query stringlengths 22 409 | positive_passages listlengths 1 1 | negative_passages listlengths 12 12 | score float64 56 100 |
|---|---|---|---|
Query: How can I calculate the area when I only know the lengths of two sides and the angle between them? | [
{
"doc_id": 17476,
"text": "\\section{Area of Triangle in Terms of Two Sides and Angle} Tags: Triangles, Area of Triangle in Terms of Two Sides and Angle, Euclidean Geometry, Area Formulas, Areas of Triangles \\begin{theorem} The area of a triangle $ABC$ is given by: :$\\displaystyle \\frac 1 2 a b \\sin C... | [
{
"doc_id": 37085,
"text": "['', 'A triangle has two sides, which measure 12cm and 14 cm, and the angle between them is 67 degrees. Find the area of the triangle using trigonometry.\\nTo find the area of the triangle, we can use the formula:\\n\\nArea = (1/2) * a * b * sin(C)\\n\\nwhere a and b are the leng... | 96 |
Query: If I have a triangle and measure two sides and the angle between them, what formula can I use to find the area? | [
{
"doc_id": 17476,
"text": "\\section{Area of Triangle in Terms of Two Sides and Angle} Tags: Triangles, Area of Triangle in Terms of Two Sides and Angle, Euclidean Geometry, Area Formulas, Areas of Triangles \\begin{theorem} The area of a triangle $ABC$ is given by: :$\\displaystyle \\frac 1 2 a b \\sin C... | [
{
"doc_id": 37085,
"text": "['', 'A triangle has two sides, which measure 12cm and 14 cm, and the angle between them is 67 degrees. Find the area of the triangle using trigonometry.\\nTo find the area of the triangle, we can use the formula:\\n\\nArea = (1/2) * a * b * sin(C)\\n\\nwhere a and b are the leng... | 96 |
Query: What is the formula for finding the area of a triangle if you only have the lengths of two sides and the included angle? | [
{
"doc_id": 17476,
"text": "\\section{Area of Triangle in Terms of Two Sides and Angle} Tags: Triangles, Area of Triangle in Terms of Two Sides and Angle, Euclidean Geometry, Area Formulas, Areas of Triangles \\begin{theorem} The area of a triangle $ABC$ is given by: :$\\displaystyle \\frac 1 2 a b \\sin C... | [
{
"doc_id": 5690,
"text": "['', 'A triangle has two sides of length 10 units and 15 units, with a included angle of 60 degrees. What is the area of the triangle?\\nTo find the area of the triangle, we can use the formula for the area of a triangle when two sides and the included angle are given:\\n\\nArea =... | 96 |
Query: Imagine I'm building a triangular garden bed and I've measured two sides and the angle between them, how do I figure out its area? | [
{
"doc_id": 17476,
"text": "\\section{Area of Triangle in Terms of Two Sides and Angle} Tags: Triangles, Area of Triangle in Terms of Two Sides and Angle, Euclidean Geometry, Area Formulas, Areas of Triangles \\begin{theorem} The area of a triangle $ABC$ is given by: :$\\displaystyle \\frac 1 2 a b \\sin C... | [
{
"doc_id": 37085,
"text": "['', 'A triangle has two sides, which measure 12cm and 14 cm, and the angle between them is 67 degrees. Find the area of the triangle using trigonometry.\\nTo find the area of the triangle, we can use the formula:\\n\\nArea = (1/2) * a * b * sin(C)\\n\\nwhere a and b are the leng... | 96 |
Query: When the sides of a triangle are 5m and 7m, and the included angle is 60 degrees, how do I find the area? | [
{
"doc_id": 17476,
"text": "\\section{Area of Triangle in Terms of Two Sides and Angle} Tags: Triangles, Area of Triangle in Terms of Two Sides and Angle, Euclidean Geometry, Area Formulas, Areas of Triangles \\begin{theorem} The area of a triangle $ABC$ is given by: :$\\displaystyle \\frac 1 2 a b \\sin C... | [
{
"doc_id": 5690,
"text": "['', 'A triangle has two sides of length 10 units and 15 units, with a included angle of 60 degrees. What is the area of the triangle?\\nTo find the area of the triangle, we can use the formula for the area of a triangle when two sides and the included angle are given:\\n\\nArea =... | 96 |
Query: How can I calculate the area of a triangle if I only know two sides and the angle between them? | [
{
"doc_id": 17476,
"text": "\\section{Area of Triangle in Terms of Two Sides and Angle} Tags: Triangles, Area of Triangle in Terms of Two Sides and Angle, Euclidean Geometry, Area Formulas, Areas of Triangles \\begin{theorem} The area of a triangle $ABC$ is given by: :$\\displaystyle \\frac 1 2 a b \\sin C... | [
{
"doc_id": 37085,
"text": "['', 'A triangle has two sides, which measure 12cm and 14 cm, and the angle between them is 67 degrees. Find the area of the triangle using trigonometry.\\nTo find the area of the triangle, we can use the formula:\\n\\nArea = (1/2) * a * b * sin(C)\\n\\nwhere a and b are the leng... | 96 |
