y(x) at -2, 0, and 2. Because y(x) is undefined at x = 0, the value is NaN. For details, see Create Symbolic Functions.
y([-2 0 2])
ans = $\left(\begin{array}{ccc}-1& \mathrm{NaN}& 1\end{array}\right)$
Set the value of a piecewise function when no condition is true (called otherwise value) by specifying an additional input argument. If an additional argument is not specified, the default otherwise value of the function is NaN.
Define the piecewise function
$y=\left\{\begin{array}{cc}-2& x<-2\\ 0& -2
syms y(x)
y(x) = piecewise(x < -2,-2,(-2 < x) & (x < 0),0,1)
y(x) =
Evaluate y(x) between -3 and 1 by generating values of x using linspace. At -2 and 0, y(x) evaluates to 1 because the other conditions are not true.
xvalues = linspace(-3,1,5)
xvalues = 1×5
-3 -2 -1 0 1 | {
"domain": "mathworks.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9755769049752756,
"lm_q1q2_score": 0.8838627399560954,
"lm_q2_score": 0.905989815306765,
"openwebmath_perplexity": 3955.442450780083,
"openwebmath_score": 0.7444276213645935,
"tags": null,
"url": "https://www.mathworks.com/help/symbolic/piecewise.html"
} |
xvalues = linspace(-3,1,5)
xvalues = 1×5
-3 -2 -1 0 1
yvalues = y(xvalues)
yvalues = $\left(\begin{array}{ccccc}-2& 1& 0& 1& 1\end{array}\right)$
Plot the following piecewise expression by using fplot.
$y=\left\{\begin{array}{cc}-2& x<-2\\ x& -22\end{array}.$
syms x
y = piecewise(x < -2,-2,-2 < x < 2,x,x > 2,2);
fplot(y)
On creation, a piecewise expression applies existing assumptions. Apply assumptions set after creating the piecewise expression by using simplify on the expression.
Assume x > 0. Then define a piecewise expression with the same condition x > 0. piecewise automatically applies the assumption to simplify the condition.
syms x
assume(x > 0)
pw = piecewise(x < 0,-1,x > 0,1)
pw = $1$
Clear the assumption on x for further computations.
assume(x,'clear')
Create a piecewise expression pw with the condition x > 0. Then set the assumption that x > 0. Apply the assumption to pw by using simplify.
pw = piecewise(x < 0,-1,x > 0,1);
assume(x > 0)
pw = simplify(pw)
pw = $1$
Clear the assumption on x for further computations.
assume(x,'clear')
Differentiate, integrate, and find limits of a piecewise expression by using diff, int, and limit respectively.
Differentiate the following piecewise expression by using diff.
$\mathit{y}=\left\{\begin{array}{ll}1/\mathit{x}& \mathit{x}<-1\\ \mathrm{sin}\left(\mathit{x}\right)/\mathit{x}& \mathit{x}\ge -1\end{array}$
syms x
y = piecewise(x < -1,1/x,x >= -1,sin(x)/x);
diffy = diff(y,x)
diffy =
Integrate y by using int.
inty = int(y,x)
inty =
Find the limits of y at 0 by using limit.
limit(y,x,0)
ans = $1$
Find the right- and left-sided limits of y at -1. For details, see limit.
limit(y,x,-1,'right')
ans = $\mathrm{sin}\left(1\right)$
limit(y,x,-1,'left')
ans = $-1$
Add, subtract, divide, and multiply two piecewise expressions. The resulting piecewise expression is only defined where the initial piecewise expressions are defined. | {
"domain": "mathworks.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9755769049752756,
"lm_q1q2_score": 0.8838627399560954,
"lm_q2_score": 0.905989815306765,
"openwebmath_perplexity": 3955.442450780083,
"openwebmath_score": 0.7444276213645935,
"tags": null,
"url": "https://www.mathworks.com/help/symbolic/piecewise.html"
} |
syms x
pw1 = piecewise(x < -1,-1,x >= -1,1);
pw2 = piecewise(x < 0,-2,x >= 0,2);
sub = pw1-pw2
sub =
mul = pw1*pw2
mul =
div = pw1/pw2
div =
Modify a piecewise expression by replacing part of the expression using subs. Extend a piecewise expression by specifying the expression as the otherwise value of a new piecewise expression. This action combines the two piecewise expressions. piecewise does not check for overlapping or conflicting conditions. Instead, like an if-else ladder, piecewise returns the value for the first true condition.
Change the condition x < 2 in a piecewise expression to x < 0 by using subs.
syms x
pw = piecewise(x < 2,-1,x > 0,1);
pw = subs(pw,x < 2,x < 0)
pw =
Add the condition x > 5 with the value 1/x to pw by creating a new piecewise expression with pw as the otherwise value.
pw = piecewise(x > 5,1/x,pw)
pw =
## Input Arguments
collapse all
Condition, specified as a symbolic condition or variable. A symbolic variable represents an unknown condition.
Example: x > 2
Value when condition is satisfied, specified as a number, vector, matrix, or multidimensional array, or as a symbolic number, variable, vector, matrix, multidimensional array, function, or expression.
Value if no conditions are true, specified as a number, vector, matrix, or multidimensional array, or as a symbolic number, variable, vector, matrix, multidimensional array, function, or expression. If otherwiseVal is not specified, its value is NaN.
## Output Arguments
collapse all
Piecewise expression or function, returned as a symbolic expression or function. The value of pw is the value val of the first condition cond that is true. To find the value of pw, use subs to substitute for variables in pw.
## Tips
• piecewise does not check for overlapping or conflicting conditions. A piecewise expression returns the value of the first true condition and disregards any following true expressions. Thus, piecewise mimics an if-else ladder.
## Version History | {
"domain": "mathworks.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9755769049752756,
"lm_q1q2_score": 0.8838627399560954,
"lm_q2_score": 0.905989815306765,
"openwebmath_perplexity": 3955.442450780083,
"openwebmath_score": 0.7444276213645935,
"tags": null,
"url": "https://www.mathworks.com/help/symbolic/piecewise.html"
} |
## Version History
Introduced in R2016b | {
"domain": "mathworks.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9755769049752756,
"lm_q1q2_score": 0.8838627399560954,
"lm_q2_score": 0.905989815306765,
"openwebmath_perplexity": 3955.442450780083,
"openwebmath_score": 0.7444276213645935,
"tags": null,
"url": "https://www.mathworks.com/help/symbolic/piecewise.html"
} |
# question on probability of winning a game of dice
QUESTION:
A and B are playing a game by alternately rolling a die, with A starting first. Each player’s score is the number obtained on his last roll. The game ends when the sum of scores of the two players is 7, and the last player to roll the die wins. What is the probability that A wins the game
MY ATTEMPT:
On the first roll it is impossible for A to win. on the other hand B can win as long as the number on A's die and the number on B's die adds up to seven. The combinations which allow B to win are: $$S=\{(1,6),(6,1),(2,5),(5,2),(3,4),(4,3)\}$$ which is 6 out of a total of 36 combinations so the probability of B winning in the first round is $\frac{1}{6}$ and the probability of A winning is 0.
In the second roll of A a has a winning probability of $\frac{1}{6}$ using the same logic as for B in roll1 and in the next roll of B again B has a winning probability of $\frac{1}{6}$. I can continue to do the above on and on but it will only cycle back to the first scenario so i stop here.
Now if B wins in round 1 then A loses and can not proceed to the second roll. Therefore the probability of a winning is probability of B not winning in the first round into the probability of A winning in the next, which is $$(1- \frac{1}{6}) (\frac{1}{6})$$ which is $\frac{5}{36}$
Unfortunately this is not the correct answer as per the answer key. Can someone please point out where I have gone wrong. Any help is appreciated.Thanks :)
EDIT: on any next attempt the probability of A winning can be written as $(\frac{5}{6})^2$ of the previous since there is a $(\frac{5}{6})$ probability that A will not win and a $(\frac{5}{6})$ chance that B will not win hence leading to the next attempt. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9845754447499796,
"lm_q1q2_score": 0.8838487482135093,
"lm_q2_score": 0.8976952989498448,
"openwebmath_perplexity": 127.69045875533342,
"openwebmath_score": 0.7995924949645996,
"tags": null,
"url": "https://math.stackexchange.com/questions/2812614/question-on-probability-of-winning-a-game-of-dice/2812717"
} |
thus the answere is $(\frac{1}{6}) ((\frac{5}{6})^2+(\frac{5}{6})^4+...)$ which is equal to $\frac{1}{6} (\frac{1}{1-(\frac{5}{6})^2)})$ which is 6/11 and the right answer. EDIT thus the answere is $(\frac{5}{36}) ((\frac{5}{6})^2+(\frac{5}{6})^4+...)$ which is equal to $\frac{1}{6} (\frac{1}{1-(\frac{5}{6})^2)})$ which is 5/11
In your last equation, the factor $1/6$ should be the $5/36$ that you found above. This gives $5/11$ not $6/11$. You can get the answer without using infinite series, as follows: Let $p$ be the probability that A wins. Consider the situation after B's first roll. B wins $1/6$ of the time. If this doesn't happen then B is in the situation that A was in at the start. So B wins the game with probability $1/6 + (5/6)p$. But B's probability is also $1-p$. This gives $p = 5/11$.
Your value of $\frac{5}{36}$ gives the probability of A winning on A's second roll. But of course there may be later chances for A to win. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9845754447499796,
"lm_q1q2_score": 0.8838487482135093,
"lm_q2_score": 0.8976952989498448,
"openwebmath_perplexity": 127.69045875533342,
"openwebmath_score": 0.7995924949645996,
"tags": null,
"url": "https://math.stackexchange.com/questions/2812614/question-on-probability-of-winning-a-game-of-dice/2812717"
} |
This point is called the point of tangency. Rules for Dealing with Chords, Secants, Tangents in Circles This page created by Regents reviews three rules that are used when working with secants, and tangent lines of circles. AB 2 = DB * CB ………… This gives the formula for the tangent. Solution This problem is similar to the previous one, except that now we don’t have the standard equation. The straight line \ (y = x + 4\) cuts the circle \ (x^ {2} + y^ {2} = 26\) at \ (P\) and \ (Q\). Tangent. A tangent line intersects a circle at exactly one point, called the point of tangency. We’re finally done. If two tangents are drawn to a circle from an external point, 3 Circle common tangents The following set of examples explores some properties of the common tangents of pairs of circles. Tangent lines to one circle. Challenge problems: radius & tangent. The line is a tangent to the circle at P as shown below. In the figure below, line B C BC B C is tangent to the circle at point A A A. Proof: Segments tangent to circle from outside point are congruent. Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher). Tangents of circles problem (example 1) Tangents of circles problem (example 2) Tangents of circles problem (example 3) Practice: Tangents of circles problems. The required equation will be x(5) + y(6) + (–2)(x + 5) + (– 3)(y + 6) – 15 = 0, or 4x + 3y = 38. This video provides example problems of determining unknown values using the properties of a tangent line to a circle. The tangent line never crosses the circle, it just touches the circle. We know that AB is tangent to the circle at A. This lesson will cover a few examples to illustrate the equation of the tangent to a circle in point form. Since the tangent line to a circle at a point P is perpendicular to the radius to that point, theorems involving tangent lines often involve radial lines and orthogonal circles. Also find the point of contact. Question 1: Give some properties of tangents | {
"domain": "exaton.hu",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513881564148,
"lm_q1q2_score": 0.8838395350696157,
"lm_q2_score": 0.8962513724408292,
"openwebmath_perplexity": 358.98978621027345,
"openwebmath_score": 0.4151161313056946,
"tags": null,
"url": "https://www.exaton.hu/qn573/c5fe24-tangent-of-a-circle-example"
} |
and orthogonal circles. Also find the point of contact. Question 1: Give some properties of tangents to a circle. The Tangent intersects the circle’s radius at $90^{\circ}$ angle. We’ve got quite a task ahead, let’s begin! Yes! 16 = x. If two segments from the same exterior point are tangent to a circle, then the two segments are congruent. Phew! Can you find ? Comparing non-tangents to the point form will lead to some strange results, which I’ll talk about sometime later. Question: Determine the equation of the tangent to the circle: $x^{2}+y^{2}-2y+6x-7=0\;at\;the\;point\;F(-2:5)$ Solution: Write the equation of the circle in the form: $\left(x-a\right)^{2}+\left(y-b\right)^{2}+r^{2}$ Note that in the previous two problems, we’ve assumed that the given lines are tangents to the circles. Example 5 Show that the tangent to the circle x2 + y2 = 25 at the point (3, 4) touches the circle x2 + y2 – 18x – 4y + 81 = 0. BY P ythagorean Theorem, LJ 2 + JK 2 = LK 2. (2) ∠ABO=90° //tangent line is perpendicular to circle. If the center of the second circle is inside the first, then the and signs both correspond to internally tangent circles. Example 4 Find the point where the line 4y – 3x = 20 touches the circle x2 + y2 – 6x – 2y – 15 = 0. This is the currently selected item. a) state all the tangents to the circle and the point of tangency of each tangent. Note; The radius and tangent are perpendicular at the point of contact. That’ll be all for this lesson. The required equation will be x(4) + y(-3) = 25, or 4x – 3y = 25. } } } On comparing the coefficients, we get x1/3 = y1/4 = 25/25, which gives the values of x1 and y1 as 3 and 4 respectively. (1) AB is tangent to Circle O //Given. Problem 1: Given a circle with center O.Two Tangent from external point P is drawn to the given circle. Answer:The tangent lin… What is the length of AB? It meets the line OB such that OB = 10 cm. 4. A circle is a set of all points that are equidistant from a fixed point, called the center, and | {
"domain": "exaton.hu",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513881564148,
"lm_q1q2_score": 0.8838395350696157,
"lm_q2_score": 0.8962513724408292,
"openwebmath_perplexity": 358.98978621027345,
"openwebmath_score": 0.4151161313056946,
"tags": null,
"url": "https://www.exaton.hu/qn573/c5fe24-tangent-of-a-circle-example"
} |
4. A circle is a set of all points that are equidistant from a fixed point, called the center, and the segment that joins the center of a circle to any point on the circle is called the radius. This means that A T ¯ is perpendicular to T P ↔. and … Consider a circle in a plane and assume that $S$ is a point in the plane but it is outside of the circle. Think, for example, of a very rigid disc rolling on a very flat surface. Therefore, to find the values of x1 and y1, we must ‘compare’ the given equation with the equation in the point form. How do we find the length of A P ¯? The angle formed by the intersection of 2 tangents, 2 secants or 1 tangent and 1 secant outside the circle equals half the difference of the intercepted arcs!Therefore to find this angle (angle K in the examples below), all that you have to do is take the far intercepted arc and near the smaller intercepted arc and then divide that number by two! Solution We’ve done a similar problem in a previous lesson, where we used the slope form. Here, I’m interested to show you an alternate method. and are tangent to circle at points and respectively. We’ll use the new method again – to find the point of contact, we’ll simply compare the given equation with the equation in point form, and solve for x1 and y1. (4) ∠ACO=90° //tangent line is perpendicular to circle. Tangent to a Circle is a straight line that touches the circle at any one point or only one point to the circle, that point is called tangency. Solved Examples of Tangent to a Circle. vidDefer[i].setAttribute('src',vidDefer[i].getAttribute('data-src')); But there are even more special segments and lines of circles that are important to know. (3) AC is tangent to Circle O //Given. Let’s begin. Now, let’s learn the concept of tangent of a circle from an understandable example here. If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. A tangent to the inner circle would be a secant of the outer | {
"domain": "exaton.hu",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513881564148,
"lm_q1q2_score": 0.8838395350696157,
"lm_q2_score": 0.8962513724408292,
"openwebmath_perplexity": 358.98978621027345,
"openwebmath_score": 0.4151161313056946,
"tags": null,
"url": "https://www.exaton.hu/qn573/c5fe24-tangent-of-a-circle-example"
} |
radius drawn to the point of tangency. A tangent to the inner circle would be a secant of the outer circle. In the circle O, P T ↔ is a tangent and O P ¯ is the radius. How to Find the Tangent of a Circle? And when they say it's circumscribed about circle O that means that the two sides of the angle they're segments that would be part of tangent lines, so if we were to continue, so for example that right over there, that line is tangent to the circle and (mumbles) and this line is also tangent to the circle. 26 = 10 + x. Subtract 10 from each side. When two segments are drawn tangent to a circle from the same point outside the circle, the segments are congruent. Worked example 13: Equation of a tangent to a circle. The tangent to a circle is perpendicular to the radius at the point of tangency. Make a conjecture about the angle between the radius and the tangent to a circle at a point on the circle. A tangent line t to a circle C intersects the circle at a single point T.For comparison, secant lines intersect a circle at two points, whereas another line may not intersect a circle at all. it represents the equation of the tangent at the point P 1 (x 1, y 1), of a circle whose center is at S(p, q). Get access to all the courses and over 150 HD videos with your subscription, Monthly, Half-Yearly, and Yearly Plans Available, Not yet ready to subscribe? Through any point on a circle , only one tangent can be drawn; A perpendicular to a tangent at the point of contact passes thought the centre of the circle. Can the two circles be tangent? Answer:The properties are as follows: 1. Knowing these essential theorems regarding circles and tangent lines, you are going to be able to identify key components of a circle, determine how many points of intersection, external tangents, and internal tangents two circles have, as well as find the value of segments given the radius and the tangent segment. The problem has given us the equation of the tangent: 3x + 4y = 25. Example. Now, | {
"domain": "exaton.hu",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513881564148,
"lm_q1q2_score": 0.8838395350696157,
"lm_q2_score": 0.8962513724408292,
"openwebmath_perplexity": 358.98978621027345,
"openwebmath_score": 0.4151161313056946,
"tags": null,
"url": "https://www.exaton.hu/qn573/c5fe24-tangent-of-a-circle-example"
} |
tangent segment. The problem has given us the equation of the tangent: 3x + 4y = 25. Example. Now, draw a straight line from point $S$ and assume that it touches the circle at a point $T$. Consider the circle below. The point of contact therefore is (3, 4). We’ll use the point form once again. its distance from the center of the circle must be equal to its radius. At the point of tangency, the tangent of the circle is perpendicular to the radius. window.onload = init; © 2021 Calcworkshop LLC / Privacy Policy / Terms of Service. The extension problem of this topic is a belt and gear problem which asks for the length of belt required to fit around two gears. And the final step – solving the obtained line with the tangent gives us the foot of perpendicular, or the point of contact as (39/5, 2/5). Solution This one is similar to the previous problem, but applied to the general equation of the circle. and are both radii of the circle, so they are congruent. We have highlighted the tangent at A. 3. // Last Updated: January 21, 2020 - Watch Video //. Sine, Cosine and Tangent are the main functions used in Trigonometry and are based on a Right-Angled Triangle. Cross multiplying the equation gives. 16 Perpendicular Tangent Converse. Sketch the circle and the straight line on the same system of axes. Suppose line DB is the secant and AB is the tangent of the circle, then the of the secant and the tangent are related as follows: DB/AB = AB/CB. Example 1 Find the equation of the tangent to the circle x2 + y2 = 25, at the point (4, -3). Examples of Tangent The line AB is a tangent to the circle at P. A tangent line to a circle contains exactly one point of the circle A tangent to a circle is at right angles to … And if a line is tangent to a circle, then it is also perpendicular to the radius of the circle at the point of tangency, as Varsity Tutors accurately states. Take Calcworkshop for a spin with our FREE limits course. function init() { Example 6 : If the line segment JK | {
"domain": "exaton.hu",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513881564148,
"lm_q1q2_score": 0.8838395350696157,
"lm_q2_score": 0.8962513724408292,
"openwebmath_perplexity": 358.98978621027345,
"openwebmath_score": 0.4151161313056946,
"tags": null,
"url": "https://www.exaton.hu/qn573/c5fe24-tangent-of-a-circle-example"
} |
for a spin with our FREE limits course. function init() { Example 6 : If the line segment JK is tangent to circle … b) state all the secants. The circle’s center is (9, 2) and its radius is 2. Since tangent AB is perpendicular to the radius OA, ΔOAB is a right-angled triangle and OB is the hypotenuse of ΔOAB. In this geometry lesson, we’re investigating tangent of a circle. Tangent, written as tan(θ), is one of the six fundamental trigonometric functions.. Tangent definitions. On solving the equations, we get x1 = 0 and y1 = 5. Because JK is tangent to circle L, m ∠LJK = 90 ° and triangle LJK is a right triangle. Proof of the Two Tangent Theorem. Therefore, the point of contact will be (0, 5). The required perpendicular line will be (y – 2) = (4/3)(x – 9) or 4x – 3y = 30. The equation of the tangent in the point for will be xx1 + yy1 – 3(x + x1) – (y + y1) – 15 = 0, or x(x1 – 3) + y(y1 – 1) = 3x1 + y1 + 15. On comparing the coefficients, we get (x1 – 3)/(-3) = (y1 – 1)/4 = (3x1 + y1 + 15)/20. var vidDefer = document.getElementsByTagName('iframe'); By using Pythagoras theorem, OB^2 = OA^2~+~AB^2 AB^2 = OB^2~-~OA^2 AB = \sqrt{OB^2~-~OA^2 } = \sqrt{10^2~-~6^2} = \sqrt{64}= 8 cm To know more about properties of a tangent to a circle, download … for (var i=0; i | {
"domain": "exaton.hu",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513881564148,
"lm_q1q2_score": 0.8838395350696157,
"lm_q2_score": 0.8962513724408292,
"openwebmath_perplexity": 358.98978621027345,
"openwebmath_score": 0.4151161313056946,
"tags": null,
"url": "https://www.exaton.hu/qn573/c5fe24-tangent-of-a-circle-example"
} |
Volvo Xc40 Plug-in Hybrid Release Date Usa, Bash Remove Trailing Slash, Perle Cotton Vs Embroidery Floss, 3d Print Troubleshooting Cura, Email Address Search, When Is The Ymca Going To Reopen, | {
"domain": "exaton.hu",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513881564148,
"lm_q1q2_score": 0.8838395350696157,
"lm_q2_score": 0.8962513724408292,
"openwebmath_perplexity": 358.98978621027345,
"openwebmath_score": 0.4151161313056946,
"tags": null,
"url": "https://www.exaton.hu/qn573/c5fe24-tangent-of-a-circle-example"
} |
In the game of Two-Finger Morra, 2 players show 1 or 2 fingers and simultaneously guess the number of fingers their opponent will show
# In the game of Two-Finger Morra, 2 players show 1 or 2 fingers and simultaneously guess the number of fingers their opponent will show
A
45 points
In the game of Two-Finger Morra, 2 players show 1 or 2 fingers and simultaneously guess the number of fingers their opponent will show. If only one of the players guesses correctly, he wins an amount (in dollars) equal to the sum of the fingers shown by him and his opponent. If both players guess correctly or if neither guesses correctly, then no money is exchanged. Consider a specified player, and denote by X the amount of money he wins in a single game of Two-Finger Morra.