Query: I'm trying to figure out the formula for finding the area of a triangle when given two side lengths and the included angle. Can you help? | [
{
"doc_id": 17476,
"text": "\\section{Area of Triangle in Terms of Two Sides and Angle} Tags: Triangles, Area of Triangle in Terms of Two Sides and Angle, Euclidean Geometry, Area Formulas, Areas of Triangles \\begin{theorem} The area of a triangle $ABC$ is given by: :$\\displaystyle \\frac 1 2 a b \\sin C... | [
{
"doc_id": 5690,
"text": "['', 'A triangle has two sides of length 10 units and 15 units, with a included angle of 60 degrees. What is the area of the triangle?\\nTo find the area of the triangle, we can use the formula for the area of a triangle when two sides and the included angle are given:\\n\\nArea =... | 96 |
Query: If I have a triangular garden bed and want to plant flowers in it, how much space will I have if I know two sides and the angle between them? | [
{
"doc_id": 17476,
"text": "\\section{Area of Triangle in Terms of Two Sides and Angle} Tags: Triangles, Area of Triangle in Terms of Two Sides and Angle, Euclidean Geometry, Area Formulas, Areas of Triangles \\begin{theorem} The area of a triangle $ABC$ is given by: :$\\displaystyle \\frac 1 2 a b \\sin C... | [
{
"doc_id": 36631,
"text": "['', \"In a garden, there are yellow and green flowers which are straight and curved. If the probability of picking a green flower is 2/3 and picking a straight flower is 1/2, then what is the probability of picking a flower which is yellow and straight\\nAnswer Choices: (A) 1/6 ... | 96 |
Query: Could you explain why the area of a triangle is calculated as half the product of two sides times the sine of the included angle? | [
{
"doc_id": 17476,
"text": "\\section{Area of Triangle in Terms of Two Sides and Angle} Tags: Triangles, Area of Triangle in Terms of Two Sides and Angle, Euclidean Geometry, Area Formulas, Areas of Triangles \\begin{theorem} The area of a triangle $ABC$ is given by: :$\\displaystyle \\frac 1 2 a b \\sin C... | [
{
"doc_id": 5690,
"text": "['', 'A triangle has two sides of length 10 units and 15 units, with a included angle of 60 degrees. What is the area of the triangle?\\nTo find the area of the triangle, we can use the formula for the area of a triangle when two sides and the included angle are given:\\n\\nArea =... | 96 |
Query: What if the angle between the two sides is obtuse? | [
{
"doc_id": 17476,
"text": "\\section{Area of Triangle in Terms of Two Sides and Angle} Tags: Triangles, Area of Triangle in Terms of Two Sides and Angle, Euclidean Geometry, Area Formulas, Areas of Triangles \\begin{theorem} The area of a triangle $ABC$ is given by: :$\\displaystyle \\frac 1 2 a b \\sin C... | [
{
"doc_id": 30079,
"text": "['', \"\\\\begin{definition}[Definition:Obtuse Angle]\\nAn '''obtuse angle''' is an angle which has a measurement between those of a right angle and a straight angle.\\n{{EuclidDefinition|book = I|def = 11|name = Obtuse Angle}}\\nNote that Euclid did not consider the concept of r... | 88 |
Query: How can I summarize a set of numbers with one value? | [
{
"doc_id": 343,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a se... | [
{
"doc_id": 14594,
"text": "['', '‘Web pagination’ and limiting the maximum number of rows are detailed in this\\nsection. For each ‘page’ of results, a SQL query is executed to get the\\nappropriate set of rows from a table. Since the query may be executed more\\nthan once, ensure to use bind variables f... | 96 |
Query: When averaging a group of temperatures, what formula do I use? | [
{
"doc_id": 343,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a se... | [
{
"doc_id": 63363,
"text": "['', \"This attitude discrepancy can be observed by asking a group of participants\\nabout their personal attitudes and beliefs on a topic as well as asking them\\nfor the average group attitude or belief on the same topic. Next, the _actual_\\ngroup attitude is compiled by avera... | 96 |
Query: If I want to quickly estimate the typical value from a large dataset without detailed calculations, which method should I choose? | [
{
"doc_id": 343,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a se... | [
{
"doc_id": 10860,
"text": "['', 'Given all of the above, what happens if we need to do synchronous-intense\\nstuff, such as making complex calculations in memory in a large dataset? Then\\nwe might have a synchronous block of code that takes a lot of time and will\\nblock the rest of the code.']",
"tit... | 84 |
Query: Imagine I'm planning a party and I need to budget for food based on guest preferences. How do I determine the average amount each person will eat? | [
{
"doc_id": 343,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a se... | [
{
"doc_id": 77276,
"text": "['', '* While eating an apple, you recognize Apple as fruit and from your knowledge, can confer its importance.\\n* When listening to the birds chirping near the window, you straight away point out the bird to be the sparrow.\\n* The calculation of the month’s grocery budget thro... | 84 |
Query: How do I find a single value that represents the center of a dataset? | [
{
"doc_id": 343,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a se... | [
{
"doc_id": 6756,
"text": "['', \"The 3rd quartile of a dataset is a measure of central tendency that separates the top 25% of the data from the bottom 75%. Suppose we have a dataset of 20 numbers with a median of 14, and the 3rd quartile is 18. What is the interquartile range (IQR) of the dataset?\\nThe in... | 96 |
Query: I have a week's worth of hourly temperature readings. How can I summarize this data into one number? | [
{
"doc_id": 343,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a se... | [
{
"doc_id": 4742,
"text": "['', 'A medical research team collected data on the blood pressure readings of 50 patients at their clinic. The blood pressure readings were as follows:\\n\\n120, 135, 128, 150, 143, 132, 124, 136, 129, 141, 131, 142, 124, 136, 141, 130, 139, 133, 121, 144, 139, 132, 137, 126, 140... | 92 |
Query: What formula should I use when averaging a group of numbers, especially if some numbers might be zero or negative? | [
{
"doc_id": 343,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a se... | [
{
"doc_id": 79310,
"text": "['', 'Here is another example along similar lines. If my nephew, the one who is doing A-level maths, says that the product of\\n-0.00518\\n17,577,797\\n26,954.56\\n-1,699\\nis\\n-4,169,844,215,820,818\\nthen I am justified in rejecting his view. I do not know what the right answe... | 88 |
Query: If I take the average of a set of numbers, does it always represent the typical value in the data? | [
{
"doc_id": 343,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a se... | [
{
"doc_id": 37274,
"text": "['', \"Here's a numerical problem for the student to solve related to outlier detection in statistics:\\n\\nA set of 10 numbers has an average of 25.5. Upon reviewing the data, there appears to be one outlier. When the outlier is removed, the average of the remaining 9 numbers is... | 88 |
Query: Can you explain the process of calculating an average for a large dataset where each value is unique and there are many of them? | [
{
"doc_id": 343,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a se... | [
{
"doc_id": 5727,
"text": "['', \"Suppose a marketing company wants to know the satisfaction level of their customers towards their product. They conducted a survey with 20 customers and got their feedback on 5 different aspects of the product. Using factor analysis, how many factors can you extract from th... | 84 |
Query: How can I find a single number that represents the center of a group of numbers? | [
{
"doc_id": 343,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a se... | [
{
"doc_id": 6627,
"text": "['', \"How many five-digit numbers can be formed using the digits 1, 2, 3, 4, and 5 if the digits 2 and 4 must not appear in the same number?\\nWe will solve this problem using the principle of inclusion-exclusion.\\n\\nFirst, let's find the total number of five-digit numbers that... | 96 |
Query: I'm analyzing exam scores from a class. Can you give me a method to get one score that shows the class performance? | [
{
"doc_id": 343,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a se... | [
{
"doc_id": 5198,
"text": "['', 'Suppose you have a dataset of test scores for a class of 30 students. The mean score is 75 and the standard deviation is 10. After analyzing the dataset, you found that one of the scores is an outlier. If the outlier has a z-score of 2.5, what is the actual test score of the... | 96 |
Query: I have a list of daily temperatures over a month. How should I calculate a summary figure that captures the average warmth? | [
{
"doc_id": 343,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a se... | [
{
"doc_id": 37954,
"text": "['', \"Suppose that the average daily temperature and the daily ice cream sales for a community are recorded over a period of one month. After analyzing the data, it is found that there is a strong positive correlation between the two variables. The average daily temperature for ... | 92 |