(a) If each player acts independently of the other, and if each player makes his choice of the number of fingers he will hold up and the number he will guess that his opponent will hold up in such a way that each of the 4 possibilities is equally likely, what are the possible values of X and what are their associated probabilities?
(b) Suppose that each player acts independently of the other. If each player decides to hold up the same number of fingers that he guesses his opponent will hold up, and if each player is equally likely to hold up 1 or 2 fingers, what are the possible values of X and their associated probabilities?
In the
8.7k points
For this type of problem, it is helpful to construct a table,
Player 1 Player 2 Random Variable
Guess Show Guess Show X
1 1 1 1 0
1 1 1 2 -3
1 1 2 1 2
1 1 2 2 0
1 2 1 1 3
1 2 1 2 0
1 2 2 1 0
1 2 2 2 -4
2 1 1 1 -2
2 1 1 2 0
2 1 2 1 0
2 1 2 2 3
2 2 1 1 0
2 2 1 2 4
2 2 2 1 -3
2 2 2 2 0
part (a)
There are 16 possibilities in the table above and so we have,
$$P(X=0)=\frac{8}{16}=\frac{1}{2}$$
$$P(X=2)=P(X=-2)=\frac{1}{16}$$
$$P(X=3)=P(X=-3)=\frac{2}{16}=\frac{1}{8}$$
$$P(X=4)=P(X=-4)=\frac{1}{16}$$
part (b) | {
"domain": "lil-help.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9877587261496031,
"lm_q1q2_score": 0.8838361059808151,
"lm_q2_score": 0.8947894689081711,
"openwebmath_perplexity": 255.606206917506,
"openwebmath_score": 0.7249987721443176,
"tags": null,
"url": "https://www.lil-help.com/questions/7852/in-the-game-of-two-finger-morra-2-players-show-1-or-2-fingers-and-simultaneously-guess-the-number-of-fingers-their-opponent-will-show"
} |
$$P(X=3)=P(X=-3)=\frac{2}{16}=\frac{1}{8}$$
$$P(X=4)=P(X=-4)=\frac{1}{16}$$
part (b)
from the table above the following set can only occur {1111,1122,2211,2222} all with output X=0, so, $P(X=0)=1$
Surround your text in *italics* or **bold**, to write a math equation use, for example, $x^2+2x+1=0$ or $$\beta^2-1=0$$
Use LaTeX to type formulas and markdown to format text. See example. | {
"domain": "lil-help.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9877587261496031,
"lm_q1q2_score": 0.8838361059808151,
"lm_q2_score": 0.8947894689081711,
"openwebmath_perplexity": 255.606206917506,
"openwebmath_score": 0.7249987721443176,
"tags": null,
"url": "https://www.lil-help.com/questions/7852/in-the-game-of-two-finger-morra-2-players-show-1-or-2-fingers-and-simultaneously-guess-the-number-of-fingers-their-opponent-will-show"
} |
Pairwise Puzzler
You may wish to take a look at Pairwise Adding before trying this problem
Alison reckons that if Charlie gives her the sums of all the pairs of numbers from any set of five numbers, she'll be able to work out the original five numbers.
Charlie gives her this set of pair sums: $0$, $2$, $4$, $4$, $6$, $8$, $9$, $11$, $13$, $15$.
Can you work out Charlie's original five numbers?
Can you work out a strategy to work out the original five numbers for any set of pair sums that you are given?
Does it help to add together all the pair sums?
Given ten randomly generated numbers, will there always be a set of five numbers whose pair sums are that set of ten?
Can two different sets of five numbers give the same set of pair sums?
Four numbers are added together in pairs to produce the following answers: $5$, $9$, $10$, $12$, $13$, $17$.
What are the four numbers?
Is there more than one possible solution?
Six numbers are added together in pairs to produce the following answers: $7$, $10$, $13$, $13$, $15$, $16$, $18$, $19$, $21$, $21$, $24$, $24$, $27$, $30$, $32$.
What are the six numbers?
Can you devise a general strategy to work out a set of six numbers when you are given their pair sums?
Pairwise Puzzler - strategy for original five numbers
The five numbers can be calculated simply by using simultaneous equations.
Let's call the lowest number a, the second lowest b etc.. and we'll place the sums from smallest to largest in series s.
We know straight away that a + b = s1, as the smallest sum must be made up of the smallest pair.
Also the largest sum is clear: d + e = s10
We can also work out that a + c = s2, as this is the second smallest sum; a and c are the smallest two numbers which haven't yet been paired.
Similarly, c + e = s9 (the second largest). | {
"domain": "maths.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9893474879039229,
"lm_q1q2_score": 0.8837934217185495,
"lm_q2_score": 0.8933094110250331,
"openwebmath_perplexity": 344.7798361841747,
"openwebmath_score": 0.6365929245948792,
"tags": null,
"url": "https://wild.maths.org/comment/1010"
} |
So we now have five unknowns and four equations... we just need one more equation.
If we look at the sum of all the sums in terms of the five original numbers ((a + b) + (a + c)...(e + f)),
this simplifies to 4 (a + b + c + d + e).
So therefore a + b + c + d + e = sum (s) / 4
Here we have the final equation, and we can now calculate the five numbers through simple algebra.
So let's make each letter in turn the subject of an equation.
We can do this easily for c, using substitution: a + b = s1, d + e = s10, and a + b + c + d + e + f = sum (s) / 4;
therefore s1 + s10 + c = sum (s) / 4
So c = sum (s) / 4 - s1 - s10
Now that we have c, all of the other letters can be expressed in terms of c:
a = s2 - c
b = s1 - (s2 - c) = s1 - s2 + c
e = s9 - c
d = s10 - (s9 - c) = s10 - s9 + c
These equations can all be written, by substituting c, in terms of the sums:
a = s1 + s2 + s10 - sum (s) / 4
b = sum (s) / 4 - s2 - s10
c = sum (s) / 4 - s1 - s10
d = sum (s) / 4 - s1 - s9
e = s1 + s9 + s10 - sum (s) / 4
So to find the original five numbers given only the sum of each pair, just arrange the sums from smallest to largest and substitute them into the equations above.
This is an excellent solution
This is an excellent solution, well done! I particularly like the way that you simplified the problem by writing the numbers as $a \leq b \leq c \leq d \leq e$, and the way that adding all the numbers together allows you to say for certain what the algebraic sum is.
Pairwise puzzlers
I thought this puzzle was set in a misleading way - at first sight it appears it only involves positive integers, whereas the solution includes a negative integer
(-1,1,3,5,10)
I suppose it does teach you to assume nothing that is not explicitly stated!
Math
The answer to the first question is 0,2,6,7,9!!!!!
Math
Julie, I can choose pairs of numbers from your set of five and add them to get 2, 6, 8, 9, 11, 13 and 15, but I can't get 0, 4 and 4.
Do have another go. | {
"domain": "maths.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9893474879039229,
"lm_q1q2_score": 0.8837934217185495,
"lm_q2_score": 0.8933094110250331,
"openwebmath_perplexity": 344.7798361841747,
"openwebmath_score": 0.6365929245948792,
"tags": null,
"url": "https://wild.maths.org/comment/1010"
} |
Do have another go.
0, 2, 4, 6, 9
or
-1, 1, 3, 5, 10
One of your sets of five numbers can be used to produce this set of pair sums 0, 2, 4, 4, 6, 8, 9, 11, 13, 15, but I'm afraid the other one can't.
Can you figure out which one is which?
A somewhat inelegant answer to the 6-number problem
Having tackled the 4- and 5-number versions of this problem, I couldn't see an obvious way to generate the 6th equation which would allow direct solution via simultaneous equations. Five equations are simple enough to find, as Mr Redman showed very clearly in his comment, but without the sixth (which I hope someone will enlighten me about at some point), the system would be under-determined. So, without an elegant solution to hand, I tried an inelegant one instead...
About 10 lines of matlab code allowed me to generate random collections of six integers (within a reasonable range to keep things somewhat manageable), generate all 15 of their pairwise combinations and sums, and test the resulting sums against the given sums. About 25 seconds of computer time gave me the following answer (which I have checked by hand just to be sure): 2 5 8 11 13 19
I know that this technique will scale very poorly as the size of problem increases - I certainly wouldn't want to attempt it for pairwise combinations of 10 numbers, for example - and would welcome any suggestions regarding a more subtle approach, but feel that a computational approach does have a place even on this website!
A somewhat inelegant answer to the 6-number problem
Those six numbers do indeed produce the pair sums 7, 10, 13, 13, 15, 16, 18, 19, 21, 21, 24, 24, 27, 30 and 32. Very well done! Could you explain in more detail the method that you programmed Matlab to use? | {
"domain": "maths.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9893474879039229,
"lm_q1q2_score": 0.8837934217185495,
"lm_q2_score": 0.8933094110250331,
"openwebmath_perplexity": 344.7798361841747,
"openwebmath_score": 0.6365929245948792,
"tags": null,
"url": "https://wild.maths.org/comment/1010"
} |
As for a more elegant solution, it is indeed difficult to find a 6th equation. In some of the cases where some of the pair sums are equal, you can deduce a 6th equation. For example if $s_3 = s_4$, then you know the value of $a+d$. Would considering the possible positions of the some of the pair sums (such as $a+d$) help you to deduce a more general approach for all cases? | {
"domain": "maths.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9893474879039229,
"lm_q1q2_score": 0.8837934217185495,
"lm_q2_score": 0.8933094110250331,
"openwebmath_perplexity": 344.7798361841747,
"openwebmath_score": 0.6365929245948792,
"tags": null,
"url": "https://wild.maths.org/comment/1010"
} |
# Probability $A$ is before $B$ when the 26 letters are arranged randomly
The 26 letters A, B, ... , Z are arrange in a random order. [Equivalently, the letters are selected sequentially at random without replacement.]
a) What is the probability that A comes before B in the random order?
b) What is the probability that A comes before Z in the random order?
c) What is the probability that A comes just before B in the random order?
Any help would be much appreciated. I was thinking that for part c the answer would be $$1/26$$ because we have $$25!$$ ways of having A right before B and $$26!$$ total arrangements. Not sure how to proceed with a and b, however.
Edit:
Thank you. For parts (a) and (b) is there a more formal way of getting $$1/2$$? Such as the formula for the total number of ways we can have $$A$$ before $$B$$ over the total number of arrangements? Would it be 25 choose 1 ... 2 choose 1 over $$26!$$ since we can have a in the first spot and B in any spot after it? Then we can have a in the 2nd spot and B in any spot after it. Also, for part (c), doesn't $$A$$ have to come immediately before $$B$$, so wouldn't the probability be $$1/26$$?
• a) 1/2 b) 1/2 . Only 2 possibilities A before or after B – mridul Dec 8 '14 at 6:43
• MAy someone shed some light on the correct answer in part c? Thanks! – user198454 Dec 8 '14 at 16:54
• Is c asking if A come right before B or that A comes right before B but not Z? – Kamster Dec 29 '14 at 5:24
• The reasoning for part (a) in chandu1729's excellent answer is quite formal actually. Let me know if you need an explanation of how to make that formal. – 6005 Dec 30 '14 at 7:02
• Yeah! Just a bounty — and here we are, a bunch of totally equal answers. – sas Dec 30 '14 at 8:15
a) From symmetry, the probability that $A$ comes before $B$ is same as the probability that $B$ comes before $A$, and the sum of these $2$ probabilities is $1$. Hence, the probability that $A$ comes before $B$ is $0.5$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9893474888461861,
"lm_q1q2_score": 0.883793416237481,
"lm_q2_score": 0.8933094046341532,
"openwebmath_perplexity": 300.8157627126416,
"openwebmath_score": 0.8802695870399475,
"tags": null,
"url": "https://math.stackexchange.com/questions/1056982/probability-a-is-before-b-when-the-26-letters-are-arranged-randomly"
} |
b) Here $B$ is replaced with $Z$ and by exchanging the roles of $B$ and $Z$, we get the same probability as in case (a): $0.5$.
c) If $A$ has to come just before $B$, $B$ shouldn't occur at the $1$st position, which has probability of $25/26$. After $B$'s position has been chosen(other than the $1$st position), there are 25 possibilities for position just before $B$ (everything except $B$) and only $1$ possibility is favourable (A coming before B). Hence the required probability is $(25/26)\cdot(1/25) = 1/26$.
In part c), $A$ comes right before $B.$ Consider a block $\boxed{\text{AB}}$ and treat it as ONE object. The other letters are $C, D, E, \ldots , Z$. In total, you have 25 objects - 24 letters and a block of two letters together. They can be arranged in $25!$ ways in total. Since the total number of arrangements is $26!$, the required probability is $$\dfrac{25!}{26!}=\dfrac1{26}$$
Now, let us consider part a). We'll formally prove the probability is $\dfrac12$
For this, we will consider 26 cases.
Case 1 = B is in the first position.
Clearly, if $B$ is in the first position, then in no possible arrangement, A comes before B. Thus, the number of way A comes before B is $0$ for the first case.
Case 2 = B is in the second position.
For A to come before B, it has to occupy the first spot. A can occupy the first spot in $1$ way. The rest 24 letters can be arranged in $24!$ ways. The number of ways A comes before B can happen is $1.24!$
Case 3 = B is in the third position.
For A to come before B, A can occupy the first two positions. The number of ways A can take the first two position is $2.$ The rest 24 letter can be again arranged in $24!$ ways. Hence, the number of ways A comes before B happens is $2.24!$
Case 4 = B is in the fourth position.
Now, by same logic as before, you can say it will happen in $3.24!$ ways.
Similarly, we go up to Case 26 where B comes in the $26^{th}$ i.e, last position. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9893474888461861,
"lm_q1q2_score": 0.883793416237481,
"lm_q2_score": 0.8933094046341532,
"openwebmath_perplexity": 300.8157627126416,
"openwebmath_score": 0.8802695870399475,
"tags": null,
"url": "https://math.stackexchange.com/questions/1056982/probability-a-is-before-b-when-the-26-letters-are-arranged-randomly"
} |
Similarly, we go up to Case 26 where B comes in the $26^{th}$ i.e, last position.
These $26$ cases exhausts all the possible arrangements. By considering all the $26$ cases and summing up the total number of ways A comes before B can happen, we get
$$0+1.24!+2.24!+\cdots 25.24! = 24!(1+2+\cdots 25)=24!*\dfrac{25.26}{2}=\dfrac{26!}{2}$$
Since the total number of arrangements is $26!$, the probability for A comes before B in a random order is $$\dfrac{26!/2}{26!}=\dfrac{1}{2}$$
• Right answer but...way too many words for part (a) :P – 6005 Dec 30 '14 at 7:01
• Thanks. Right, it bothered me too! I could've provided the same thing in a pictorial proof, which would have been way easier to comprehend, but IDK much of Geogebra. :/ – Dipanjan Pal Dec 30 '14 at 8:17
Here's a proof that the probability in part (a) is 1/2. It's the same as the one in The Math Troll's and chandu1729's answers, but with more details filled in. I want to emphasize that it really is a proof, as rigorous as anything you're likely to encounter in an introductory probability course, even though it doesn't use any formulas or complicated calculations.
Let's say two arrangements are partners if exchanging the positions of A and B turns one into the other. Here are some examples of arrangements which are partners:
ABCDEFGHIJKLMNOPQRSTUVWXYZ and BACDEFGHIJKLMNOPQRSTUVWXYZ AEIOUYBCDFGHJKLMNPQRSTVWXZ and BEIOUYACDFGHJKLMNPQRSTVWXZ THEQUICKBROWNFXJMPSVLAZYDG and THEQUICKAROWNFXJMPSVLBZYDG
You should be able to convince yourself that:
• Every arrangement has exactly one partner.
• If an arrangement has A before B, its partner has B before A, and vice versa.
Now, line up all the arrangements in two lines, with each arrangement standing next to its partner. The lines with be the same length, because each arrangement has exactly one partner. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9893474888461861,
"lm_q1q2_score": 0.883793416237481,
"lm_q2_score": 0.8933094046341532,
"openwebmath_perplexity": 300.8157627126416,
"openwebmath_score": 0.8802695870399475,
"tags": null,
"url": "https://math.stackexchange.com/questions/1056982/probability-a-is-before-b-when-the-26-letters-are-arranged-randomly"
} |
In each pair of partners, tell the arrangement with A before B to stand on the left, and the arrangement with B before A to stand on the right. Now, the left line has all the arrangements with A before B, and the right line has all the arrangements with B before A. Since the lines are the same length, it's now obvious that the numbers of A-before-B arrangements and B-before-A arrangements are equal!
That means exactly half the arrangements have A before B, and exactly half have B before A. Therefore, if you pick an arrangement at random, the probability of getting A before B is 1/2.
You can use exactly the same argument for part (b).
The simplest way to answer these questions:
a) What is the probability that A comes before B in the random order?
In half of the cases A comes before B and in the other half B comes before A, so the answer is $\frac12$
b) What is the probability that A comes before Z in the random order?
In half of the cases A comes before Z and in the other half Z comes before A, so the answer is $\frac12$
c) What is the probability that A comes just before B in the random order?
• The total number of permutations is $26!$
• Join the letters A and B into a symbol AB, and the total number of permutations is $25!$
• Hence the answer is $\frac{25!}{26!}=\frac{1}{26}$
A more "formal" way to answer to the first (as well as the second) question:
• The probability that A is at the 1st place and B is at the 2nd to 26th places is $\frac{1}{26}\cdot\frac{25}{25}=\frac{25}{26\cdot25}$
• The probability that A is at the 2nd place and B is at the 3rd to 26th places is $\frac{1}{26}\cdot\frac{24}{25}=\frac{24}{26\cdot25}$
• The probability that A is at the 3rd place and B is at the 4th to 26th places is $\frac{1}{26}\cdot\frac{23}{25}=\frac{23}{26\cdot25}$
• $\dots$
• The probability that A is at the 25th place and B is at the 26th place is $\frac{1}{26}\cdot\frac{1}{25}=\frac{1}{26\cdot25}$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9893474888461861,
"lm_q1q2_score": 0.883793416237481,
"lm_q2_score": 0.8933094046341532,
"openwebmath_perplexity": 300.8157627126416,
"openwebmath_score": 0.8802695870399475,
"tags": null,
"url": "https://math.stackexchange.com/questions/1056982/probability-a-is-before-b-when-the-26-letters-are-arranged-randomly"
} |
• So the probability that A comes before B is $\sum\limits_{n=1}^{25}\frac{26-n}{26\cdot25}=\frac12$
a) By symmetry, there are as many permutations with $A$ before $B$ as with $A$ after $B$.
$$p=\frac12$$
b) By symmetry, there are as many permutations with $A$ before $Z$ as with $A$ after $Z$.
$$p=\frac12$$
c) Consider the $24!$ permutations of $CDE\dots Z$. You can insert $AB$ at $25$ different places to form all distinct configurations. $$p=\frac{25\cdot24!}{26!}=\frac1{26}$$
for parts a and b then the answer is $1/2$. One way to look at is is to note that for any arrangement of the letters, say "DEFGBAC", there is another arrangment the order of the letters reversed, in this case "CABGFED". It is clear that if $A$ comes before $B$ in the first, then it will come after the $B$ in the second, and vice versa. So we can take all the random arrangements and arrange them in to pairs in this manner. Since each arrangement with $A$ before $B$ is paired with another unique arrangment with $B$ before $A$, there must be the same number of strings with $A$ before $B$ and $B$ before $A$. So selecting one of the possible arrangements at random has a probability of $1/2$ of having the $A$ before the $B$.
For part c, There are 26 postions where the $A$ can go, and, since that position is filled, 25 positions when the $B$ can go. Now for 25 of the 26 positions where the $A$ can go, the position immediately following it is vacant. In this case there is a change of $1/25$ that the $B$ randomly falls into that position. So in $25/26$ of the cases (where $A$ is not in the last position), there is a chance of $1/25$ of the $B$ immediately following the $A$.
However, in 1/26 of the cases, the $A$ is in the last place, and the $B$ cannot follow the $A$, so the probability is $0$.
So from the first situation we have $25/26 \times 1/25 = 1/26$, and from the second $1/26\times0 = 0$
Adding up the two, $1/26 + 0 = 1/26$. So you appear to be correct that the probability is $1/26$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9893474888461861,
"lm_q1q2_score": 0.883793416237481,
"lm_q2_score": 0.8933094046341532,
"openwebmath_perplexity": 300.8157627126416,
"openwebmath_score": 0.8802695870399475,
"tags": null,
"url": "https://math.stackexchange.com/questions/1056982/probability-a-is-before-b-when-the-26-letters-are-arranged-randomly"
} |
Adding up the two, $1/26 + 0 = 1/26$. So you appear to be correct that the probability is $1/26$.
For parts a and b, we can construct a bijection between A before B and B before A (proceed similarly for part b).
Suppose we have one that is A before B. Then we switch the A and B then we get a B before A. The other direction of the bijection will be similar.
Now for part c. If A is the last, then there is 0 chance. This happens 1/26 of the time. If A isn't the last, then there will be one immediately after it. There will be 25 possibilities (and they are random), so the chance is 1/25. This happens 25/26 of the time Hence 1/25*25/26+0=1/26.
• Your answer for part C is wrong! it is 1/52 – mridul Dec 8 '14 at 10:21
• Then what is the problebility that A and B together? 1/13? Please use good words. – mridul Dec 8 '14 at 10:24
• Note that there are 26*25 positions for A and B. For A to come before B, there are 25 positions. Thus the probability is 1/26. Please check your solution first. – user198454 Dec 8 '14 at 10:25
Total 26! rearrangements are possible. In 1st case A has only 2 relative possitions after or before B. Therefore probability is 1/2.
In 3rd case : Total possible arrangements such that A and B are together is 25!. Therefore Total possible arrangements such that A comes just before B is 25! / 2.
a) 1/2
b) 1/2
c) 1/52
Edit:
For parts (a) and (b),
Let A at 1st possition, total number of arrangements(A before B) is 25!.
Let A at 26th possition, total number of arrangements is 0. together we got 25!.
Then take 2nd and 25th possitions.
Let A at 2st possition, total number of arrangements(A before B) is 24*24!. (B has 24 possitions such that B after A)
Let A at 25st possition, total number of arrangements(A before B) is 1*24!. (B has only 1 possition such that B after A)
24*24! + 1*24! = 25*24! =25!
Then take 3rd and 24th possitions.
Let A at 3st possition, total number of arrangements(A before B) is 23*24!. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9893474888461861,
"lm_q1q2_score": 0.883793416237481,
"lm_q2_score": 0.8933094046341532,
"openwebmath_perplexity": 300.8157627126416,
"openwebmath_score": 0.8802695870399475,
"tags": null,
"url": "https://math.stackexchange.com/questions/1056982/probability-a-is-before-b-when-the-26-letters-are-arranged-randomly"
} |
Let A at 3st possition, total number of arrangements(A before B) is 23*24!.
Let A at 24st possition, total number of arrangements(A before B) is 2*24!.
Together we got 25!.
etc .. upto 13th,14th possitions.
Add all together we got 13*25! arrangements.
• Thank you. For parts (a) and (b) is there a more formal way of getting $1/2$? Such as the formula for the total number of ways we can have $A$ before $B$ over the total number of arrangements $(26!)$? Also, for part (c), doesn't $A$ have to come immediately before $B$, so wouldn't the probability be $1/26$? – mylasthope Dec 8 '14 at 7:08
• In the case of part c answer is 1/52. For parts (a) and (b), let A at 1st possition, total number of arrangements(A before B) is 25!. let A at 26th possition, total number of arrangements is 0. together we got 25!. Then take 2nd and 25th possitions together we got 25!. etc. Add all together we got 13*25! arrangements. – mridul Dec 8 '14 at 7:43 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9893474888461861,
"lm_q1q2_score": 0.883793416237481,
"lm_q2_score": 0.8933094046341532,
"openwebmath_perplexity": 300.8157627126416,
"openwebmath_score": 0.8802695870399475,
"tags": null,
"url": "https://math.stackexchange.com/questions/1056982/probability-a-is-before-b-when-the-26-letters-are-arranged-randomly"
} |
# probability of a pair in cards
I have a simple problem involving probability of drawing at least 1 pair of cards in a four card hand. I am not getting the right answer but I dont understand the flaw in my logic. Can anyone explain to me why my approach is wrong?
The problem:
Bill has a small deck of 12 playing cards made up of only 2 suits of 6 cards each. Each of the 6 cards within a suit has a different value from 1 to 6; thus, for each value from 1 to 6, there are two cards in the deck with that value. Bill likes to play a game in which he shuffles the deck, turns over 4 cards, and looks for pairs of cards that have the same value. What is the chance that Bill finds at least one pair of cards that have the same value?
My solution:
Probability of 1 or more pair = ((6) * (10 choose 2))/(12 choose 4). i.e. the # of hands with at least 1 pair are 6 (the number of ways to create a pair) * (10 choose 2) (the # of ways to select 2 cards from the remaining 10 cards after the pair). This simplifies to 6/11, however, the correct answer is 17/33.
Any help understanding would be greatly appreciated. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8837692004349541,
"lm_q2_score": 0.9032942041005327,
"openwebmath_perplexity": 241.87707813867704,
"openwebmath_score": 0.6800057291984558,
"tags": null,
"url": "https://stats.stackexchange.com/questions/50274/probability-of-a-pair-in-cards"
} |
Any help understanding would be greatly appreciated.
• You make some interesting implicit assumptions in your calculation. First, it focuses only on the possibility that the pair occurs among the first two within a sequence of four cards: there are other ways a pair can occur, so this underestimates the probability. Second, it ignores the sequence among the second two cards, thereby overestimating the probability. Fortunately, the two mistakes do not cancel, revealing the fact there is a problem! – whuber Feb 18 '13 at 22:17
• Realized I was double counting. Cant simply multiply 6 * (10 choose 2). 2 pair instances will be double counted. e.g. pair of 1s in the 6 and then pair of 2s in the 10 choose 2 versus pair of 2s in the 6 and then pair of 1s in the 10 choose 2. The better way to do it is to separate the cases of exactly 1 pair, exactly 2 pair. ways to create exactly 1 pair = 6 (ways to create pair) * ( (10*8)/2 ) + ways to create exactly 2 pair = (6*5)/2 Total # = 255 Denominator = (12 choose 4) = (12*11*10*9)/(4*3*2) = 11*5*9 = 455. Probability = 255/455 = 51/99 = 17/33. – Evan V Feb 18 '13 at 22:24
• Another way to do this is simply to remove the double counting directly. There are 6*(10 choose 2) = 270 ways to create 1 or 2 pairs where the instances of 2 pairs are double counted. If you subtract the # of ways to make exactly 2 pairs (6*5)/2 then you get 270 - 15 = 255. – Evan V Feb 18 '13 at 22:28 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8837692004349541,
"lm_q2_score": 0.9032942041005327,
"openwebmath_perplexity": 241.87707813867704,
"openwebmath_score": 0.6800057291984558,
"tags": null,
"url": "https://stats.stackexchange.com/questions/50274/probability-of-a-pair-in-cards"
} |
To see 0 pairs there are 12 possibilities for the 1st card, but only 10 for the second card (since the 1st chosen card is no longer possible and the card that matches the 1st would form a pair) and 8 cards for the 3rd and 6 for the 4th, this multiplies to 5760. The total number of possible hands (with order mattering) is $12 \times 11 \times 10 \times 9 = 11880$, so the complimentary number which is the number of (ordered) hands that contain at least 1 pair is $11880 - 5760 = 6120$, divide that by 11880 and it reduces to $17/33$. We could also do this with order not mattering, but that would be more complicated and the extra pieces would all end up cancelling each other. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846684978779,
"lm_q1q2_score": 0.8837692004349541,
"lm_q2_score": 0.9032942041005327,
"openwebmath_perplexity": 241.87707813867704,
"openwebmath_score": 0.6800057291984558,
"tags": null,
"url": "https://stats.stackexchange.com/questions/50274/probability-of-a-pair-in-cards"
} |
# I Questions about linear transformations
1. Oct 3, 2016
### Chan Pok Fung
We learnt that the condition of a linear transformation is
1. T(v+w) = T(v)+T(w)
2. T(kv) = kT(v)
I am wondering if there is any transformation which only fulfil either one and fails another condition. As obviously, 1 implies 2 for rational number k.
Could anyone give an example of each case? (Fulfilling 1 but 2 and 2 but 1)
Thanks!
2. Oct 3, 2016
3. Oct 3, 2016
### andrewkirk
Consider the real numbers $\mathbb R$ as a vector space over the rationals, and the operator $T:\mathbb R\to \mathbb R$ that is the identity on the rationals and maps to zero on the irrationals. Then T satisfies the second axiom, since it is a linear operator on $\mathbb Q$ considered as a subspace of $\mathbb R$ and, for $q\in\mathbb Q-\{0\},x\in\mathbb R$, $qx$ is in $\mathbb Q\cup\{0\}=\ker\ T$ iff $x$ is.
But the addition axiom does not hold, because $T(1+(\sqrt2-1))=T(\sqrt 2)=0$ but $T(1)+T(\sqrt2-1)=1\neq 0$.
4. Oct 3, 2016
### Stephen Tashi
Define $M((x,y))$ by
if $x \ne y$ then $M((x,y)) = (x,y)$
if $x = y$ then $M((x,y)) = (2x, 2y)$
$M$ satisfies 2, but not 1
5. Oct 4, 2016
### Chan Pok Fung
This really gives me new insight into linear transformation. thanks all!
6. Oct 4, 2016
### Chan Pok Fung
I am sorry but I don't quite understand. How can we construct a transformation like that?
7. Oct 4, 2016
### mathman
Unfortunately Hamel basis exists, but it is not constructable - existence is equivalent to axiom of choice.
8. Oct 4, 2016
### andrewkirk
I'm still trying to think of a scenario with a map that satisfies 1 (additivity) but not 2 (scalar mult). Can anybody think of one?
All the examples I come up with either end up satisfying neither 1 nor 2, or satisfying 2 but not 1.
I assume there must be one, otherwise some texts would specify 1 as the sole requirement and derive 2 as a consequence of 1.
9. Oct 4, 2016
### Chan Pok Fung | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846634557752,
"lm_q1q2_score": 0.8837692003378307,
"lm_q2_score": 0.9032942086563877,
"openwebmath_perplexity": 1244.7749528275344,
"openwebmath_score": 0.8395922780036926,
"tags": null,
"url": "https://www.physicsforums.com/threads/questions-about-linear-transformations.887744/"
} |
9. Oct 4, 2016
### Chan Pok Fung
Andrew, I am thinking that as we have to apply the transformation to a vector space, and vectors in vector space obeys kv is also in the space. As we can have k be any real number, it seems that it somehow implies axiom2. The transformation only satisfy axiom1 must be of a very weird form.
10. Oct 4, 2016
### Stephen Tashi
On the "talk" page for the current Wikipedia article on "Linear transformation", I found:
11. Oct 4, 2016
### Chan Pok Fung
That's a clear and direct example! | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846634557752,
"lm_q1q2_score": 0.8837692003378307,
"lm_q2_score": 0.9032942086563877,
"openwebmath_perplexity": 1244.7749528275344,
"openwebmath_score": 0.8395922780036926,
"tags": null,
"url": "https://www.physicsforums.com/threads/questions-about-linear-transformations.887744/"
} |
# Multiplication of Moduli in Modular Congruences
Lately it's come up in my discrete mathematics class that proving things for smaller congruences, namely those where the modulus is a prime is much easier than attempting to do so for larger congruences.
For example, one such problem was to prove that $a^5 \equiv a \pmod{10}$ for $a$ $\varepsilon$ $\mathbb Z^+\$This problem on its own becomes difficult only because we are not guaranteed that for every a $(a,10)=1$. The solution was to factorize 10 in terms of primes $p_1, p_2... p_n$ s.t for each prime $p_i$ $(a,p_i)=1$ for all $a \ \varepsilon \ \mathbb Z^+\$, resulting in n congruences under modulo the given prime and then apply Euler's Theorem. In this case we find the following: $$10 = 2 \cdot 5$$ so $$a^{\phi(5)} \equiv 1 \pmod{5} \\ a^{\phi(2)} \equiv 1 \pmod{2}$$ since $\phi(5)=4, \phi(2)=1$ this becomes: $$a^{4} \equiv 1 \pmod{5} \implies a^{5} \equiv a \pmod{5}\\ a^{1} \equiv 1 \pmod{2} \implies a^{5} \equiv a^4 \pmod{2}$$ but since a is its own inverse modulo 2, we can transform the 2nd congruence into: $$a^5 \equiv a \mod{2}$$ Then the part that I am unclear on occurs. It seems that we can just multiply the 2 moduli together and the desired congruence falls out that: $$a^5 \equiv a \pmod{10}$$
So my question is whether or not $a \equiv b \pmod{n}$ and $a \equiv b \pmod{m}$ always$\implies$ $a \equiv b \pmod{mn}$.
Edit (Extension):
The answer to my question has been stated to be yes provided that $(m,n) = 1$ but I also noticed that if my 2 relations had been left stated as $$a^4 \equiv 1 \pmod{2}$$ $$a^4 \equiv 1 \pmod{5}$$ Applying the conjecture I made gives: $$a^4 \equiv 1 \pmod{10}$$ which is demonstrably false given that $$2^4 \equiv 6 \pmod{10} \implies 2^4 \not\equiv 1 \pmod{10}$$ Why is this and what prevents this identity from being true as well? I see that 2 would then not be coprime to 10 but why does it work when I multiply both sides by a then?
Excellent question!
The short answer is no. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683513421314,
"lm_q1q2_score": 0.8836657606079439,
"lm_q2_score": 0.8947894689081711,
"openwebmath_perplexity": 147.5086358032251,
"openwebmath_score": 0.9810587167739868,
"tags": null,
"url": "https://math.stackexchange.com/questions/2668314/multiplication-of-moduli-in-modular-congruences"
} |
Excellent question!
The short answer is no.
For example, $4 \equiv 16 \pmod 6$ and $4 \equiv 16 \pmod 4$, but $4 \not \equiv 16 \pmod{24}$.
However if $n$ and $m$ are relatively prime, then the answer is yes.
This is pretty straightforward to see, as if $a \equiv b \pmod n$ then $n \mid (b-a)$, and if $a \equiv b \pmod m$, then $m \mid (b-a)$. Then one notes (or proves, if it's not clear) that $n \mid (b-a)$, $m \mid (b-a)$, and $\gcd(m,n) = 1$ implies that $mn \mid (b-a)$.
In fact, this is at the edge of a deeper theorem called the Chinese Remainder Theorem, which says roughly that knowing the structure of $x$ mod $n$ and $m$ for $m,n$ relatively prime is equivalent to knowing the structure of $x$ mod $mn$ --- or perhaps with two or three or more moduli all taken together. Look up the Chinese Remainder Theorem on the web and on this site for more.
• If I might add to this question a bit. This works when i have the $a^5$ and a for a and b but if i take b = 1 it clearly doesn't work since $2^4 \equiv 6 \pmod{10}$ but $2^5 \equiv 2 \pmod{10}$ which makes sense since with a not coprime to 10 when a=2 it wouldn't be invertible mod 10. How is one to catch this beforehand though and what makes it work when it's multiplied on both sides by a? – rjm27trekkie Feb 27 '18 at 1:16
You require $\gcd(m,n)=1$, otherwise \begin{eqnarray*} 1 \equiv 13 \pmod{4} \\ 1 \equiv 13 \pmod{6} \\ \end{eqnarray*} but \begin{eqnarray*} 1 \neq 13 \pmod{24}. \\ \end{eqnarray*}
• Understood. Otherwise, is the conjecture correct given that (m,n) = 1? – rjm27trekkie Feb 27 '18 at 1:05
• That's right ... – Donald Splutterwit Feb 27 '18 at 1:09
This is true if (and only if) $m$ and $n$ are coprime – which is the case ot two distinct primes. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683513421314,
"lm_q1q2_score": 0.8836657606079439,
"lm_q2_score": 0.8947894689081711,
"openwebmath_perplexity": 147.5086358032251,
"openwebmath_score": 0.9810587167739868,
"tags": null,
"url": "https://math.stackexchange.com/questions/2668314/multiplication-of-moduli-in-modular-congruences"
} |
This is true if (and only if) $m$ and $n$ are coprime – which is the case ot two distinct primes.
This is the statement of the Chinese remainder theorem: if $m, n$ are coprime integers, the map \begin{align} \mathbf Z/mn\mathbf Z&\longrightarrow\mathbf Z/m\mathbf Z\times \mathbf Z/n\mathbf Z\\ a\bmod mn&\longmapsto(a\bmod m,a\bmod n) \end{align} is an isomorphism.
$x \equiv a \mod n$ means $x=a+kn$ for some $k$. Now $k=qm+r$ for some value of $q$ and $0\le r <m$. So $x=a +rn + q*mn$ so $x=a+rn\mod mn$. However we have no idea yet what $r$ is. we just know it is between $0$(inclusively) and $m$(exclusively).
Likewise if $x\equiv a\mod m$ by the same reasoning we know $x\equiv a+sm\mod mn$. We don't know what $s$ is yet.
But $a+sm <mn$ and $a+rn <mn$ and $a+sm\equiv a+rn\mod mn$ so $sm=rn$.
now if $m$ and $n$ are not relatively prime there might be non trivial solutions. For example maybe $r=\frac m {\gcd (m,n)}$ and $s=\frac n {\gcd (m,n)}$
But if $m$ and $n$ are relatively prime, then $sm=rn$ means $m|r$ and $n|s$. But $r <m$ and $s <n$ and the only way that can happen is if $sm=rn=0$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683513421314,
"lm_q1q2_score": 0.8836657606079439,
"lm_q2_score": 0.8947894689081711,
"openwebmath_perplexity": 147.5086358032251,
"openwebmath_score": 0.9810587167739868,
"tags": null,
"url": "https://math.stackexchange.com/questions/2668314/multiplication-of-moduli-in-modular-congruences"
} |
# Combinatorics Challenge
#### anemone
##### MHB POTW Director
Staff member
For $n=1,2,...,$ set $$\displaystyle S_n=\sum_{k=0}^{3n} {3n\choose k}$$ and $$\displaystyle T_n=\sum_{k=0}^{n} {3n\choose 3k}$$.