Query: What if I only have two numbers? | [
{
"doc_id": 343,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a se... | [
{
"doc_id": 64418,
"text": "['', 'def count_odds(low, high):\\n \"\"\"A sequence of numbers is called **arithmetic** if it consists of at least two elements, and the difference between every two consecutive elements is the same. More formally, a sequence `s` is arithmetic if and only if `s[i+1] - s[i] ==... | 88 |
Query: I have hourly weather data for a week. What kind of math do I use to find a single value that reflects the typical weather? | [
{
"doc_id": 343,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a se... | [
{
"doc_id": 19089,
"text": "['', ' \\n # Finally, we compile it! \\n # This compiles it into a LangChain Runnable, \\n # meaning you can use it as you would any other runnable \\n app = workflow.compile() \\n \\n\\n### Use it! \\u200b\\n\\nWe can now use it! This now exposes the [ s... | 84 |
Query: If I have five test scores and want to know their average score, what formula should I use? | [
{
"doc_id": 343,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a se... | [
{
"doc_id": 5198,
"text": "['', 'Suppose you have a dataset of test scores for a class of 30 students. The mean score is 75 and the standard deviation is 10. After analyzing the dataset, you found that one of the scores is an outlier. If the outlier has a z-score of 2.5, what is the actual test score of the... | 96 |
Query: I have daily rainfall measurements for a month. How can I summarize this data with one value that captures the average amount of rain per day? | [
{
"doc_id": 343,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a se... | [
{
"doc_id": 37954,
"text": "['', \"Suppose that the average daily temperature and the daily ice cream sales for a community are recorded over a period of one month. After analyzing the data, it is found that there is a strong positive correlation between the two variables. The average daily temperature for ... | 92 |
Query: I've recorded temperatures hourly for a week. What's the process to get one number that summarizes the typical temperature for the week? | [
{
"doc_id": 343,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a se... | [
{
"doc_id": 5467,
"text": "['', 'Betty has a tray of cookies and a tray of brownies. She has a real sweet tooth and eats 3 cookies a day and 1 brownie a day. If she starts with 60 cookies and 10 brownies, how many more cookies than brownies does she have after a week of eating like this?\\nFirst find the ... | 92 |
Query: I'm trying to decide which single number best describes my investment portfolio's performance over the past year. What math operation should I perform on the monthly returns? | [
{
"doc_id": 343,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a se... | [
{
"doc_id": 8773,
"text": "['', \"Hi All, I'm trying to create a SDF from COLLADA files I have already. However,\\nthe files I'm trying to link to are located in different files (I'm trying to\\nload a sensor on the robot). In URDFs I could use the ($find package_name)\\nmethod to get a location. How would ... | 84 |
Query: How do I calculate a single value that summarizes a group of numbers? | [
{
"doc_id": 343,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a se... | [
{
"doc_id": 3359,
"text": "['', \"A researcher wants to compare the heights of two groups of people, Group A and Group B. However, the data collected does not follow a normal distribution. Calculate the median height of Group A and Group B, and use a non-parametric test to determine whether there is a signi... | 96 |
Query: I have a bunch of test scores. How can I find a single score that represents the class average? | [
{
"doc_id": 343,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a se... | [
{
"doc_id": 3623,
"text": "['', \"Suppose a math student wants to compare the test scores of two different classes, Class A and Class B. They collected 10 test scores from each class and computed the mean and median scores for each class. The results are as follows: \\n\\nClass A - Mean score = 85, Median s... | 96 |
Query: When averaging numbers, why is dividing by the count of numbers used important? | [
{
"doc_id": 343,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a se... | [
{
"doc_id": 15656,
"text": "['', '## Example of using the simulated numbers from rpois()\\n\\nLet’s explore the Poisson distribution a little more, seeing how the\\ndistribution changes when the mean of the distribution changes. Being able to\\nlook at how the Poisson distribution changes with the mean via... | 96 |
Query: If I have 5 different measurements and want to find their middle point, what math operation should I use? | [
{
"doc_id": 343,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a se... | [
{
"doc_id": 12751,
"text": "['', ' Arguments ¶\\n` _ percentile _ `\\nThe percentile of the value that you want to find. The percentile must be a\\nconstant between 0.0 and 1.0. For example, if you want to find the value at\\nthe 90th percentile, specify 0.9.\\n` _ order_by_expr _ `\\nThe expression (ty... | 84 |
Query: I'm analyzing hourly weather data over a week. How should I reduce it to a single metric that captures the general climate trend? | [
{
"doc_id": 343,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a se... | [
{
"doc_id": 20094,
"text": "['', 'It’s difficult to get good temperature estimates around the whole planet.\\nAccording to [ _NASA Climate_ ](https://climate.nasa.gov/ask-nasa-\\nclimate/3071/the-raw-truth-on-global-temperature-records/) , global\\ntemperature data comes from thousands of observation statio... | 84 |
Query: When measuring the length of a part of a circle, what is the relationship between the radius, the angle, and the length? | [
{
"doc_id": 17549,
"text": "\\section{Arc Length of Sector} Tags: Circles \\begin{theorem} Let $\\CC = ABC$ be a circle whose center is $A$ and with radii $AB$ and $AC$. Let $BAC$ be the sector of $\\CC$ whose angle between $AB$ and $AC$ is $\\theta$. :300px Then the length $s$ of arc $BC$ is given by: :$s... | [
{
"doc_id": 3550,
"text": "['', \"A circle with a radius of 2 cm is moving along a straight line that is tangent to its center, and its speed is 3 cm/s. What is the rate at which the angle between the radius vector and the tangent is changing when the circle has rolled 4 cm along the line?\\nLet's denote th... | 96 |
Query: Imagine I'm designing a track for a robotic car that follows a circular path for 90 degrees. If the radius of the path is 10 meters, how far will the car travel along the track? | [
{
"doc_id": 17549,
"text": "\\section{Arc Length of Sector} Tags: Circles \\begin{theorem} Let $\\CC = ABC$ be a circle whose center is $A$ and with radii $AB$ and $AC$. Let $BAC$ be the sector of $\\CC$ whose angle between $AB$ and $AC$ is $\\theta$. :300px Then the length $s$ of arc $BC$ is given by: :$s... | [
{
"doc_id": 5245,
"text": "['', \"A car traveling along a circular track of radius 100 meters has an average speed of 25 m/s over the first half of one lap. Using the mean value theorem, find the minimum speed of the car during that lap.\\nThe Mean Value Theorem states that if a function is continuous on a ... | 96 |
Query: How do I calculate the length of an arc when I only know the angle it spans and the radius? | [
{
"doc_id": 17549,
"text": "\\section{Arc Length of Sector} Tags: Circles \\begin{theorem} Let $\\CC = ABC$ be a circle whose center is $A$ and with radii $AB$ and $AC$. Let $BAC$ be the sector of $\\CC$ whose angle between $AB$ and $AC$ is $\\theta$. :300px Then the length $s$ of arc $BC$ is given by: :$s... | [
{
"doc_id": 5793,
"text": "['', \"A sector of a circle has a radius of 8cm and an angle of 60 degrees. Calculate the perimeter of the sector.\\nTo calculate the perimeter of the sector, we need to find the length of the arc and then add the two radii.\\n\\nFirst, let's find the length of the arc. The formul... | 96 |
Query: If I have a circular pizza slice and I know its central angle, how can I find its edge length? | [
{
"doc_id": 17549,
"text": "\\section{Arc Length of Sector} Tags: Circles \\begin{theorem} Let $\\CC = ABC$ be a circle whose center is $A$ and with radii $AB$ and $AC$. Let $BAC$ be the sector of $\\CC$ whose angle between $AB$ and $AC$ is $\\theta$. :300px Then the length $s$ of arc $BC$ is given by: :$s... | [
{
"doc_id": 36875,
"text": "['', \"A pizza has a diameter of 16 inches. If a slice of pizza is cut in the shape of a sector with a central angle of 45 degrees, what is the perimeter of the slice of pizza?\\nTo find the perimeter of the pizza slice, we need to find the length of the arc and the two radii.\\n... | 92 |
Query: What if the angle is given in degrees instead of radians? | [
{
"doc_id": 17549,
"text": "\\section{Arc Length of Sector} Tags: Circles \\begin{theorem} Let $\\CC = ABC$ be a circle whose center is $A$ and with radii $AB$ and $AC$. Let $BAC$ be the sector of $\\CC$ whose angle between $AB$ and $AC$ is $\\theta$. :300px Then the length $s$ of arc $BC$ is given by: :$s... | [