Prove that $|S_n-3T_n|=2$.
#### MarkFL
Staff member
My solution:
For the first sum, we may simply apply the binomial theorem to obtain the closed form:
$$\displaystyle S_n=\sum_{k=0}^{3n}{3n \choose k}=(1+1)^{3n}=8^n$$
For the second sum, I looked at the first 5 values:
$$\displaystyle T_1=2,\,T_2=22,\,T_3=170,\,T_4=1366,\,T_5=10922$$
and determined the recursion:
$$\displaystyle T_{n+1}=7T_{n}+8T_{n-1}$$
The characteristic equation for this recursion is:
$$\displaystyle r^2-7r-8=(r+1)(r-8)=0$$
and so the closed form is:
$$\displaystyle T_{n}=k_1(-1)^n+k_28^n$$
Using the initial values to determine the parameters, we may write:
$$\displaystyle T_{1}=-k_1+8k_2=2$$
$$\displaystyle T_{2}=k_1+64k_2=22$$
Adding the two equations, we find:
$$\displaystyle 72k_2=24\implies k_2=\frac{1}{3}\implies k_1=\frac{2}{3}$$
Hence:
$$\displaystyle T_{n}=\frac{1}{3}\left(2(-1)^n+8^n \right)$$
And so we find:
$$\displaystyle \left|S_n-3T_n \right|=\left|8^n-3\left(\frac{1}{3}\left(2(-1)^n+8^n \right) \right) \right|=\left|8^n-2(-1)^n-8^n \right|=\left|2(-1)^{n+1} \right|=2$$
Shown as desired.
#### anemone
##### MHB POTW Director
Staff member
My solution:
For the first sum, we may simply apply the binomial theorem to obtain the closed form:
$$\displaystyle S_n=\sum_{k=0}^{3n}{3n \choose k}=(1+1)^{3n}=8^n$$
For the second sum, I looked at the first 5 values:
$$\displaystyle T_1=2,\,T_2=22,\,T_3=170,\,T_4=1366,\,T_5=10922$$
and determined the recursion:
$$\displaystyle T_{n+1}=7T_{n}+8T_{n-1}$$
The characteristic equation for this recursion is:
$$\displaystyle r^2-7r-8=(r+1)(r-8)=0$$
and so the closed form is:
$$\displaystyle T_{n}=k_1(-1)^n+k_28^n$$
Using the initial values to determine the parameters, we may write: | {
"domain": "mathhelpboards.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9925393565360598,
"lm_q1q2_score": 0.8836520561561116,
"lm_q2_score": 0.8902942239389252,
"openwebmath_perplexity": 1185.4687731455952,
"openwebmath_score": 0.9442697167396545,
"tags": null,
"url": "https://mathhelpboards.com/threads/combinatorics-challenge.6224/"
} |
Using the initial values to determine the parameters, we may write:
$$\displaystyle T_{1}=-k_1+8k_2=2$$
$$\displaystyle T_{2}=k_1+64k_2=22$$
Adding the two equations, we find:
$$\displaystyle 72k_2=24\implies k_2=\frac{1}{3}\implies k_1=\frac{2}{3}$$
Hence:
$$\displaystyle T_{n}=\frac{1}{3}\left(2(-1)^n+8^n \right)$$
And so we find:
$$\displaystyle \left|S_n-3T_n \right|=\left|8^n-3\left(\frac{1}{3}\left(2(-1)^n+8^n \right) \right) \right|=\left|8^n-2(-1)^n-8^n \right|=\left|2(-1)^{n+1} \right|=2$$
Shown as desired.
Hey thanks for participating MarkFL! And I'm so impressed that you were so fast in cracking this problem!
#### caffeinemachine
##### Well-known member
MHB Math Scholar
For $n=1,2,...,$ set $$\displaystyle S_n=\sum_{k=0}^{3n} {3n\choose k}$$ and $$\displaystyle T_n=\sum_{k=0}^{n} {3n\choose 3k}$$.
Prove that $|S_n-3T_n|=2$.
Define $f(x)=\sum_{k=0}^{3n}{3n\choose k}x^k$.
Then $f(1)+f(\omega)+f(\omega^2)=3T_n$, where $\omega$ is a complex cube root of unity. Note that $f(1)=S_n$. So we get $S_n-3T_n=-[(1+\omega)^{3n}+(1+\omega^2)^{3n}]=-2$
#### anemone
##### MHB POTW Director
Staff member
Define $f(x)=\sum_{k=0}^{3n}{3n\choose k}x^k$.
Then $f(1)+f(\omega)+f(\omega^2)=3T_n$, where $\omega$ is a complex cube root of unity. Note that $f(1)=S_n$. So we get $S_n-3T_n=-[(1+\omega)^{3n}+(1+\omega^2)^{3n}]=-2$
Hi caffeinemachine,
Thanks for participating and I really appreciate you adding another good solution to this problem and my thought to solve this problem revolved around the idea that you used too! | {
"domain": "mathhelpboards.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9925393565360598,
"lm_q1q2_score": 0.8836520561561116,
"lm_q2_score": 0.8902942239389252,
"openwebmath_perplexity": 1185.4687731455952,
"openwebmath_score": 0.9442697167396545,
"tags": null,
"url": "https://mathhelpboards.com/threads/combinatorics-challenge.6224/"
} |
What is the difference between $\lim_{h\to0}\frac{0}{h}$ and $\lim_{h\to\infty}\frac{0}{h}$?
What is the difference (if any) between-
$$\lim_{h\to0}\frac{0}{h} \text{ and } \lim_{h\to\infty}\frac{0}{h}$$
I argue that both must be $=0$ since the numerator is exactly $0$. But one fellow refuses to agree and argues that the first limit can't be $0$ as anything finite by something tending to $0$ is always $\infty$.
So,how can I explain it to the person? Also, if possible can anyone provide some good reference on this particular issue (Apostol perhaps)?
Thanks for any help!
• Graph the function $f(x) = \dfrac{0}{x}$. Perhaps by seeing it, it will make more sense to the 'fellow'. Explain the geometric interpretation of a limit. – InterstellarProbe Aug 21 '18 at 17:27
• "Anything finite by something tending to 0 is always $\infty$" is not true (at least as long as $0$ counts as finite) -- as this very example shows. Case closed. – Henning Makholm Aug 21 '18 at 17:41
• @tatan: If they don't want to consider your arguments, you can't. Give it up; life is too short for some things. – Henning Makholm Aug 21 '18 at 17:43
• Let "the fellow" read this. – drhab Aug 21 '18 at 18:09
• It's not that the numerator is zero, it's that the fraction is zero. Does your "fellow" agree that $\lim_{h\to0}0=0$. If he does, but doesn't then agree that $\lim_{h\to0}0/h=0$, then there is little hope for him. – Lord Shark the Unknown Aug 21 '18 at 18:22
In both cases you are dividing zero by a nonzero number .
Thus your fraction is identically zero and as a result the limit is zero.
For the first one let me make an intuitive explanation of what's going on with $\lim_{h\to 0}\frac{0}{h}$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9859363729567545,
"lm_q1q2_score": 0.8836468342323038,
"lm_q2_score": 0.8962513793687401,
"openwebmath_perplexity": 328.29831887739505,
"openwebmath_score": 0.8766734004020691,
"tags": null,
"url": "https://math.stackexchange.com/questions/2890157/what-is-the-difference-between-lim-h-to0-frac0h-and-lim-h-to-infty"
} |
The above limit means "What value does the above expression get while $h$ gets arbitrarily close to $0$", meaning that $h$ is number very close to $0$ but not equal to that. Suppose $h=0.000001$. Then $\frac{0}{h}=\frac{0}{0.000001}=0$. Even if this number gets even closer to $0$ you see that the expression continuous to be equal to $0$. That's because $0$ divided by any number obsiously gives $0$ as the answer. Hence $\lim_{h\to 0}\frac{0}{h}=0$.
I wish I helped!
When in doubt take it back to the definition.
$\lim_\limits{x\to 0} f(x) = 0$ means
$\forall \epsilon >0, \exists \delta >0 : 0<|x|<\delta \implies |f(x)|<\epsilon$
As $f(x) = 0$ for all $x \ne 0$ the definition above is satisfied. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9859363729567545,
"lm_q1q2_score": 0.8836468342323038,
"lm_q2_score": 0.8962513793687401,
"openwebmath_perplexity": 328.29831887739505,
"openwebmath_score": 0.8766734004020691,
"tags": null,
"url": "https://math.stackexchange.com/questions/2890157/what-is-the-difference-between-lim-h-to0-frac0h-and-lim-h-to-infty"
} |
# $f(x) = \lfloor x \rfloor - \left\{ x \right\}:$ Increasing, Decreasing, Even , Odd, And/Or Invertible?
Define $\{x\} = x-\lfloor x \rfloor$. That is to say, $\{x\}$ is the "fractional part" of $x$. If you were to expand the number $x$ as a decimal, $\{x\}$ is the stuff after the decimal point. For example $\left\{\frac{3}{2}\right\} = 0.5$ and $\{\pi\} = 0.14159\dots$
Now, using the above definition, determine if the function below is increasing, decreasing, even, odd, and/or invertible on its natural domain:
$$f(x) = \lfloor x \rfloor - \left\{ x \right\}$$
I think that it is invertible only, but I can't seem to find the inverse. Am I correct saying that it is only invertible? Is it also increasing, decreasing, even, and/or odd?
• You are indeed correct that $f$ is injective. It is also surjective, and hence invertible. But proving these facts, and finding an inverse, will take a bit of work. You should try graphing this function first - on a small region, say $[0, 3]$. This will also help with the other questions. – Noah Schweber Aug 26 '16 at 16:15
• Thanks for the confirmation! What about other properties? Is the funtion also increasing, decreasing, even, and/or odd? – Dreamer Aug 26 '16 at 16:18
• Have you tried graphing it? I think that will make the situation much clearer . . . – Noah Schweber Aug 26 '16 at 16:19
• I find it easier to visualise it as $f(x) = -x + 2 \lfloor x \rfloor$ – Shai Aug 26 '16 at 16:22
• @Regina Yes, that is correct. – Noah Schweber Aug 26 '16 at 16:27 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.98593637543616,
"lm_q1q2_score": 0.883646830307043,
"lm_q2_score": 0.8962513731336204,
"openwebmath_perplexity": 142.6961857718192,
"openwebmath_score": 0.8596678376197815,
"tags": null,
"url": "https://math.stackexchange.com/questions/1904493/fx-lfloor-x-rfloor-left-x-right-increasing-decreasing-even"
} |
We verify that $$g(x)=-f(-x)=\{-x\}-\lfloor -x \rfloor$$ is the inverse of $f$. In fact if $x\in \mathbb{Z}$ then $$f(g(x))=\lfloor \{-x\}-\lfloor -x \rfloor \rfloor -\{\{-x\}-\lfloor -x \rfloor\}=\lfloor 0+x \rfloor -\{0+x\}=x.$$ If $x\not\in \mathbb{Z}$ then $$\lfloor -x \rfloor=-1-\lfloor x \rfloor\quad\mbox{and}\quad\{-x\}=1-\{x\}$$ and $$f(g(x))=\lfloor \{-x\}-\lfloor -x \rfloor \rfloor -\{\{-x\}-\lfloor -x \rfloor\}\\ =\lfloor 1-\{x\}-(-1-\lfloor x \rfloor) \rfloor -\{1-\{x\}-(-1-\lfloor x \rfloor)\}\\ =\lfloor 2-\{x\}+\lfloor x \rfloor \rfloor -\{2-\{x\}+\lfloor x \rfloor\}\\ =1+\lfloor x \rfloor-(1-\{x\})=\lfloor x \rfloor+\{x\}=x.$$
$f(-1/2)=-3/2$ and $f(1/2)=-1/2$ so $f$ is neither even nor odd.
$f(-1/2)=-3/2$ and $f(0)=0$ and $f(1/2)=-1/2$ so $f$ is neither increasing nor decreasing.
$f$ maps $[0,1)$ bijectively onto $(-1,0]$ because $f(x)=-x$ for $x\in [0,1).$
$f(x+1)=2+f(x)$ so if $x-y=n\in \mathbb Z$ then $f(x)=2n+f(y).$ So for $n\in \mathbb Z,$ the function $f$ maps $[n,n+1)$ bijectively onto $(2n-1,2n].$ So $f$ is 1-to-1.
And $\cup_{n\in \mathbb Z}(2n-1,2n]=\mathbb R.$ So $f:\mathbb R\to \mathbb R$ is a surjection. A 1-to-1 surjection is a bijection, and is invertible.
Remarks: In case it is unclear that $f$ is 1-to-1, note that (1) when $n\in \mathbb Z$ and $x,y\in [n,n+1)$ then $f(x)\ne f(y)$ because $f$ is 1-to-1 on $[n,n+1)$. And (2) when $m,n$ are unequal integers and $x\in [m,m+1)$ and $y\in [n,n+1)$ then $f(x)\in (2m-1,2m]$ while $f(y)\in (2m-1,2m],$ and $(2m-1,2m]\cap (2n-1,2n]$ is empty, so $f(x)\ne f(y).$
It may be helpful to sketch the graph of $f.$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.98593637543616,
"lm_q1q2_score": 0.883646830307043,
"lm_q2_score": 0.8962513731336204,
"openwebmath_perplexity": 142.6961857718192,
"openwebmath_score": 0.8596678376197815,
"tags": null,
"url": "https://math.stackexchange.com/questions/1904493/fx-lfloor-x-rfloor-left-x-right-increasing-decreasing-even"
} |
# Finding Composition Series of Groups
#### Poirot
##### Banned
My sum total of knowledge of composition series is: the definition, the jordan holder theorem and the fact that the product of the indices must equal the order of the group.
With this in mind, can someone help with me with finding a composition series for the following:
(1) Z60
(2) D12 (dihedral group)
(3) S10 (symmetric group)
I am not looking for just an answer but actually how to go about finding a series.
#### Opalg
##### MHB Oldtimer
Staff member
My sum total of knowledge of composition series is: the definition, the jordan holder theorem and the fact that the product of the indices must equal the order of the group.
With this in mind, can someone help with me with finding a composition series for the following:
(1) Z60
(2) D12 (dihedral group)
(3) S10 (symmetric group)
I am not looking for just an answer but actually how to go about finding a series.
As a general strategy, start with the whole group, look for a maximal normal subgroup. Then repeat the process.
(1) should be easy, because $\mathbb{Z}_{60}$ is abelian and so every subgroup is normal. You start by looking for a maximal subgroup. For example, you could take the subgroup generated by 2, which is (isomorphic to) $\mathbb{Z}_{30}.$ Now repeat the process: find a maximal subgroup of $\mathbb{Z}_{30}.$ And so on.
For (2), any dihedral group $D_{2n}$ has a subgroup of index 2 (therefore necessarily normal), consisting of all the rotations in $D_{2n}$ and isomorphic to $\mathbb{Z}_{n}$. That subgroup is abelian, so you can proceed as in (1).
For (3), here's a hint.
#### Poirot
##### Banned
Ok thanks, it seems it should be quite easy but I have across an example which I don't understand. I am told that {0},<12>,<4>,Z48 is a composition series of Z48 but couldn't <24> be inserted between {0} and <12> since it is of order 2 in Z48? (sorry don't know how to do triangles denoting normal subgroup)
#### Opalg | {
"domain": "mathhelpboards.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9632305339244013,
"lm_q1q2_score": 0.8835739305630267,
"lm_q2_score": 0.9173026595857203,
"openwebmath_perplexity": 508.8522199901578,
"openwebmath_score": 0.8786824941635132,
"tags": null,
"url": "https://mathhelpboards.com/threads/finding-composition-series-of-groups.2483/"
} |
#### Opalg
##### MHB Oldtimer
Staff member
Ok thanks, it seems it should be quite easy but I have across an example which I don't understand. I am told that {0},<12>,<4>,Z48 is a composition series of Z48 but couldn't <24> be inserted between {0} and <12> since it is of order 2 in Z48? (sorry don't know how to do triangles denoting normal subgroup)
What you say is quite correct. The standard definition of a composition series requires that each component should be maximal normal in the next one. The series $\{0\}\lhd \langle12\rangle \lhd \langle4\rangle \lhd \mathbb{Z}_{48}$ fails that test on two counts. You could put $\langle24\rangle$ between $\{0\}$ and $\langle12\rangle$; and you could put $\langle2\rangle$ between $\langle4\rangle$ and $\mathbb{Z}_{48}$.
#### Poirot
##### Banned
oh sorry, I misread the text (it just said it was a normal series, not maximal). Well I will go away and follow your hints and advice. By the way, in dihedral groups, is it the case that the rotations are in a conjugacy class on their own?, as this would definitly ease the burden working out if a group is closed under conjugacy. Also, are they any rules about what the subgroups of a dihedral group are. Thanks again, and also I would like to say (doubt this is particularly controversial) that I think you are the best mathematician on this site.
#### Deveno | {
"domain": "mathhelpboards.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9632305339244013,
"lm_q1q2_score": 0.8835739305630267,
"lm_q2_score": 0.9173026595857203,
"openwebmath_perplexity": 508.8522199901578,
"openwebmath_score": 0.8786824941635132,
"tags": null,
"url": "https://mathhelpboards.com/threads/finding-composition-series-of-groups.2483/"
} |
#### Deveno
##### Well-known member
MHB Math Scholar
oh sorry, I misread the text (it just said it was a normal series, not maximal). Well I will go away and follow your hints and advice. By the way, in dihedral groups, is it the case that the rotations are in a conjugacy class on their own?, as this would definitly ease the burden working out if a group is closed under conjugacy. Also, are they any rules about what the subgroups of a dihedral group are. Thanks again, and also I would like to say (doubt this is particularly controversial) that I think you are the best mathematician on this site.
the conjugacy class of a rotation will always contain just other rotations (because the rotations form a normal subgroup of the dihedral group), but not all rotations are conjugate (necessarily).
the reason for this is that for some n, D2n may have a non-trivial center, and central elements only contain themselves in their conjugacy class.
for example, in D8 (the symmetries of a square), we have:
[1] = {1}
[r] = {r,r3}
[r2] = {r2} (where the square brackets mean the conjugacy class of an element).
even if we have a trivial center (like with D10 the symmetries of a regular pentagon), we still have:
(rk)r(rk)-1= r (for k = 0,1,2,3,4)
(rks)r(rks)-1 = (rks)r(rks) = (rk)(sr)(rks) = (rk)(r4s)(rks) = (rk)(r4s)(sr-k) = r4
which shows that [r] = {r,r4}.
that is, conjugacy classes of a normal subgroup may still form a (non-trivial) partition of that normal subgroup (as another example, two cycles of the same cycle type may be conjugate in Sn, but not be conjugate in An). | {
"domain": "mathhelpboards.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9632305339244013,
"lm_q1q2_score": 0.8835739305630267,
"lm_q2_score": 0.9173026595857203,
"openwebmath_perplexity": 508.8522199901578,
"openwebmath_score": 0.8786824941635132,
"tags": null,
"url": "https://mathhelpboards.com/threads/finding-composition-series-of-groups.2483/"
} |
# What is the cdf for a partially non-continuous pdf?
Suppose there is a pdf/pmf (?!) which places an atom of size 0.5 on x = 0 and randomizes uniformly with probability 0.5 over the interval [0.5,1].
Such that...
$$f(x)= \begin{cases} 0.5, & \text{if}\ x=0 \\ {1\over (1-0.5)}, & \text{if}\ 0.5 ≤ x ≤ 1 \\ 0, & \text{otherwise} \end{cases}$$
Does the corresponding cdf then look like the following?
$$F(x)= \begin{cases} 0.5, & \text{if}\ x < 0.5 \\ 0.5+0.5\cdot{(x-0.5)\over (1-0.5)}, & \text{if}\ 0.5 ≤ x ≤ 1 \\ 1, & \text{if}\ x > 1 \end{cases}$$
And how to calculate the expected value of this cdf formally? I suppose that $$E(x)={3\over 8}$$ ...but I dont know exactly how to formally deal with the intervalls as f(x) is not continuous.
You can describe the probability density of this using the Dirac delta, viz. $\int_\mathbb{R}\delta(x)g(x)dx=g(0)$. In your case, $f(x)=\frac{1}{2}\delta(x)+1_{[\frac{1}{2},\,1]}(x)$. The second term is an indicator function, and integrates to the half of the probability the delta term doesn't cover; your $\frac{1}{1-0.5}$ factor is unnecessary. Notice I said probability density, not probability density function, because no function has the defining properties of $\delta$. (If you want some terminology, it's a measure. The popular name "Dirac delta function", used in the above link, is misleading.)
Integrating gives the CDF $\frac{1}{2}\Theta(x)+(x-\frac{1}{2})(\Theta(x-\frac{1}{2})-\Theta(x-1))$, where $\Theta(y):=\int_{-\infty}^y\delta(z)dz$ is called the Heaviside step function, which really is a function. (Wikipedia denotes it $H$, but I've often seen people use $\Theta$ or $\theta$.) Equivalently, $\Theta\left(z\right):=\left\{ \begin{array}{cc} 0 & z<0\\ 1 & z\ge0 \end{array}\right.$. The CDF is $$\left\{ \begin{array}{cc} 0 & x<0\\ \tfrac{1}{2} & x\in\left[0,\,\tfrac{1}{2}\right)\\ x & x\in\left[\tfrac{1}{2},\,1\right)\\ 1 & x\ge1 \end{array}\right..$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9811668684574637,
"lm_q1q2_score": 0.8835699980558419,
"lm_q2_score": 0.9005297941266013,
"openwebmath_perplexity": 379.2986891035858,
"openwebmath_score": 0.8759428262710571,
"tags": null,
"url": "https://math.stackexchange.com/questions/2879619/what-is-the-cdf-for-a-partially-non-continuous-pdf"
} |
• How does your answer change (especially the resulting cdf) if the probability density randomizes uniformly over the interval [(1/3),1] with probability 0.5, ceteris paribus? – Moritz Sch Aug 11 '18 at 18:42
• @MoritzSch The PDF (CDF) would be $\frac{1}{2}\delta(x)+\frac{3}{4}1_{[\frac{1}{3},\,1]}(x)$ ($\frac{1}{2}\Theta(x)+\frac{3x-1}{4}(\Theta(x-\frac{1}{3})-\Theta(x-1))$). – J.G. Aug 11 '18 at 18:45
• Thank you! I see, I have to read up about Dirac delta, to understand this. – Moritz Sch Aug 11 '18 at 19:14
• Also I've realised these CDF formulae of mine simply won't work after $x=1$. – J.G. Aug 11 '18 at 19:29
In what sense this is a density function will bear examination. Normally one says $f$ is a probability density function for the distribution of a random variable $X$ if $$\Pr(a<X<b) = \int_a^b f(x) \, dx$$ for all values of $a,b.$ But $\displaystyle \int_a^b \cdots\cdots\,dx$ means an integral with respect to Lebesgue measure, which is the measure that assigns to every interval $(c,d),$ for $c<d,$ its length $d-c.$ That measure assigns $0$ to an interval that is only a point, so the integral of any function over that point is $0.$
However, suppose one integrates with respect to a measure that assigns measure $1$ to the one-point set $\{0\}$ and assigns the length of every interval to that interval if $0$ is not a member of the interval. Then what you've got is a density.
But there's no need to go into that in order to answer your question about the expected value. The c.d.f. is $$F(x) = \Pr(X\le x) = \begin{cases} 0 & \text{if } x<0, \\ 1/2 & \text{if } 0\le x \le 1/2, \\ x & \text{if } 1/2<x\le 1, \\ 1 & \text{if } x\ge1. \end{cases}$$ The expected value is $$\operatorname E(X) = 0 \cdot\Pr(X=0) + \int_{1/2}^1 x\cdot\left(\frac 1 2 \, dx\right) = \frac 3 8.$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9811668684574637,
"lm_q1q2_score": 0.8835699980558419,
"lm_q2_score": 0.9005297941266013,
"openwebmath_perplexity": 379.2986891035858,
"openwebmath_score": 0.8759428262710571,
"tags": null,
"url": "https://math.stackexchange.com/questions/2879619/what-is-the-cdf-for-a-partially-non-continuous-pdf"
} |
Suppose $m$ is the measure described above, assigning measure $1/2$ to $\{0\}$ and the length of each interval to the interval if it does not contain $0.$ Then one can write $$\operatorname E(X) = \int_{-\infty}^\infty xf(x)\,dm(x)$$ and its value will be $3/8.$
• How does your answer change (especially the resulting cdf) if the probability density randomizes uniformly over the interval [0.25,1] with probability 0.5, c.p.? Does it then look like: $$F(x)= \begin{cases} 0.5, & \text{if}\ x < 0.25/ \\ 0.5+0.5\cdot{(x-0.25)\over (1-0.25)}, & \text{if}\ 0.25 ≤ x ≤ 1 \\ 1, & \text{if}\ x > 1 \end{cases}$$ And how does calculation of the expected value change? Is it: $$E(x)= 0*\text{Pr}(X=0)+\int_{1\over 4}^1\ (...)={3\over 16}$$ What is the (...) – Moritz Sch Aug 11 '18 at 19:00
• E(x)=(...)=5/16 Sorry for the mistake, my bad... But still the (...) and the actual form of the cdf in my original reply to your answer is not clear. Thank you in advance! – Moritz Sch Aug 11 '18 at 19:13
• ....And F(x) = 0 if x<0 is also missing in my reply (cdf) to your comment. But I understand this part. I also messed up the inequality signs, but no further explanation needed on that... Things to clarify would be: actual form of E(x) and the form of F(x) if 0.25 < x </= 1 – Moritz Sch Aug 11 '18 at 19:19
• @MoritzSch : In the expression $\displaystyle \int_a^b \cdots\cdots \, dx,$ the dots just mean that what is said is true regardless of which function goes where those dots are. – Michael Hardy Aug 12 '18 at 15:14 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9811668684574637,
"lm_q1q2_score": 0.8835699980558419,
"lm_q2_score": 0.9005297941266013,
"openwebmath_perplexity": 379.2986891035858,
"openwebmath_score": 0.8759428262710571,
"tags": null,
"url": "https://math.stackexchange.com/questions/2879619/what-is-the-cdf-for-a-partially-non-continuous-pdf"
} |
# Simplify $\sqrt[4]{\frac{162x^6}{16x^4}}$ is $\frac{3\sqrt[4]{2x^2}}{2}$
(I've been posting a lot today and yesterday, not sure if too many posts are frowned upon or not. I am studying and making sincere efforts to solve on my own and only post here as a last resort)
I'm asked to simplify $$\sqrt[4]{\frac{162x^6}{16x^4}}$$ and am provided the text book solution $$\frac{3\sqrt[4]{2x^2}}{2}$$.
I arrived at $$\frac{3\sqrt[4]{2x^6}}{2x^4}$$. I cannot tell if this is right and that the provided solution is just a further simplification of where I've gotten to, or if I'm off track entirely.
Here is my working:
$$\sqrt[4]{\frac{162x^6}{16x^4}}$$ = $$\frac{\sqrt[4]{162x^6}}{\sqrt[4]{16x^4}}$$
Denominator: $$\sqrt[4]{16x^4}$$ I think can be simplified to $$2x^4$$ since $$2^4$$ = 16
Numerator: $$\sqrt[4]{162x^6}$$ I was able to simplify (or over complicate) to $$3\sqrt[4]{2}\sqrt[4]{x^6}$$ since:
$$\sqrt[4]{162x^6}$$ = $$\sqrt[4]{81}$$ * $$\sqrt[4]{2}$$ * $$\sqrt[4]{x^6}$$ = $$3 * \sqrt[4]{2} * \sqrt[4]{x^6}$$
Thus I got: $$\frac{3\sqrt[4]{2}\sqrt[4]{x^6}}{2x^4}$$ which I think is equal to $$\frac{3\sqrt[4]{2x^6}}{2x^4}$$ (product of the radicals in the numerator).
How ca I arrive at the provided solution $$\frac{3\sqrt[4]{2x^2}}{2}$$?
• Hint, $\sqrt[4]{16x^4}$ does not simplify to $2x^4$. Rather, it becomes $2x$. Jan 6 '19 at 21:05
• On your first comment. Posting a lot is fine as long as you are actually working on each of the questions you ask (as you clearly are on this one), Jan 6 '19 at 21:13
$$\sqrt[4]{16x^4} = 2\vert x\vert$$
because $$\sqrt[4]{16x^4} = \sqrt[4]{(2x)^4}$$. (Note the absolute value sign since the value returned is positive regardless of whether $$x$$ itself is positive or negative.) The rest is fine, so from here, you get
$$\frac{3\sqrt[4]{2x^6}}{2\vert x\vert} = \frac{3\sqrt[4]{2x^4x^2}}{2x} = \frac{3\vert x\vert\sqrt[4]{2x^2}}{2\vert x\vert} = \frac{3\sqrt[4]{2x^2}}{2}$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9766692359451417,
"lm_q1q2_score": 0.8835443475420282,
"lm_q2_score": 0.9046505357435624,
"openwebmath_perplexity": 278.30161274951274,
"openwebmath_score": 0.7980544567108154,
"tags": null,
"url": "https://math.stackexchange.com/questions/3064391/simplify-sqrt4-frac162x616x4-is-frac3-sqrt42x22"
} |
As shown in the other answer, it is usually better to simplify within the radical so you don’t mess up with absolute values (for even indices).
$$\sqrt[4]{\frac{162x^6}{16x^4}} = \sqrt[4]{\frac{2\cdot3^4x^2}{2^4}} = \frac{3\sqrt[4]{2x^2}}{2}$$
• In your second approach you have $\sqrt[4]{\frac{2\cdot3^4x^2}{2^4}}$. Why is it not $x^6$ in the numerator there? Jan 6 '19 at 21:24
• I simplified $\frac{x^6}{x^4}$ first. Jan 6 '19 at 21:25
• In your first answer, your numerator goes from $3\sqrt[4]{2x^4x^2}$ to $3\vert x\vert\sqrt[4]{2x^2}$. I see the benefit of pulling an x out in front of the radical but cannot see how you did that? Would it be possible to expand on that part if you have a minute? Jan 6 '19 at 21:40
• Sure. $\sqrt[4]{x^4} = \vert x\vert$, like how $\sqrt{x^2} = \vert x\vert$, $\sqrt[3]{x^3} = x$, etc. (As another point, note the use of absolute values when the index is even because the answer is always positive. For an odd index, as in cube roots, sign is preserved, so no absolute value is used.) Jan 6 '19 at 21:47
• A good way of thinking about it: $$\sqrt[4]{2x^6} = \sqrt[4]{2x^2}\cdot\sqrt[4]{x^4}$$ The first quartic root can’t be simplified, and it is therefore left as it is. The second simplifies to $\vert x\vert$. So, the $x$ comes from $x^4$. The $2x^2$ stays inside the radical. (I think your confusion is coming from the $2$ in front of the $x^4$. Bringing out $\sqrt[4]{2}$ would literally be... $\sqrt[4]{2}$ again, so you don’t bring it out as it won’t simplify anything. The exponent of the base must be greater than $4$ for it to be brought out.) Jan 6 '19 at 22:00
$$\sqrt[4]{\frac{162x^6}{16x^4}}=\sqrt[4]{\frac{81\cdot2x^2}{16}}=\frac{3\sqrt[4]{2x^2}}{2}.$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9766692359451417,
"lm_q1q2_score": 0.8835443475420282,
"lm_q2_score": 0.9046505357435624,
"openwebmath_perplexity": 278.30161274951274,
"openwebmath_score": 0.7980544567108154,
"tags": null,
"url": "https://math.stackexchange.com/questions/3064391/simplify-sqrt4-frac162x616x4-is-frac3-sqrt42x22"
} |
×
# Binomial coefficients
$$\text{Binomial coefficients} { n\choose k}, n,k \in \mathbb{N}_0, k \leq n, \text{are defined as}$$
${n \choose i}=\frac{n!}{i!(n-i)!}$.
$\text{They satisfy}{n \choose i}+{n \choose i-1}={n+1 \choose i} \text{ for } i > 0$
$\text{and also} {n \choose 0}+{n\choose 1}+\cdots+{n \choose n}=2^{n},$
${n \choose 0}-{n\choose 1}+\cdots+(-1)^{n}{n \choose n}=0,$
${n+m \choose k}=\sum\limits_{i=0}^k {n \choose i} {m \choose k-i}.$
$$\text{How do I prove that}$$
${n+m \choose k}=\sum\limits_{i=0}^k {n \choose i} {m \choose k-i}?$
$$\text{(Edit: This is also known as the Vandermonde's Identity.)}$$
$$\text{Help would be greatly appreciated. (I came across this in a book)}$$
$$\text{Victor}$$
$$\text{By the way, I used LaTeX to type the whole note :)}$$
Note by Victor Loh
3 years, 5 months ago
Sort by:
Here's a combinatorial proof.
Question:
From a group of $$m+n$$ students consisting of $$n$$ boys and $$m$$ girls, how many ways are there to form a team of $$k$$ students?
${n+m}\choose{k}$.
If that team has $$i$$ boys, then it'll have $$k-i$$ girls. How many ways are there to choose $$i$$ boys from $$n$$ boys? $$n\choose i$$.
How many ways are there to choose $$k-i$$ girls from $$m$$ girls? $$m\choose{k-i}$$.
So for a fixed $$i$$, there are $${n\choose i}{m\choose {k-i}}$$ ways to form a team of $$k$$ students.
Since $$i$$ could be any number from $$0$$ to $$k$$, we add the number ways to form the team for different values of $$i$$.
So, our final count is,
$\displaystyle \sum_{i=0}^k {n\choose i}{m\choose{k-i}}$
Since answers 1 and 2 are counting the same thing, they must be equal.
[proved]
- 3 years, 5 months ago
yeah, that is the actual proof...by counting in 2 ways.
- 3 years, 4 months ago
Wow
- 3 years, 5 months ago | {
"domain": "brilliant.org",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9940889309686339,
"lm_q1q2_score": 0.8835052789118895,
"lm_q2_score": 0.888758793492457,
"openwebmath_perplexity": 593.7462214112378,
"openwebmath_score": 0.9628685116767883,
"tags": null,
"url": "https://brilliant.org/discussions/thread/binomial-coefficients-2/"
} |
- 3 years, 4 months ago
Wow
- 3 years, 5 months ago
Deriving Vandermonde's identity is very simple . ( I am going to tell just the procedure ) First what you need to do is just right binomial expansion of $$(1+x)^{ n }$$ . again rewrite the binomial expansion of$$(x+1)^{ m }$$ . Note that you should expand $$(x+1)^{ m }$$ not $$(1+x)^{ m }$$. now multiplying these two expansions . we get the above summation which is equal to the one of the binomial coefficient in the expansion of $$(x+1)^{ m+n }$$ Hence prooved
- 3 years, 4 months ago
The last identity is known as Vandermonde's Identity.
- 3 years, 5 months ago
How do I prove it in a non-combinatorial way?
- 3 years, 5 months ago
Consider the coeff of $$x^k$$ of both sides in $$(1+x)^n (1+x)^m=(1+x)^{n+m}$$
- 3 years, 5 months ago
Thank you.