{
"doc_id": 1213,
"text": "\\section{Value of Radian in Degrees} Tags: Units of Measurement, Definitions: Geometry, Trigonometry, Definitions: Trigonometry, Angles, Definitions: Units of Measurement, Definitions: Angles \\begin{theorem} The value of a radian in degrees is given by: :$1 \\radians = \\dfrac ... | 88 |
Query: I'm trying to figure out how much ribbon I need to wrap around a circular gift box with a specific angle. Can you help me with the math? | [
{
"doc_id": 17549,
"text": "\\section{Arc Length of Sector} Tags: Circles \\begin{theorem} Let $\\CC = ABC$ be a circle whose center is $A$ and with radii $AB$ and $AC$. Let $BAC$ be the sector of $\\CC$ whose angle between $AB$ and $AC$ is $\\theta$. :300px Then the length $s$ of arc $BC$ is given by: :$s... | [
{
"doc_id": 10354,
"text": "['', \"I need to make a demo where I can see how the robot in Gazebo moves in the\\nsame way the robot in Unity. I'm using in this case a turtlebot3 burger. For\\nthe communication between ROS and Unity i'm using ROS-TCP-Connector and tcp\\nendpoint. I'm using: -Ubuntu 18.04.5 -W... | 88 |
Query: I have a circle's radius and center coordinates, how can I write its position using cos and sin? | [
{
"doc_id": 14440,
"text": "\\section{Equation of Circle/Parametric} Tags: Circles, Equation of Circle, Examples of Curves \\begin{theorem} The equation of a circle embedded in the Cartesian plane with radius $R$ and center $\\tuple {a, b}$ can be expressed as a parametric equation: :$\\begin {cases} x = a... | [
{
"doc_id": 437,
"text": "\\section{Equation of Circle/Polar} Tags: Circles, Equation of Circle, Examples of Curves \\begin{theorem} The equation of a circle with radius $R$ and center $\\polar {r_0, \\varphi}$ can be expressed in polar coordinates as: : $r^2 - 2 r r_0 \\map \\cos {\\theta - \\varphi} + \\... | 96 |
Query: If I know a circle's center is at (3,4) and it has a radius of 5, how can I find points on its edge using trigonometry? | [
{
"doc_id": 14440,
"text": "\\section{Equation of Circle/Parametric} Tags: Circles, Equation of Circle, Examples of Curves \\begin{theorem} The equation of a circle embedded in the Cartesian plane with radius $R$ and center $\\tuple {a, b}$ can be expressed as a parametric equation: :$\\begin {cases} x = a... | [
{
"doc_id": 2998,
"text": "['', \"Calculate the geodesics on the surface of a sphere with radius 3 units, passing through two polar opposite points on its surface.\\nTo find the geodesics on the surface of a sphere with radius 3 units, passing through two polar opposite points, we can use the concept of gre... | 96 |
Query: I'm tracking the movement of a robot in a circular path. How do I express its position over time if I know the center and the radius? | [
{
"doc_id": 14440,
"text": "\\section{Equation of Circle/Parametric} Tags: Circles, Equation of Circle, Examples of Curves \\begin{theorem} The equation of a circle embedded in the Cartesian plane with radius $R$ and center $\\tuple {a, b}$ can be expressed as a parametric equation: :$\\begin {cases} x = a... | [
{
"doc_id": 38059,
"text": "['', 'A metal rod of length 10 cm is bent to form a torus. If the radius of the circle forming the torus is 3 cm, find the volume of the torus.\\nTo find the volume of the torus, we need to know the major radius (R) and the minor radius (r). The major radius is the distance from ... | 96 |
Query: I have data points that form a circular pattern. Can you explain how to use cosine and sine functions to represent these points mathematically? | [
{
"doc_id": 14440,
"text": "\\section{Equation of Circle/Parametric} Tags: Circles, Equation of Circle, Examples of Curves \\begin{theorem} The equation of a circle embedded in the Cartesian plane with radius $R$ and center $\\tuple {a, b}$ can be expressed as a parametric equation: :$\\begin {cases} x = a... | [
{
"doc_id": 10397,
"text": "['', 'Can you explain here a bit more what are you doing? Are you sending the full\\ntrajectory through action client or individual points? What is the source of\\nthose points? How are they interpoalted? Is it possible that robot reaches the\\npoints and start to slow-down befor... | 96 |
Query: What if I only have points in the third quadrant of a unit circle and need to find the horizontal distance to the y-axis? | [
{
"doc_id": 1996,
"text": "\\begin{definition}[Definition:Cosine/Definition from Circle/Third Quadrant] Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P = \\tuple {x, y}$ be the point on $C$ in the third quadrant such that $\\theta$ is the angle made by $OP$ with ... | [