- 3 years, 5 months ago
Yay thanks :)
- 3 years, 5 months ago
That's funny that you used $$\LaTeX$$ on the whole thing. Yes, I will surely work on this cool identity. :D
- 3 years, 5 months ago | {
"domain": "brilliant.org",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9940889309686339,
"lm_q1q2_score": 0.8835052789118895,
"lm_q2_score": 0.888758793492457,
"openwebmath_perplexity": 593.7462214112378,
"openwebmath_score": 0.9628685116767883,
"tags": null,
"url": "https://brilliant.org/discussions/thread/binomial-coefficients-2/"
} |
1$Find two irrational numbers between two given rational numbers. The ellipsis $(\dots)$ means that this number does not stop. $\sqrt{36}$ The comment about$2^x$still holds for$\sqrt{2}$, but if you were using a larger irrational number you might have to pick a bigger base than$2$. Solution: The numbers you would have form the set of rational numbers. stops or repeats, the number is rational. Roots of all numbers that are not perfect squares (NPS) are irrational, as are some useful values like #pi# and #e#.. To find the irrational numbers between two numbers like #2 and 3# we need to first find squares of the two numbers which in this case are #2^2=4 and 3^2=9#. Number System Notes. How to Write Irrational Numbers as Decimals. A rational number is a number that can be written as a ratio. Roots of all numbers that are not perfect squares (NPS) are irrational, as are some useful values like #pi# and #e#.. To find the irrational numbers between two numbers like #2 and 3# we need to first find squares of the two numbers which in this case are #2^2=4 and 3^2=9#. Now let us take any two numbers, say a and b. Learn how to find the approximate values of square roots. 1 answer. So we're saying between any two of those rational numbers, you can always find an irrational number. The definition of an irrational number is a number that cannot be written as a ratio of two integers. Yes. In this video, let us learn how to find irrational numbers between any two fractional numbers. We’ve already seen that integers are rational numbers. Similarly, the decimal representations of square roots of numbers that are not perfect squares never stop and never repeat. Example: Find two irrational numbers between 2 and 3. And we’ll practice using them in ways that we’ll use when we solve equations and complete other procedures in algebra. There is no repeating pattern of digits. Before we go ahead to adding, first you have to understand what makes a number irrational. Irrational numbers are | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
ahead to adding, first you have to understand what makes a number irrational. Irrational numbers are the real numbers that cannot be represented as a simple fraction. . Conclusion After reviewing the above points, it is quite clear that the expression of rational numbers can be possible in both fraction and decimal form. An irrational number is a number that cannot be written as the ratio of two integers. 1/7 = 0. $0.475$ Therefore $\sqrt{36}$ is rational. All fractions, both positive and negative, are rational numbers. Which means that the only way to find the next digit is to calculate it. So if we think about the interval between 0 and 1, we know that there are irrational numbers there. 2. Let’s summarize a method we can use to determine whether a number is rational or irrational. Decimals, fractions, and irrational numbers are all closely related. Conversely, irrational numbers include those numbers whose decimal expansion is infinite, non-repetitive and shows no pattern. Learn the difference between rational and irrational numbers, and watch a video about ratios and rates Rational Numbers. So, clearly, some decimals are rational. Many people are surprised to know that a repeating decimal is a rational number. asked Dec 18, 2017 in Class IX Maths by ashu Premium (930 points) 0 votes. $\sqrt{44}$. 1. Click here to get an answer to your question ️ How to find irrational numbers 1. how to find 4 irrational numbers between 3 and 4 - Mathematics - TopperLearning.com | 8p3p2bgg This decimal stops after the $5$, so it is a rational number. 2/7 = 0. So what is an irrational number, anyway? 1. Its decimal form does not stop and does not repeat. Many people are surprised to know that a repeating decimal is a rational number. | EduRev Class 9 Question is disucussed on EduRev Study Group by 114 Class 9 Students. Conversely, irrational numbers include those numbers whose decimal expansion is infinite, non-repetitive and shows no pattern. We have also seen that every fraction | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
expansion is infinite, non-repetitive and shows no pattern. We have also seen that every fraction is a rational number. If the decimal form of a number, Identify each of the following as rational or irrational: 3. Irrational Numbers on a Number Line. Rational and Irrational numbers both are real numbers but different with respect to their properties. Transcript. But choosing an irrational number in an interval, e.g. Let’s look at a few to see if we can write each of them as the ratio of two integers. Example 10 Find an irrational number between 1/7 and 2/7. Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball. (285714) ̅. In short, rational numbers are whole numbers, fractions, and decimals — the numbers we use in our daily lives.. To decide if an integer is a rational number, we try to write it as a ratio of two integers. Learn the difference between rational and irrational numbers, and watch a video about ratios and rates Rational Numbers. Decimals, fractions, and irrational numbers are all closely related. is irrational since exact value of it cannot be obtained. Join now. Let x be any number between a and b. Then, We have a < x < b….. let this be equation (1) Now, subtract √2 from both the sides of equation (1) 1. In this chapter, we’ll make sure your skills are firmly set. It is a contradiction of rational numbers.. Irrational numbers are expressed usually in the form of R\Q, where the backward slash symbol denotes ‘set minus’. 1 answer. test cases, where taking the square root of the number is True for irrational / complex, and False if the square root is a float or int. Decimal Forms $0.8,-0.875,3.25,-6.666\ldots,-6.\overline{66}$ A decimal that does not stop and does not repeat cannot be written as the ratio of integers. We’ll take another look at the kinds of numbers we have worked with in all previous chapters. Follow. Introduction In Rational and Irrational Numbers post, we have | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
with in all previous chapters. Follow. Introduction In Rational and Irrational Numbers post, we have discussed that is irrational. When placing irrational numbers on a number line, note that your placement will not be exact, but a very close estimation. Real Life Math SkillsLearn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball. If you are only looking for the square-root, you could use the. You can't fully write one down because it'd have to go on forever, so you’d need an infinite amount of paper. All right reserved. But an irrational number cannot be written in the form of simple fractions. In fact, any terminating decimal (decimal that stops after a set number of digits) or repeating decimal (decimal in which one or several digits repeat over and over a… $3=\frac{3}{1}-8=\frac{-8}{1}0=\frac{0}{1}$. 1/7 = 0. Are they rational? If the decimal form of a number. Proof that square root of 5 is irrational. An Irrational Number is a real number that cannot be written as a simple fraction.. Irrational means not Rational. We have seen that every integer is a rational number, since $a=\frac{a}{1}$ for any integer, $a$. Try it with the following problem, to make sure you have it right. Remember that all the counting numbers and all the whole numbers are also integers, and so they, too, are rational. What type of numbers would you get if you started with all the integers and then included all the fractions? Top-notch introduction to physics. Ask your question. As we can see, irrational numbers can also be represented as decimals. A radical sign is a math symbol that looks almost like the letter v and is placed in front of a number to indicate that the root should be taken: Not all radicals are irrational. As we can see, irrational numbers can also be represented as decimals. DOWNLOAD IMAGE. how to find 4 irrational numbers between 3 and 4 - Mathematics - TopperLearning.com | 8p3p2bgg between 0 and 1 is | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
irrational numbers between 3 and 4 - Mathematics - TopperLearning.com | 8p3p2bgg between 0 and 1 is not always impossible, it just depends on what you want to do. Let us consider an example √2 and √3 are irrational numbers √2 = 1.4142 (nearly) √3 = 1.7321 (nealry) Now we have to find an irrational number which should lie between 1.4142 and 1.7321 A rational number is a number that can be written as a ratio of two integers. Can we write it as a ratio of two integers? }[/latex] one simple way would be rounding the irrational numbers to a certain place, say, the millionths place, and then find the average of the "trimmed-up" numbers. pavitra2 pavitra2 11.06.2016 Math Secondary School +5 pts. A rational number is the one which can be represented in the form of P/Q where P and Q are integers and Q ≠ 0. Remember that ${6}^{2}=36$ and ${7}^{2}=49$, so $44$ is not a perfect square. And we're going to start thinking about it by just thinking about the interval between 0 and 1. asked Dec 18, 2017 in Class IX Maths by ashu Premium (930 points) 0 votes. A rational number is a number that can be written as a ratio. About me :: Privacy policy :: Disclaimer :: Awards :: DonateFacebook page :: Pinterest pins, Copyright © 2008-2019. ⅔ is an example of rational numbers whereas √2 is an irrational number. Irrational numbers don't have a pattern. The key is to find any like terms, and then add the coefficients together. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The examples used in this video are √32, √55, and √123. Everything you need to prepare for an important exam!K-12 tests, GED math test, basic math tests, geometry tests, algebra tests. Do you remember what the difference is among these types of numbers? How to find out if a radical is irrational There are a couple of ways to check if a number is rational: If you can quickly | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
is irrational There are a couple of ways to check if a number is rational: If you can quickly find a root for the radical, the radical is rational. 2. Tough Algebra Word Problems.If you can solve these problems with no help, you must be a genius! Find irrational numbers between two numbers. Stack Exchange Network. To find if the square root of a number is irrational or not, check to see if its prime factors all have even exponents. $\sqrt{5}=\text{2.236067978…..}$ So the number 1.25, for example, would be rational because it could be written as 5/4. For example, there is no number among integers and fractions that equals the square root of 2. But an irrational number cannot be written in the form of simple fractions. Find irrational numbers between two numbers Class 9. That means it can be written as a fraction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers. Is disucussed on EduRev Study Group by 114 Class 9 Question is disucussed EduRev... Video are √32, √55, and each one tends to go on forever with repeating! So [ latex ] \sqrt { 36 } [ /latex ] 2 Identify each of the fractions we considered. | EduRev Class 9 Question is disucussed on EduRev Study Group by 114 Class 9 is! On a number Line adding, first you have to understand what are rational numbers must... Numbers you would have form the set of rational numbers whereas √2 is an example of rational numbers all. Integer to a decimal by adding a decimal or repeats of square roots and —! Our daily lives rational and irrational numbers, you could use the going to thinking! The interval between 0 and 1, are rational roots of perfect never..., -6.666\ldots, -6.\overline { 66 } [ /latex ] 2 approximate its value ; for example, is... Among integers and then add the coefficients together: Awards:::! ( 930 points ) 0 votes at the counting numbers, and decimals — the numbers we at! ] -8.0 [ /latex ] paying taxes, mortgage loans, and irrational numbers | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
decimals — the numbers we at! ] -8.0 [ /latex ] paying taxes, mortgage loans, and irrational numbers include those numbers whose decimal is! Rates rational numbers shapesMath problem solver a deep understanding of important concepts in physics, Area of irregular problem! Could use the place value of it can not be exact, but a very close estimation that fraction... Help, you can quickly find a root for the radical, the more the! Note that your placement will not be represented as decimals, the form... These problems with no help, you could use the place value of the fractions stops or.. Not perfect squares are always approximations of a ratio two rational number, which is usually abbreviated 2.71828. Find two irrational … irrational numbers can also be represented as decimals t repeat, it just depends what. For the square-root, you could use the place value of it can not be expressed as the denominator writing. Calculator or a computer, we can use to determine whether a is. Find irrational number between 1/7 and 2/7 on forever as a ratio of integers! ; for example,, if you started with all the fractions we considered... Number 0.3333333 ( with a repeating decimal is a rational number about math... 2.71828 but also continues how to find irrational numbers to the right of the following as rational or irrational: 1 far verify..., [ latex ] \sqrt { 36 } =6 [ /latex ] therefore [ latex ] \sqrt { }... All closely related you remember what the difference is among these types of numbers you! You need any other stuff in … Transcript squares of whole numbers, you must be a genius from! Shorthand symbol representing the actual number for an important exam following as rational or.. Numbers you would have form the set of rational numbers are all related... All the numbers we use in our daily lives that your placement will not be expressed as ratio... Answer to your Question ️ How to find the next digit is to find irrational numbers there how to find irrational numbers. Video we | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
to find the next digit is to find irrational numbers there how to find irrational numbers. Video we show more examples of How to find irrational numbers there, -2, -1,0,1,2,3,4\dots [ /latex these! The denominator when writing the decimal form does not stop and does not stop or repeat Natural, integer calculator. Bitterne Park School Homework, Is Gasworks Park Open, Kirkland Signature Organic Quinoa Nutrition Facts, What Color Is Boron, Hp Pavilion I5 7th Generation 8gb Ram Price, When To Prune Brunnera, Book Of Mormon Audio Cd, 12995 8th Rd, Garden, Mi 49835, Diwali Subhakankshalu Telugu, Best Skinfood Mask, Pediatric Residency Lifestyle, "> 1$ Find two irrational numbers between two given rational numbers. The ellipsis $(\dots)$ means that this number does not stop. $\sqrt{36}$ The comment about $2^x$ still holds for $\sqrt{2}$, but if you were using a larger irrational number you might have to pick a bigger base than $2$. Solution: The numbers you would have form the set of rational numbers. stops or repeats, the number is rational. Roots of all numbers that are not perfect squares (NPS) are irrational, as are some useful values like #pi# and #e#.. To find the irrational numbers between two numbers like #2 and 3# we need to first find squares of the two numbers which in this case are #2^2=4 and 3^2=9#. Number System Notes. How to Write Irrational Numbers as Decimals. A rational number is a number that can be written as a ratio. Roots of all numbers that are not perfect squares (NPS) are irrational, as are some useful values like #pi# and #e#.. To find the irrational numbers between two numbers like #2 and 3# we need to first find squares of the two numbers which in this case are #2^2=4 and 3^2=9#. Now let us take any two numbers, say a and b. Learn how to find the approximate values of square roots. 1 answer. So we're saying between any two of those rational numbers, you can always find an irrational number. The definition of an irrational number is a number that | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
you can always find an irrational number. The definition of an irrational number is a number that cannot be written as a ratio of two integers. Yes. In this video, let us learn how to find irrational numbers between any two fractional numbers. We’ve already seen that integers are rational numbers. Similarly, the decimal representations of square roots of numbers that are not perfect squares never stop and never repeat. Example: Find two irrational numbers between 2 and 3. And we’ll practice using them in ways that we’ll use when we solve equations and complete other procedures in algebra. There is no repeating pattern of digits. Before we go ahead to adding, first you have to understand what makes a number irrational. Irrational numbers are the real numbers that cannot be represented as a simple fraction. . Conclusion After reviewing the above points, it is quite clear that the expression of rational numbers can be possible in both fraction and decimal form. An irrational number is a number that cannot be written as the ratio of two integers. 1/7 = 0. $0.475$ Therefore $\sqrt{36}$ is rational. All fractions, both positive and negative, are rational numbers. Which means that the only way to find the next digit is to calculate it. So if we think about the interval between 0 and 1, we know that there are irrational numbers there. 2. Let’s summarize a method we can use to determine whether a number is rational or irrational. Decimals, fractions, and irrational numbers are all closely related. Conversely, irrational numbers include those numbers whose decimal expansion is infinite, non-repetitive and shows no pattern. Learn the difference between rational and irrational numbers, and watch a video about ratios and rates Rational Numbers. So, clearly, some decimals are rational. Many people are surprised to know that a repeating decimal is a rational number. asked Dec 18, 2017 in Class IX Maths by ashu Premium (930 points) 0 votes. $\sqrt{44}$. 1. Click here to get an | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
2017 in Class IX Maths by ashu Premium (930 points) 0 votes. $\sqrt{44}$. 1. Click here to get an answer to your question ️ How to find irrational numbers 1. how to find 4 irrational numbers between 3 and 4 - Mathematics - TopperLearning.com | 8p3p2bgg This decimal stops after the $5$, so it is a rational number. 2/7 = 0. So what is an irrational number, anyway? 1. Its decimal form does not stop and does not repeat. Many people are surprised to know that a repeating decimal is a rational number. | EduRev Class 9 Question is disucussed on EduRev Study Group by 114 Class 9 Students. Conversely, irrational numbers include those numbers whose decimal expansion is infinite, non-repetitive and shows no pattern. We have also seen that every fraction is a rational number. If the decimal form of a number, Identify each of the following as rational or irrational: 3. Irrational Numbers on a Number Line. Rational and Irrational numbers both are real numbers but different with respect to their properties. Transcript. But choosing an irrational number in an interval, e.g. Let’s look at a few to see if we can write each of them as the ratio of two integers. Example 10 Find an irrational number between 1/7 and 2/7. Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball. (285714) ̅. In short, rational numbers are whole numbers, fractions, and decimals — the numbers we use in our daily lives.. To decide if an integer is a rational number, we try to write it as a ratio of two integers. Learn the difference between rational and irrational numbers, and watch a video about ratios and rates Rational Numbers. Decimals, fractions, and irrational numbers are all closely related. is irrational since exact value of it cannot be obtained. Join now. Let x be any number between a and b. Then, We have a < x < b….. let this be equation (1) Now, subtract √2 from both the sides of equation (1) 1. In this chapter, we’ll make sure | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
(1) Now, subtract √2 from both the sides of equation (1) 1. In this chapter, we’ll make sure your skills are firmly set. It is a contradiction of rational numbers.. Irrational numbers are expressed usually in the form of R\Q, where the backward slash symbol denotes ‘set minus’. 1 answer. test cases, where taking the square root of the number is True for irrational / complex, and False if the square root is a float or int. Decimal Forms $0.8,-0.875,3.25,-6.666\ldots,-6.\overline{66}$ A decimal that does not stop and does not repeat cannot be written as the ratio of integers. We’ll take another look at the kinds of numbers we have worked with in all previous chapters. Follow. Introduction In Rational and Irrational Numbers post, we have discussed that is irrational. When placing irrational numbers on a number line, note that your placement will not be exact, but a very close estimation. Real Life Math SkillsLearn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball. If you are only looking for the square-root, you could use the. You can't fully write one down because it'd have to go on forever, so you’d need an infinite amount of paper. All right reserved. But an irrational number cannot be written in the form of simple fractions. In fact, any terminating decimal (decimal that stops after a set number of digits) or repeating decimal (decimal in which one or several digits repeat over and over a… $3=\frac{3}{1}-8=\frac{-8}{1}0=\frac{0}{1}$. 1/7 = 0. Are they rational? If the decimal form of a number. Proof that square root of 5 is irrational. An Irrational Number is a real number that cannot be written as a simple fraction.. Irrational means not Rational. We have seen that every integer is a rational number, since $a=\frac{a}{1}$ for any integer, $a$. Try it with the following problem, to make sure you have it right. Remember that all the counting numbers and all the whole numbers are also integers, | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
have it right. Remember that all the counting numbers and all the whole numbers are also integers, and so they, too, are rational. What type of numbers would you get if you started with all the integers and then included all the fractions? Top-notch introduction to physics. Ask your question. As we can see, irrational numbers can also be represented as decimals. A radical sign is a math symbol that looks almost like the letter v and is placed in front of a number to indicate that the root should be taken: Not all radicals are irrational. As we can see, irrational numbers can also be represented as decimals. DOWNLOAD IMAGE. how to find 4 irrational numbers between 3 and 4 - Mathematics - TopperLearning.com | 8p3p2bgg between 0 and 1 is not always impossible, it just depends on what you want to do. Let us consider an example √2 and √3 are irrational numbers √2 = 1.4142 (nearly) √3 = 1.7321 (nealry) Now we have to find an irrational number which should lie between 1.4142 and 1.7321 A rational number is a number that can be written as a ratio of two integers. Can we write it as a ratio of two integers? }[/latex] one simple way would be rounding the irrational numbers to a certain place, say, the millionths place, and then find the average of the "trimmed-up" numbers. pavitra2 pavitra2 11.06.2016 Math Secondary School +5 pts. A rational number is the one which can be represented in the form of P/Q where P and Q are integers and Q ≠ 0. Remember that ${6}^{2}=36$ and ${7}^{2}=49$, so $44$ is not a perfect square. And we're going to start thinking about it by just thinking about the interval between 0 and 1. asked Dec 18, 2017 in Class IX Maths by ashu Premium (930 points) 0 votes. A rational number is a number that can be written as a ratio. About me :: Privacy policy :: Disclaimer :: Awards :: DonateFacebook page :: Pinterest pins, Copyright © 2008-2019. ⅔ is an example of rational numbers whereas √2 is an irrational number. Irrational numbers don't have a pattern. The | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
of rational numbers whereas √2 is an irrational number. Irrational numbers don't have a pattern. The key is to find any like terms, and then add the coefficients together. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The examples used in this video are √32, √55, and √123. Everything you need to prepare for an important exam!K-12 tests, GED math test, basic math tests, geometry tests, algebra tests. Do you remember what the difference is among these types of numbers? How to find out if a radical is irrational There are a couple of ways to check if a number is rational: If you can quickly find a root for the radical, the radical is rational. 2. Tough Algebra Word Problems.If you can solve these problems with no help, you must be a genius! Find irrational numbers between two numbers. Stack Exchange Network. To find if the square root of a number is irrational or not, check to see if its prime factors all have even exponents. $\sqrt{5}=\text{2.236067978…..}$ So the number 1.25, for example, would be rational because it could be written as 5/4. For example, there is no number among integers and fractions that equals the square root of 2. But an irrational number cannot be written in the form of simple fractions. Find irrational numbers between two numbers Class 9. That means it can be written as a fraction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers. Is disucussed on EduRev Study Group by 114 Class 9 Question is disucussed EduRev... Video are √32, √55, and each one tends to go on forever with repeating! So [ latex ] \sqrt { 36 } [ /latex ] 2 Identify each of the fractions we considered. | EduRev Class 9 Question is disucussed on EduRev Study Group by 114 Class 9 is! On a number Line adding, first you have to understand what are rational numbers must... | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
Class 9 is! On a number Line adding, first you have to understand what are rational numbers must... Numbers you would have form the set of rational numbers whereas √2 is an example of rational numbers all. Integer to a decimal by adding a decimal or repeats of square roots and —! Our daily lives rational and irrational numbers, you could use the going to thinking! The interval between 0 and 1, are rational roots of perfect never..., -6.666\ldots, -6.\overline { 66 } [ /latex ] 2 approximate its value ; for example, is... Among integers and then add the coefficients together: Awards:::! ( 930 points ) 0 votes at the counting numbers, and decimals — the numbers we at! ] -8.0 [ /latex ] paying taxes, mortgage loans, and irrational numbers include those numbers whose decimal is! Rates rational numbers shapesMath problem solver a deep understanding of important concepts in physics, Area of irregular problem! Could use the place value of it can not be exact, but a very close estimation that fraction... Help, you can quickly find a root for the radical, the more the! Note that your placement will not be represented as decimals, the form... These problems with no help, you could use the place value of the fractions stops or.. Not perfect squares are always approximations of a ratio two rational number, which is usually abbreviated 2.71828. Find two irrational … irrational numbers can also be represented as decimals t repeat, it just depends what. For the square-root, you could use the place value of it can not be expressed as the denominator writing. Calculator or a computer, we can use to determine whether a is. Find irrational number between 1/7 and 2/7 on forever as a ratio of integers! ; for example,, if you started with all the fractions we considered... Number 0.3333333 ( with a repeating decimal is a rational number about math... 2.71828 but also continues how to find irrational numbers to the right of the following as rational or irrational: 1 far verify..., [ latex | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
irrational numbers to the right of the following as rational or irrational: 1 far verify..., [ latex ] \sqrt { 36 } =6 [ /latex ] therefore [ latex ] \sqrt { }... All closely related you remember what the difference is among these types of numbers you! You need any other stuff in … Transcript squares of whole numbers, you must be a genius from! Shorthand symbol representing the actual number for an important exam following as rational or.. Numbers you would have form the set of rational numbers are all related... All the numbers we use in our daily lives that your placement will not be expressed as ratio... Answer to your Question ️ How to find the next digit is to find irrational numbers there how to find irrational numbers. Video we show more examples of How to find irrational numbers there, -2, -1,0,1,2,3,4\dots [ /latex these! The denominator when writing the decimal form does not stop and does not stop or repeat Natural, integer calculator. Bitterne Park School Homework, Is Gasworks Park Open, Kirkland Signature Organic Quinoa Nutrition Facts, What Color Is Boron, Hp Pavilion I5 7th Generation 8gb Ram Price, When To Prune Brunnera, Book Of Mormon Audio Cd, 12995 8th Rd, Garden, Mi 49835, Diwali Subhakankshalu Telugu, Best Skinfood Mask, Pediatric Residency Lifestyle, " /> | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
# how to find irrational numbers | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
The two irrational numbers between 2 and 2.5 are 2.101001000100001-----and 2.201 001 0001 00001-----Related questions 0 votes. hi, I can give you an easier and simple method to find irrational number between any two whole numbers. The table below shows the numbers we looked at expressed as a ratio of integers and as a decimal. Transcript. (142857) ̅. A non- terminating and non-recurring decimal is an irrational number.For example, 0.424344445 The number is also an irrational number. If you are only looking for the square-root, you could use the square root algorithm. How to find out if a radical is irrational There are a couple of ways to check if a number is rational: If you can quickly find a root for the radical, the radical is rational. Rational,Irrational,Natural,Integer Property Calculator. In general, any decimal that ends after a number of digits such as $7.3$ or $-1.2684$ is a rational number. These decimals either stop or repeat. $0.58\overline{3}$ It is a rational number. Conclusion After reviewing the above points, it is quite clear that the expression of rational numbers can be possible in both fraction and decimal form. 1. How To Find Irrational Numbers Between Two Decimals DOWNLOAD IMAGE. For this reason, there will usually be some shorthand symbol representing the actual number. Definition: Can be expressed as the quotient of two integers (ie a fraction) with a denominator that is not zero.. Ask your question. Hence 3 √2 is irrational number. Log in. So we're saying between any two of those rational numbers, you can always find an irrational number. Since any integer can be written as the ratio of two integers, all integers are rational numbers. This may be the best way to check. Finding Irrational Numbers between Numbers Find 2 different irrational numbers between numbers 2.6 and 2.8 We know that Irrational Number has Non-Terminating, Non-Repeating Expansion So, 2 different irrational numbers can be 2.6206200620006200006200000…. You could use a | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
Expansion So, 2 different irrational numbers can be 2.6206200620006200006200000…. You could use a calculator. Join now. How to find irrational numbers Ask for details ; Follow Report by EWAPAHUJA 10.03.2019 Log in to add a comment Your email is safe with us. I recently did that, but the application of those numbers just that they had to be compared and some decisions of … Solution: $0.58\overline{3}$ Identify each of the following as rational or irrational: does not stop and does not repeat, the number is irrational. 2 and 3 are rational numbers and is not a perfect square. Any real number that cannot be expressed as a ratio of integers, i.e., any real number that cannot be expressed as simple fraction is called an irrational number. The two irrational numbers between 2 and 2.5 are 2.101001000100001-----and 2.201 001 0001 00001-----Related questions 0 votes. Any irrational number will work with this method, it's just a question of making sure you can easily demonstrate that the number produces is irrational. Everything you need to prepare for an important exam! We will only use it to inform you about new math lessons. What about decimals? For example, there is no number among integers and fractions that equals the square root of 2. Adding irrational numbers is actually quite simple, once you get the hang of it. The number $\pi$ (the Greek letter pi, pronounced ‘pie’), which is very important in describing circles, has a decimal form that does not stop or repeat. Because $7.3$ means $7\frac{3}{10}$, we can write it as an improper fraction, $\frac{73}{10}$. We will now look at the counting numbers, whole numbers, integers, and decimals to make sure they are rational. 2. Let’s look at the decimal form of the numbers we know are rational. The square roots, cube roots, etc of natural numbers are irrational numbers, if their exact values cannot be obtained. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Every rational number can be written both as a ratio of integers and as a decimal that either stops or repeats. Example 10 Find an irrational number between 1/7 and 2/7. $\frac{4}{5},-\frac{7}{8},\frac{13}{4},\frac{-20}{3}$, $\frac{4}{5},\frac{-7}{8},\frac{13}{4},\frac{-20}{3}$, $\frac{-2}{1},\frac{-1}{1},\frac{0}{1},\frac{1}{1},\frac{2}{1},\frac{3}{1}$, $0.8,-0.875,3.25,-6.\overline{6}$, Identify rational numbers from a list of numbers, Identify irrational numbers from a list of numbers. How to use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions, examples and step by step solutions, videos, worksheets, activities that are suitable for Common Core Grade 8, 8.ns.2, estimate rational numbers, number line I recently did that, but the application of those numbers just that they had to be compared and some decisions of the algorithm depended on the these comparisions. (142857) ̅. Transcript. Euler's number, which is usually abbreviated as 2.71828 but also continues infinitely to the right of the decimal point. $3.605551275\dots$ Note: Of course the two irrational numbers must be sufficiently distant, thats to say, not all-same-digits up to the millionths place. After having gone through the stuff given above, we hope that the students would have understood "How to Prove the Given Number is Irrational". Think about the decimal $7.3$. The bar above the $3$ indicates that it repeats. (285714) ̅. Email: donsevcik@gmail.com Tel: 800-234-2933; There's an infinite number of rational numbers. Irrational numbers are always approximations of a value, and each one tends to go on forever. The venn diagram below shows examples of all the different types of rational, irrational | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
on forever. The venn diagram below shows examples of all the different types of rational, irrational numbers including integers, whole numbers, repeating decimals and more. Irrational numbers don't have a pattern. It also shows us there must be irrational numbers (such as … Irrational Numbers. What do these examples tell you? An irrational number is a number that cannot be written as the ratio of two integers. Aside from its radical form, using a calculator or a computer, we can approximate its value; for example, . Find an irrational number between √2 and√3. Stack Exchange Network. https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th- In other words, irrational numbers require an infinite number of decimal digits to write—and these digits never form patterns that allow you to predict what the next one will be. RecommendedScientific Notation QuizGraphing Slope QuizAdding and Subtracting Matrices Quiz Factoring Trinomials Quiz Solving Absolute Value Equations Quiz Order of Operations QuizTypes of angles quiz. $0.475$ Find an irrational number between √2 and√3. CC licensed content, Specific attribution, $\dots -3,-2,-1,0,1,2,3,4\dots$. This means $\sqrt{44}$ is irrational. Irrational numbers have decimals that go on forever with no repeating pattern. Rational,Irrational,Natural,Integer Property Video. So $\sqrt{36}=6$. [IN 1] 20 [OUT 1] True [IN 2] 25 [OUT 2] False [IN 3] -1 [OUT 3] True [IN 4] -20 [OUT 4] True [IN 5] 6.25 [OUT 5] False … We can also change any integer to a decimal by adding a decimal point and a zero. The integer $-8$ could be written as the decimal $-8.0$. 3. So if we think about the interval between 0 and 1, we know that there are irrational numbers there. We need to look at all the numbers we have used so far and verify that they are rational. Let’s summarize a method we can use to determine whether a number is rational or irrational. ii) An irrational number between and . A Rational Number can be written as a Ratio of two integers (ie a | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
An irrational number between and . A Rational Number can be written as a Ratio of two integers (ie a simple fraction). 6 months ago | 1 view. A rational number is a number that can be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q\ne o$. Example: Identify the number as ration… For instance, when placing √15 (which is 3.87), it is best to place the dot on the number line at a place in between 3 and 4 (closer to 4), and then write √15 above it. Find three irrational numbers between 2 and 2.5 . These decimal numbers stop. By the way, of course, there is still the possibility of inputing two irrational numbers, one for each frequency, and having a rational result. Square roots of perfect squares are always whole numbers, so they are rational. Transcript. answered How to find irrational numbers 2 Decimal $-2.0,-1.0,0.0,1.0,2.0,3.0$ Let’s think about square roots now. Apart from the stuff given in this section, if you need any other stuff in … You may already be familiar with two very famous irrational numbers: π or "pi," which is almost always abbreviated as 3.14 but in fact continues infinitely to the right of the decimal point; and "e," a.k.a. Nov 22,2020 - how to find an irrational number between two rational numbers? Clearly all fractions are of that After having gone through the stuff given above, we hope that the students would have understood "How to Prove the Given Number is Irrational". For example. Look at the decimal form of the fractions we just considered. If you can quickly find a root for the radical, the radical is rational. Ratio of Integers $\frac{4}{5},\frac{7}{8},\frac{13}{4},\frac{20}{3}$. Hence 3 √2 is irrational number. We have already described numbers as counting numbers, whole numbers, and integers. Irrational number, any real number that cannot be expressed as the quotient of two integers. In mathematics, the irrational numbers are all the real numbers which are not rational numbers.That is, irrational numbers cannot | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
numbers are all the real numbers which are not rational numbers.That is, irrational numbers cannot be expressed as the ratio of two integers.When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no "measure" in common, that is, there is no … If you can solve these problems with no help, you must be a genius! List Of Irrational Numbers 1 100. The more powerful the computer, the more accurate we can approximate. Irrational Numbers on a Number Line. | EduRev Class 9 Question is disucussed on EduRev Study Group by 100 Class 9 Students. In the following video we show more examples of how to determine whether a number is irrational or rational. Its decimal form does not stop and does not repeat. The technique used is to compare the squares of whole numbers to the number we're taking the square root of. 1 answer. Find an irrational number between 3 and 4 Answer If a and b are two positive rational numbers such that ab is not a perfect square of a rational number, then a b is an irrational number lying between a and b. Solution: If a and b are two positive numbers such that ab is not a perfect square then : i ) A rational number between and . To study irrational numbers one has to first understand what are rational numbers. When placing irrational numbers on a number line, note that your placement will not be exact, but a very close estimation. Rational and Irrational numbers both are real numbers but different with respect to their properties. Therefore, $0.58\overline{3}$ is a repeating decimal, and is therefore a rational number. A counterpart problem in measurement would be to find the length of the diagonal of a square whose side is one unit long; there is no subdivision of the unit length that will divide evenly into the … 2/7 = 0. But the decimal forms of square roots of numbers that are not perfect squares never stop and never repeat, so these square roots are | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
roots of numbers that are not perfect squares never stop and never repeat, so these square roots are irrational. Enter Number you would like to test for, you can enter sqrt(50) for square roots or 5^4 for exponents or 6/7 for fractions . Nov 02,2020 - how to find irrational number between two rational number. Irrational number, any real number that cannot be expressed as the quotient of two integers. A few examples are, $\frac{4}{5},-\frac{7}{8},\frac{13}{4},\text{and}-\frac{20}{3}$. 2. DOWNLOAD IMAGE. You can't fully write one down because it'd have to go on forever, so you’d need an infinite amount of paper. In mathematics, a number is rational if you can write it as a ratio of two integers, in other words in a form a/b where a and b are integers, and b is not zero. The more powerful the computer, the more accurate we can approximate. $3.605551275\dots$. Let us consider an example √2 and √3 are irrational numbers √2 = 1.4142 (nearly) √3 = 1.7321 (nealry) Now we have to find an irrational number which should lie between 1.4142 and 1.7321 The venn diagram below shows examples of all the different types of rational, irrational numbers including integers, whole numbers, repeating decimals and more. Determine Rational or Irrational Numbers (Square Roots and Decimals Only). We’ll work with properties of numbers that will help you improve your number sense. Are there any decimals that do not stop or repeat? If you are only looking for the square-root, … To find if the square root of a number is irrational or not, check to see if its prime factors all have even exponents. The number 0.3333333 (with a repeating 3) could be written as 1/3. This is why such a check becomes helpful. Irrational numbers are always approximations of a value, and each one tends to go on forever. Finding Irrational Numbers between Numbers Find 2 different irrational numbers between numbers 2.6 and 2.8 We know that Irrational Number has Non-Terminating, Non-Repeating Expansion So, 2 different | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
and 2.8 We know that Irrational Number has Non-Terminating, Non-Repeating Expansion So, 2 different irrational numbers can be 2.6206200620006200006200000…. So $7.3$ is the ratio of the integers $73$ and $10$. Irrational numbers have decimals that go on forever with no repeating pattern. Irrational number between 1/7 and 2/7 should have a non – terminating & non-repeating expansion Eg: 0.150150015000150000….., 0.160160016000160000016000000…. The number $36$ is a perfect square, since ${6}^{2}=36$. For example. ⅔ is an example of rational numbers whereas √2 is an irrational number. For instance, when placing √15 (which is 3.87), it is best to place the dot on the number line at a place in between 3 and 4 (closer to 4), and then write √15 above it. Irrational number between 1/7 and 2/7 should have a non – terminating & non-repeating expansion Eg: 0.150150015000150000….., 0.160160016000160000016000000…. There's an infinite number of rational numbers. That means it can be written as a fraction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers. EDIT 1: As an addition, I guess the aformentioned may have mislead most of you to believe that I wanted MATLAB to tell me if a number is rational or not only by its double or int value. It cannot be expressed in the form of a ratio, such as p/q, where p and q are integers, q≠0. 1 answer. Basic-mathematics.com. How to use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions, examples and step by step solutions, videos, worksheets, activities that are suitable for Common Core Grade 8, 8.ns.2, estimate rational numbers, number line Definition: Can be expressed as the quotient of two integers (ie a fraction) with a denominator that is not zero.. Rational Numbers. Each numerator and each denominator is an integer. We can use the place value of the | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
Numbers. Each numerator and each denominator is an integer. We can use the place value of the last digit as the denominator when writing the decimal as a fraction. In this lesson, we will examine those relationships, and look at how to convert between these types of numbers … For this reason, there will usually be some shorthand symbol representing the actual number. Since the number doesn’t stop and doesn’t repeat, it is irrational. The definition of rational numbers tells us that all fractions are rational. New Proof Settles How To Approximate Numbers Like Pi Quanta Magazine. between 0 and 1 is not always impossible, it just depends on what you want to do. Kavita Taneja. And we're going to start thinking about it by just thinking about the interval between 0 and 1. Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here. Log in. Find two irrational … One stop resource to a deep understanding of important concepts in physics, Area of irregular shapesMath problem solver. Aside from its radical form, using a calculator or a computer, we can approximate its value; for example, . 1. An easy way to do this is to write it as a fraction with denominator one. Convert the mixed number to an improper fraction. A rational number is the one which can be represented in the form of P/Q where P and Q are integers and Q ≠ 0. We call this kind of number an irrational number. $\pi =\text{3.141592654……. But choosing an irrational number in an interval, e.g. Introduction In Rational and Irrational Numbers post, we have discussed that is irrational. Are integers rational numbers? Find three irrational numbers between 2 and 2.5 . Write the integer as a fraction with denominator 1. I am taking the square root of a number, and I want to find if it is irrational or complex, and return True or False. \lceil \alpha\rceil should work as a base for any irrational \alpha>1 Find two irrational numbers between two given rational | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
work as a base for any irrational \alpha>1 Find two irrational numbers between two given rational numbers. The ellipsis [latex](\dots)$ means that this number does not stop. $\sqrt{36}$ The comment about $2^x$ still holds for $\sqrt{2}$, but if you were using a larger irrational number you might have to pick a bigger base than $2$. Solution: The numbers you would have form the set of rational numbers. stops or repeats, the number is rational. Roots of all numbers that are not perfect squares (NPS) are irrational, as are some useful values like #pi# and #e#.. To find the irrational numbers between two numbers like #2 and 3# we need to first find squares of the two numbers which in this case are #2^2=4 and 3^2=9#. Number System Notes. How to Write Irrational Numbers as Decimals. A rational number is a number that can be written as a ratio. Roots of all numbers that are not perfect squares (NPS) are irrational, as are some useful values like #pi# and #e#.. To find the irrational numbers between two numbers like #2 and 3# we need to first find squares of the two numbers which in this case are #2^2=4 and 3^2=9#. Now let us take any two numbers, say a and b. Learn how to find the approximate values of square roots. 1 answer. So we're saying between any two of those rational numbers, you can always find an irrational number. The definition of an irrational number is a number that cannot be written as a ratio of two integers. Yes. In this video, let us learn how to find irrational numbers between any two fractional numbers. We’ve already seen that integers are rational numbers. Similarly, the decimal representations of square roots of numbers that are not perfect squares never stop and never repeat. Example: Find two irrational numbers between 2 and 3. And we’ll practice using them in ways that we’ll use when we solve equations and complete other procedures in algebra. There is no repeating pattern of digits. Before we go ahead to adding, first you have to understand what | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
is no repeating pattern of digits. Before we go ahead to adding, first you have to understand what makes a number irrational. Irrational numbers are the real numbers that cannot be represented as a simple fraction. . Conclusion After reviewing the above points, it is quite clear that the expression of rational numbers can be possible in both fraction and decimal form. An irrational number is a number that cannot be written as the ratio of two integers. 1/7 = 0. $0.475$ Therefore $\sqrt{36}$ is rational. All fractions, both positive and negative, are rational numbers. Which means that the only way to find the next digit is to calculate it. So if we think about the interval between 0 and 1, we know that there are irrational numbers there. 2. Let’s summarize a method we can use to determine whether a number is rational or irrational. Decimals, fractions, and irrational numbers are all closely related. Conversely, irrational numbers include those numbers whose decimal expansion is infinite, non-repetitive and shows no pattern. Learn the difference between rational and irrational numbers, and watch a video about ratios and rates Rational Numbers. So, clearly, some decimals are rational. Many people are surprised to know that a repeating decimal is a rational number. asked Dec 18, 2017 in Class IX Maths by ashu Premium (930 points) 0 votes. $\sqrt{44}$. 1. Click here to get an answer to your question ️ How to find irrational numbers 1. how to find 4 irrational numbers between 3 and 4 - Mathematics - TopperLearning.com | 8p3p2bgg This decimal stops after the $5$, so it is a rational number. 2/7 = 0. So what is an irrational number, anyway? 1. Its decimal form does not stop and does not repeat. Many people are surprised to know that a repeating decimal is a rational number. | EduRev Class 9 Question is disucussed on EduRev Study Group by 114 Class 9 Students. Conversely, irrational numbers include those numbers whose decimal expansion is infinite, non-repetitive and shows | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
numbers include those numbers whose decimal expansion is infinite, non-repetitive and shows no pattern. We have also seen that every fraction is a rational number. If the decimal form of a number, Identify each of the following as rational or irrational: 3. Irrational Numbers on a Number Line. Rational and Irrational numbers both are real numbers but different with respect to their properties. Transcript. But choosing an irrational number in an interval, e.g. Let’s look at a few to see if we can write each of them as the ratio of two integers. Example 10 Find an irrational number between 1/7 and 2/7. Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball. (285714) ̅. In short, rational numbers are whole numbers, fractions, and decimals — the numbers we use in our daily lives.. To decide if an integer is a rational number, we try to write it as a ratio of two integers. Learn the difference between rational and irrational numbers, and watch a video about ratios and rates Rational Numbers. Decimals, fractions, and irrational numbers are all closely related. is irrational since exact value of it cannot be obtained. Join now. Let x be any number between a and b. Then, We have a < x < b….. let this be equation (1) Now, subtract √2 from both the sides of equation (1) 1. In this chapter, we’ll make sure your skills are firmly set. It is a contradiction of rational numbers.. Irrational numbers are expressed usually in the form of R\Q, where the backward slash symbol denotes ‘set minus’. 1 answer. test cases, where taking the square root of the number is True for irrational / complex, and False if the square root is a float or int. Decimal Forms $0.8,-0.875,3.25,-6.666\ldots,-6.\overline{66}$ A decimal that does not stop and does not repeat cannot be written as the ratio of integers. We’ll take another look at the kinds of numbers we have worked with in all previous chapters. Follow. Introduction In | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
look at the kinds of numbers we have worked with in all previous chapters. Follow. Introduction In Rational and Irrational Numbers post, we have discussed that is irrational. When placing irrational numbers on a number line, note that your placement will not be exact, but a very close estimation. Real Life Math SkillsLearn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball. If you are only looking for the square-root, you could use the. You can't fully write one down because it'd have to go on forever, so you’d need an infinite amount of paper. All right reserved. But an irrational number cannot be written in the form of simple fractions. In fact, any terminating decimal (decimal that stops after a set number of digits) or repeating decimal (decimal in which one or several digits repeat over and over a… $3=\frac{3}{1}-8=\frac{-8}{1}0=\frac{0}{1}$. 1/7 = 0. Are they rational? If the decimal form of a number. Proof that square root of 5 is irrational. An Irrational Number is a real number that cannot be written as a simple fraction.. Irrational means not Rational. We have seen that every integer is a rational number, since $a=\frac{a}{1}$ for any integer, $a$. Try it with the following problem, to make sure you have it right. Remember that all the counting numbers and all the whole numbers are also integers, and so they, too, are rational. What type of numbers would you get if you started with all the integers and then included all the fractions? Top-notch introduction to physics. Ask your question. As we can see, irrational numbers can also be represented as decimals. A radical sign is a math symbol that looks almost like the letter v and is placed in front of a number to indicate that the root should be taken: Not all radicals are irrational. As we can see, irrational numbers can also be represented as decimals. DOWNLOAD IMAGE. how to find 4 irrational numbers between 3 and 4 - Mathematics - | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
as decimals. DOWNLOAD IMAGE. how to find 4 irrational numbers between 3 and 4 - Mathematics - TopperLearning.com | 8p3p2bgg between 0 and 1 is not always impossible, it just depends on what you want to do. Let us consider an example √2 and √3 are irrational numbers √2 = 1.4142 (nearly) √3 = 1.7321 (nealry) Now we have to find an irrational number which should lie between 1.4142 and 1.7321 A rational number is a number that can be written as a ratio of two integers. Can we write it as a ratio of two integers? }[/latex] one simple way would be rounding the irrational numbers to a certain place, say, the millionths place, and then find the average of the "trimmed-up" numbers. pavitra2 pavitra2 11.06.2016 Math Secondary School +5 pts. A rational number is the one which can be represented in the form of P/Q where P and Q are integers and Q ≠ 0. Remember that ${6}^{2}=36$ and ${7}^{2}=49$, so $44$ is not a perfect square. And we're going to start thinking about it by just thinking about the interval between 0 and 1. asked Dec 18, 2017 in Class IX Maths by ashu Premium (930 points) 0 votes. A rational number is a number that can be written as a ratio. About me :: Privacy policy :: Disclaimer :: Awards :: DonateFacebook page :: Pinterest pins, Copyright © 2008-2019. ⅔ is an example of rational numbers whereas √2 is an irrational number. Irrational numbers don't have a pattern. The key is to find any like terms, and then add the coefficients together. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The examples used in this video are √32, √55, and √123. Everything you need to prepare for an important exam!K-12 tests, GED math test, basic math tests, geometry tests, algebra tests. Do you remember what the difference is among these types of numbers? How to find out if a radical is irrational There are a couple of ways to check if | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
types of numbers? How to find out if a radical is irrational There are a couple of ways to check if a number is rational: If you can quickly find a root for the radical, the radical is rational. 2. Tough Algebra Word Problems.If you can solve these problems with no help, you must be a genius! Find irrational numbers between two numbers. Stack Exchange Network. To find if the square root of a number is irrational or not, check to see if its prime factors all have even exponents. $\sqrt{5}=\text{2.236067978…..}$ So the number 1.25, for example, would be rational because it could be written as 5/4. For example, there is no number among integers and fractions that equals the square root of 2. But an irrational number cannot be written in the form of simple fractions. Find irrational numbers between two numbers Class 9. That means it can be written as a fraction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers. Is disucussed on EduRev Study Group by 114 Class 9 Question is disucussed EduRev... Video are √32, √55, and each one tends to go on forever with repeating! So [ latex ] \sqrt { 36 } [ /latex ] 2 Identify each of the fractions we considered. | EduRev Class 9 Question is disucussed on EduRev Study Group by 114 Class 9 is! On a number Line adding, first you have to understand what are rational numbers must... Numbers you would have form the set of rational numbers whereas √2 is an example of rational numbers all. Integer to a decimal by adding a decimal or repeats of square roots and —! Our daily lives rational and irrational numbers, you could use the going to thinking! The interval between 0 and 1, are rational roots of perfect never..., -6.666\ldots, -6.\overline { 66 } [ /latex ] 2 approximate its value ; for example, is... Among integers and then add the coefficients together: Awards:::! ( 930 points ) 0 votes at the counting numbers, and decimals — the numbers we at! ] -8.0 [ /latex ] paying | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
points ) 0 votes at the counting numbers, and decimals — the numbers we at! ] -8.0 [ /latex ] paying taxes, mortgage loans, and irrational numbers include those numbers whose decimal is! Rates rational numbers shapesMath problem solver a deep understanding of important concepts in physics, Area of irregular problem! Could use the place value of it can not be exact, but a very close estimation that fraction... Help, you can quickly find a root for the radical, the more the! Note that your placement will not be represented as decimals, the form... These problems with no help, you could use the place value of the fractions stops or.. Not perfect squares are always approximations of a ratio two rational number, which is usually abbreviated 2.71828. Find two irrational … irrational numbers can also be represented as decimals t repeat, it just depends what. For the square-root, you could use the place value of it can not be expressed as the denominator writing. Calculator or a computer, we can use to determine whether a is. Find irrational number between 1/7 and 2/7 on forever as a ratio of integers! ; for example,, if you started with all the fractions we considered... Number 0.3333333 ( with a repeating decimal is a rational number about math... 2.71828 but also continues how to find irrational numbers to the right of the following as rational or irrational: 1 far verify..., [ latex ] \sqrt { 36 } =6 [ /latex ] therefore [ latex ] \sqrt { }... All closely related you remember what the difference is among these types of numbers you! You need any other stuff in … Transcript squares of whole numbers, you must be a genius from! Shorthand symbol representing the actual number for an important exam following as rational or.. Numbers you would have form the set of rational numbers are all related... All the numbers we use in our daily lives that your placement will not be expressed as ratio... Answer to your Question ️ How to find the next digit is to find irrational numbers | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
as ratio... Answer to your Question ️ How to find the next digit is to find irrational numbers there how to find irrational numbers. Video we show more examples of How to find irrational numbers there, -2, -1,0,1,2,3,4\dots [ /latex these! The denominator when writing the decimal form does not stop and does not stop or repeat Natural, integer calculator. | {
"domain": "com.br",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130554263733,
"lm_q1q2_score": 0.8834946686344102,
"lm_q2_score": 0.8933094096048377,
"openwebmath_perplexity": 396.32674951472035,
"openwebmath_score": 0.7736655473709106,
"tags": null,
"url": "https://crank.com.br/pages/674a61-how-to-find-irrational-numbers"