{
"doc_id": 3447,
"text": "['', 'Find the radian measure of the angle on the unit circle that corresponds to a cosine value of -1/2.\\nTo find the radian measure of the angle on the unit circle that corresponds to a cosine value of -1/2, we can use the properties of the unit circle and trigonometric functio... | 96 |
Query: If I'm working on a project that involves trigonometry and I only have data points in the third quadrant, how can I determine the cosine of the angle formed with the x-axis? | [
{
"doc_id": 1996,
"text": "\\begin{definition}[Definition:Cosine/Definition from Circle/Third Quadrant] Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P = \\tuple {x, y}$ be the point on $C$ in the third quadrant such that $\\theta$ is the angle made by $OP$ with ... | [
{
"doc_id": 37095,
"text": "['', 'Find the exact values of cosine, sine, and tangent of an angle whose reference angle is 45 degrees, and which lies in the third quadrant.\\nIn the third quadrant, both the x and y coordinates are negative. Since the reference angle is 45 degrees, we can use the properties o... | 96 |
Query: When plotting a function, why might the cosine of an angle in the third quadrant be negative? | [
{
"doc_id": 1996,
"text": "\\begin{definition}[Definition:Cosine/Definition from Circle/Third Quadrant] Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P = \\tuple {x, y}$ be the point on $C$ in the third quadrant such that $\\theta$ is the angle made by $OP$ with ... | [
{
"doc_id": 2817,
"text": "['', 'A math student needs to find the value of arccos(-0.5) in radians.\\nTo find the value of arccos(-0.5) in radians, we need to find the angle whose cosine is -0.5. \\n\\nRecall that the cosine function is positive in the first and fourth quadrants and negative in the second a... | 92 |
Query: I'm trying to calculate the horizontal component of a force applied at an angle in the third quadrant. What's the formula for that? | [
{
"doc_id": 1996,
"text": "\\begin{definition}[Definition:Cosine/Definition from Circle/Third Quadrant] Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P = \\tuple {x, y}$ be the point on $C$ in the third quadrant such that $\\theta$ is the angle made by $OP$ with ... | [
{
"doc_id": 5295,
"text": "['', \"A crate is pushed with a force of 100 Newtons across a warehouse floor, covering a distance of 20 meters. The force is applied at an angle of 30 degrees to the horizontal. Find the amount of work done in moving the crate using integration.\\nTo find the work done in moving ... | 92 |
Query: Can you give me a step-by-step guide on how to measure the horizontal distance from a point in the third quadrant to the y-axis of a unit circle? | [
{
"doc_id": 1996,
"text": "\\begin{definition}[Definition:Cosine/Definition from Circle/Third Quadrant] Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P = \\tuple {x, y}$ be the point on $C$ in the third quadrant such that $\\theta$ is the angle made by $OP$ with ... | [
{
"doc_id": 3447,
"text": "['', 'Find the radian measure of the angle on the unit circle that corresponds to a cosine value of -1/2.\\nTo find the radian measure of the angle on the unit circle that corresponds to a cosine value of -1/2, we can use the properties of the unit circle and trigonometric functio... | 92 |
Query: How do I calculate the cosine of a double angle using only the cosine of the original angle? | [
{
"doc_id": 14784,
"text": "\\section{Double Angle Formulas/Cosine/Corollary 1} Tags: Double Angle Formula for Cosine, Cosine Function \\begin{theorem} : $\\cos \\left({2 \\theta}\\right) = 2 \\ \\cos^2 \\theta - 1$ where $\\cos$ denotes cosine. \\end{theorem} \\begin{proof} {{begin-eqn}} {{eqn | l = \\co... | [
{
"doc_id": 6285,
"text": "['', 'What is the cosine value of the angle 30 degrees using the unit circle in trigonometry?\\nTo find the cosine value of a 30-degree angle using the unit circle, we can use the fact that a 30-degree angle is equivalent to π/6 radians. In the unit circle, the cosine value repres... | 96 |
Query: If I know the cosine of an angle, can I directly find the cosine of twice that angle without calculating the sine first? | [
{
"doc_id": 14784,
"text": "\\section{Double Angle Formulas/Cosine/Corollary 1} Tags: Double Angle Formula for Cosine, Cosine Function \\begin{theorem} : $\\cos \\left({2 \\theta}\\right) = 2 \\ \\cos^2 \\theta - 1$ where $\\cos$ denotes cosine. \\end{theorem} \\begin{proof} {{begin-eqn}} {{eqn | l = \\co... | [
{
"doc_id": 5766,