} |
# How does divisibility test using congruence work?
In the book, it said:
Let $n = a_{k}10^{k} + a_{k-1}10^{k-1} + a_{k-2}10^{k-2} + ... + a_110 + a_0$
Then, because $10 \equiv 0 \pmod{2}$ it follows that $10^j \equiv 0 \pmod{2^j}$
What congruence property did they use in this case? Is that:
If $a \equiv b \pmod{k_1}$ and $c \equiv d \pmod{k_2}$ then, $ab \equiv cd \pmod{k_1k_2}$ ?
I saw one property in the book, which is:
$a \equiv b \pmod{k}$ and $c \equiv d \pmod{k}$then, $ab \equiv cd \pmod{k}$ But I really don't understand how this property relates to the one above it. Any idea?
-
To typeset moduli for congruences, use \pmod{k}. For example, $a\equiv b \pmod{c}$ produces $a\equiv b \pmod{c}$. – Arturo Magidin Feb 26 '11 at 4:00
Maybe it will help to notice that $10^j=2^j5^j$, so clearly $2^j|10^j$, and thus $10^j\equiv 0\pmod{2^j}$.
Essentially for that particular case, you have $10\equiv 0\pmod{2}$, which says $2|10$. It follows that $10^j\equiv 0\pmod{2^j}$ because $10^j$ has $j$ factors of $10$, each of which is divisible by $2$, and thus you can divide by $j$ factors of $2$. That is $2^j|10^j$, or $10^j\equiv 0\pmod{2^j}$.
Does that make it more clear?
-
Thanks, it's clear now ;) However, is there a property like If $a \equiv b ( mod \ \ k_1 )$ and $c \equiv d ( mod \ \ k_2 )$ then, $ab \equiv cd ( mod \ \ k_1k_2 )$ ? – Chan Feb 26 '11 at 2:04
@Chan, I'm afraid there isn't such a property. For a counterexample, notice $5\equiv 2\pmod{3}$ and $7\equiv 2\pmod{5}$. However, $10\not\equiv 14\pmod{15}$. By the way, \pmod{k_1} will typeset to $\pmod{k_1}$, not $(mod\ k_1)$ if you prefer the mod text to not be italicized. – yunone Feb 26 '11 at 2:12
hehe what a similar example! – milcak Feb 26 '11 at 2:13
@milcak, ah I just saw your comment on your answer! How coincidental. – yunone Feb 26 '11 at 2:19
Thanks for your clear explanation. – Chan Feb 26 '11 at 2:35
You you need to use this property $j$ times, since: | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130557502124,
"lm_q1q2_score": 0.8834946583892588,
"lm_q2_score": 0.8933093989533708,
"openwebmath_perplexity": 488.43683317997164,
"openwebmath_score": 0.9607234597206116,
"tags": null,
"url": "http://math.stackexchange.com/questions/23780/how-does-divisibility-test-using-congruence-work"
} |
You you need to use this property $j$ times, since:
$10 \equiv 0 \mod{2}$ and $10 \equiv 0 \mod{2}$, then $10 \cdot 10 \equiv 10^2 \equiv 0 \mod{2^2}$
You know that $2|10$ so it must be that $2^2 | 10^2$ (factorization). Repeat again:
$10 \equiv 0 \mod{2}$ and $10^2 \equiv 0 \mod{2^2}$, then $10 \cdot 10^2 \equiv 10^3 \equiv 0 \mod{2^3}$
So in the end you get:
$10 \equiv 0 \mod{2}$ and $10^{j-1} \equiv 0 \mod{2^{j-1}}$, then $10 \cdot 10^{j-1} \equiv 10^j \equiv 0 \mod{2^j}$
-
So the the 2 inside the mod part can be multiplied? – Chan Feb 26 '11 at 2:06
@Chan Here yes (the reason why is contained in yunone's answer). But in general you cannot always do this: consider $7 \equiv 2 \mod{5}$ and $2 \equiv 2 \mod{3}$, but $7\cdot 2 \equiv 14 \equiv -1 \mod{15}$ but $2\cdot 2 \equiv 4 \mod{15}$. – milcak Feb 26 '11 at 2:12
how did you guys come up with exactly the same example! Amazing ^_^! – Chan Feb 26 '11 at 2:36
It's no coincidence that the congruence sign $\equiv$ resembles the equality sign $=\:$. This notation was explicitly devised to help remind you of the fact that congruence relations share many of the same properties as the equality relation. In particular, just like equations in the ring of integers, ring congruences can be added, multiplied, scaled, etc. Thus, considering this analogy, how would you prove that $\rm\ n = 0\ \Rightarrow\ n^{\:j} = 0\$ for $\rm\:n\:$ an integer? Precisely the same proof works for congruences.
For completeness, here is a proof of the congruence product rule
LEMMA $\rm\ \ A\equiv a,\ B\equiv b\ \Rightarrow\ AB\equiv ab\ \ (mod\ m)$
Proof $\rm\ \ m\: |\: A-a,\:\:\ B-b\ \Rightarrow\ m\ |\ (A-a)\ B + a\ (B-b)\ =\ AB - ab$
- | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9890130557502124,
"lm_q1q2_score": 0.8834946583892588,
"lm_q2_score": 0.8933093989533708,
"openwebmath_perplexity": 488.43683317997164,
"openwebmath_score": 0.9607234597206116,
"tags": null,
"url": "http://math.stackexchange.com/questions/23780/how-does-divisibility-test-using-congruence-work"
} |
What is the value of the measure of the segment $MN$?
In an ABC triangle. plot the height AH, then $$HM \perp AB$$ and $$HN \perp AC$$. Calculate $$MN$$. if the perimeter of the pedal triangle (DEH) of the triangle ABC is 26 (Answer:13)
My progress: I made the drawing and I believe that the solution must lie in the parallelism and relationships of an cyclic quadrilateral
If we reflect $$H$$ across $$AB$$ and $$AC$$ we get two new points $$F$$ and $$G$$.
Since $$BE$$ and $$CD$$ are angle bisector for $$\angle DEH$$ and $$\angle HDE$$ we see $$D,E,F$$ and $$G$$ are collinear. Now $$MN$$ is midle line in the triangle $$HGF$$ with respect to $$FG$$ which lenght is \begin{align}FG &= FD+DE+EG\\ &= DH+DE +EH\\&=26 \end{align} so $$MN = {1\over 2}FG = 13$$
• excellent..thank you for the help Sep 23 at 20:03
• would not be MN is midle line in the triangle HGF? Sep 23 at 20:23
• "Since BE and CD are angle bisector"...How did you reach this conclusion? Sep 23 at 20:28
• Last one is pretty known property of the pedal triangle with respect to $H$. Try to google it.
– Aqua
Sep 23 at 20:33
• Did not know this property ... thank you Sep 23 at 21:28
If you know that the orthocenter of the parent triangle is the incenter of the pedal triangle then the work can be made easier. Otherwise as you mentioned, we can always show it using the inscribed angle theorem and the midpoint theorem but it is not as quick as the other answer.
I will refer to the angles of $$\triangle ABC$$ as $$\angle A, \angle B$$ and $$\angle C$$.
We see quadrilateral $$BDOH$$ is cyclic.
$$\angle OHD = \angle OBD = 90^\circ - \angle A$$
$$\angle DHM = 90^\circ - \angle OHD - \angle BHM$$
$$= 90^\circ - (90^\circ - \angle A) - (90^\circ - \angle B) = \angle A + \angle B - 90^\circ$$
$$= 180^0 - \angle C - 90^\circ = 90^\circ - \angle C$$
Also given $$AMHN$$ is cyclic,
$$\angle HMN = \angle HAN = 90^\circ - \angle C$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9873750518329435,
"lm_q1q2_score": 0.8834927927023618,
"lm_q2_score": 0.8947894632969137,
"openwebmath_perplexity": 353.21336819743357,
"openwebmath_score": 0.9031890630722046,
"tags": null,
"url": "https://math.stackexchange.com/questions/4258502/what-is-the-value-of-the-measure-of-the-segment-mn"
} |
Also given $$AMHN$$ is cyclic,
$$\angle HMN = \angle HAN = 90^\circ - \angle C$$
In right triangle $$\triangle DMH$$, $$\angle HMN = \angle DHM$$ so $$P$$ must be circumcenter of the triangle.
Similarly, I will leave it for you to show that $$Q$$ is the circumcenter of $$\triangle ENH$$.
Once you show that, $$P$$ and $$Q$$ are midpoints of $$DH$$ and $$EH$$ respectively, it follows that
$$PQ = \frac{DE}{2}, MP = \frac{DH}{2}, NQ = \frac{EH}{2}$$
Adding them, $$MN = 13$$
• actually with the ownership of incenter it's much easier,,,great explanation Sep 23 at 21:30 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9873750518329435,
"lm_q1q2_score": 0.8834927927023618,
"lm_q2_score": 0.8947894632969137,
"openwebmath_perplexity": 353.21336819743357,
"openwebmath_score": 0.9031890630722046,
"tags": null,
"url": "https://math.stackexchange.com/questions/4258502/what-is-the-value-of-the-measure-of-the-segment-mn"
} |
# Converting English to Quantifiers: 'There is no greatest prime'
I'm working on an exercise that appears rather simple, but the answer I keep coming up with differs from what the instructor found.
Say I want to convert the sentence 'there is no greatest prime' to quantifier notation, and I'm to work with two english variables, $a$ and $b$, within the universe of discourse of $\mathbb{N}$, and a predicate, $\text{prime$x$}$ that corresponds to "$x$ a prime."
My approach was: this sentence is equivalent to saying that, for any prime number, we can always find some other prime greater than it. So, take $a$ and $b$ to be naturals, and with $a$ we quantify over the entire universe of naturals. We need only find one larger prime, so we can allow an existential quantifer for $b$. Then, we apply the prime predicate to both $a$ and $b$, and reason that we can always choose a $b$ so that $b > a$. So, I come up with: $$\forall b, \exists a, \left(\text{prime a} \wedge \text{prime b} \wedge \left(b > a\right)\right).$$ This seemed to make sense, and I believe follows from the relatively famous proof by contradiction that there is no greatest prime.
However, this answer was apparently wrong, and I can't quite figure out why. I'd greatly appreciate any insights on this.
REVISION: Thank you all for the very helpful answers. For reference for anyone who may look up this problem, people have highlighted two fundamental mistakes in my above constructions. First, I incorrectly suggested, with prime $b$, that every natural number is prime, which is surely not the case: this should be framed as an implication, with antecedent "$b$ is prime." From there, that $b$ is prime would guarantee the existence of some prime, $a$, such that $a > b$. This was the second mistake, as I inadvertently reversed the inequality sign. This could be framed either with $p \implies q$ or, as with one answer, the logically equivalent expression $-p \lor q$.
Thanks again. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9780517469248846,
"lm_q1q2_score": 0.8834684711248909,
"lm_q2_score": 0.9032942008463507,
"openwebmath_perplexity": 191.66395129879,
"openwebmath_score": 0.8620302677154541,
"tags": null,
"url": "https://math.stackexchange.com/questions/2849740/converting-english-to-quantifiers-there-is-no-greatest-prime"
} |
Thanks again.
• You need $\forall b \big(\text{prime}(b) \to \exists a(\text{prime}(a) \land a>b)\big)$ Jul 13 '18 at 14:52
There's one mistake that just looks like a typo: it seems that you meant $a > b$ rather than $b > a$.
More fundamentally, what goes wrong is that you (in particular) claim that any $b$ is prime. Even if we forget all the conditions on $a$, your sentence still claims that $\forall b(\mathrm{prime}(b))$. What you probably mean is that if $b$ is prime, then there is a larger prime $a$. For example, $$\forall b(\mathrm{prime}(b) \to \exists a(\mathrm{prime}(a) \land a > b)).$$
There are 2 problems with your statement: $$\forall b, \exists a, \left(\text{prime a} \wedge \text{prime b} \wedge \left(b > a\right)\right)$$ which says: for every $b$, there is $a$, such that $b$ is prime, $a$ is prime, and $a<b$. First, why must $b$ be prime? Second, $a<b$ is in the wrong direction. I would fix it as follows $$\forall b, (\neg\text{ prime }b \vee (\exists a,\text{ prime a} \wedge \text{prime }b \wedge (a>b))).$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9780517469248846,
"lm_q1q2_score": 0.8834684711248909,
"lm_q2_score": 0.9032942008463507,
"openwebmath_perplexity": 191.66395129879,
"openwebmath_score": 0.8620302677154541,
"tags": null,
"url": "https://math.stackexchange.com/questions/2849740/converting-english-to-quantifiers-there-is-no-greatest-prime"
} |
# how to find radius from area | {
"domain": "eliterugservices.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180677531124,
"lm_q1q2_score": 0.8834511758437261,
"lm_q2_score": 0.8962513772903669,
"openwebmath_perplexity": 419.2476583347293,
"openwebmath_score": 0.9071367979049683,
"tags": null,
"url": "https://www.eliterugservices.com/css/unchain-blockchain-geaajfz/archive.php?c53898=how-to-find-radius-from-area"
} |
Dimensions of a circle: O - origin, R - radius, D - diameter, C - circumference ( Wikimedia) Area, on the other hand, is all the space contained inside the circle. The diameter is always double the radius. … The radius is half the diameter, so the radius is 5 feet, or r = 5. When we connect a point on the circumference of a circle to the exact centre, then the line segment made is called the radius of the ring. A sector is an area formed between the two segments also called as radii, which meets at the center of the circle. [2] X Research source The symbol π{\displaystyle \pi } ("pi") is a special number, roughly equal to 3.14. or, when you know the Circumference: A = C2 / 4π. Finding the arc width and height The surface area of a sphere is derived from the equation A = 4πr 2. Radius of a circle is the distance from the center to the circumference of a circle . So, Area = lr/ 2 = 618.75 cm 2 (275 ⋅ r)/2 = 618.75. r = 45 cm Here is an interactive widget to help you learn about finding the radius of a circle from its area. The radius is an identifying trait, and from it other measurements of the sphere can be calculated, including its circumference, surface area and volume. (2)\ diameter:\hspace{40px} R=2r\\. Just plug that value into the formula for the area of a circle and solve. r=c/2\pi r = c/2π. How Do You Find the Area of a Circle if You Know the Radius? Just plug that value into the formula for the area of a circle and solve. Just plug that value into the formula for the area of a circle and solve. sin is the sine function calculated in degrees If you know the radius of a circle, you can use it to find the area of that circle. Example 2 : Find the radius, central angle and perimeter of a sector whose arc length and area are 27.5 cm and 618.75 cm 2 respectively. Click in the Button Draw a Circle, then Click on map to place the center of the circle and drag at same time to start creating the circle. Find the radius of a circle whose area is equal to the area of | {
"domain": "eliterugservices.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180677531124,
"lm_q1q2_score": 0.8834511758437261,
"lm_q2_score": 0.8962513772903669,
"openwebmath_perplexity": 419.2476583347293,
"openwebmath_score": 0.9071367979049683,
"tags": null,
"url": "https://www.eliterugservices.com/css/unchain-blockchain-geaajfz/archive.php?c53898=how-to-find-radius-from-area"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.