"text": "['', 'What is the sine and cosine of the angle 120 degrees when measured in the standard position using the unit circle?\\nTo find the sine and cosine of 120 degrees, we first need to convert the angle to radians. To do this, we use the conversion factor: 1 radian = 180/π degrees.... | 96 |
Query: When I'm trying to figure out the cosine of a doubled angle based solely on the cosine of the initial angle, what steps should I follow? | [
{
"doc_id": 14784,
"text": "\\section{Double Angle Formulas/Cosine/Corollary 1} Tags: Double Angle Formula for Cosine, Cosine Function \\begin{theorem} : $\\cos \\left({2 \\theta}\\right) = 2 \\ \\cos^2 \\theta - 1$ where $\\cos$ denotes cosine. \\end{theorem} \\begin{proof} {{begin-eqn}} {{eqn | l = \\co... | [
{
"doc_id": 5766,
"text": "['', 'What is the sine and cosine of the angle 120 degrees when measured in the standard position using the unit circle?\\nTo find the sine and cosine of 120 degrees, we first need to convert the angle to radians. To do this, we use the conversion factor: 1 radian = 180/π degrees.... | 96 |
Query: How do I simplify the expression cos(2θ) to only involve cos(θ)? | [
{
"doc_id": 14784,
"text": "\\section{Double Angle Formulas/Cosine/Corollary 1} Tags: Double Angle Formula for Cosine, Cosine Function \\begin{theorem} : $\\cos \\left({2 \\theta}\\right) = 2 \\ \\cos^2 \\theta - 1$ where $\\cos$ denotes cosine. \\end{theorem} \\begin{proof} {{begin-eqn}} {{eqn | l = \\co... | [
{
"doc_id": 3575,
"text": "['', \"A student wants to know the distance between two trees in a park. She uses a clinometer to measure the angle of elevation from her position to the top of each tree. The angle of elevation to the first tree is 35 degrees, and the angle of elevation to the second tree is 48 d... | 96 |
Query: I'm measuring angles in a right triangle. If I double the angle, how can I calculate the new cosine without knowing the sine? | [
{
"doc_id": 14784,
"text": "\\section{Double Angle Formulas/Cosine/Corollary 1} Tags: Double Angle Formula for Cosine, Cosine Function \\begin{theorem} : $\\cos \\left({2 \\theta}\\right) = 2 \\ \\cos^2 \\theta - 1$ where $\\cos$ denotes cosine. \\end{theorem} \\begin{proof} {{begin-eqn}} {{eqn | l = \\co... | [
{
"doc_id": 3446,
"text": "['', 'A right triangle has a hypotenuse of length 15 cm and one acute angle measuring 30 degrees. Calculate the area of the triangle using trigonometry. Round your answer to the nearest hundredth.\\nTo find the area of the triangle, we first need to find the lengths of the other t... | 96 |
Query: When doubling an angle, why does the formula 2*cos²(θ)-1 work for finding the new cosine value? | [
{
"doc_id": 14784,
"text": "\\section{Double Angle Formulas/Cosine/Corollary 1} Tags: Double Angle Formula for Cosine, Cosine Function \\begin{theorem} : $\\cos \\left({2 \\theta}\\right) = 2 \\ \\cos^2 \\theta - 1$ where $\\cos$ denotes cosine. \\end{theorem} \\begin{proof} {{begin-eqn}} {{eqn | l = \\co... | [
{
"doc_id": 5439,
"text": "['', 'A math student needs to find the value of the inverse cosine of 0.5 (arccos(0.5)), using the concept of Finding the inverse trigonometric functions in Trigonometry.\\nTo find the value of the inverse cosine of 0.5 (arccos(0.5)), we need to find the angle whose cosine is 0.5.... | 96 |
Query: If I know the cosine of an angle θ, how can I find the cosine of twice that angle using only the cosine information? | [
{
"doc_id": 14784,
"text": "\\section{Double Angle Formulas/Cosine/Corollary 1} Tags: Double Angle Formula for Cosine, Cosine Function \\begin{theorem} : $\\cos \\left({2 \\theta}\\right) = 2 \\ \\cos^2 \\theta - 1$ where $\\cos$ denotes cosine. \\end{theorem} \\begin{proof} {{begin-eqn}} {{eqn | l = \\co... | [
{
"doc_id": 6552,
"text": "['', \"Sure, here's a question:\\n\\nTwo vectors A and B have magnitudes of 3 and 5, respectively. The angle between them is 60 degrees. Using trigonometry, find the cosine of the angle between A and B.\\nTo find the cosine of the angle between two vectors A and B, we can use the ... | 96 |
Query: When plotting a point in the second quadrant on a unit circle, why is the x-coordinate always negative? | [
{
"doc_id": 1995,
"text": "\\begin{definition}[Definition:Cosine/Definition from Circle/Second Quadrant] Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P = \\tuple {x, y}$ be the point on $C$ in the second quadrant such that $\\theta$ is the angle made by $OP$ wit... | [
{
"doc_id": 2817,
"text": "['', 'A math student needs to find the value of arccos(-0.5) in radians.\\nTo find the value of arccos(-0.5) in radians, we need to find the angle whose cosine is -0.5. \\n\\nRecall that the cosine function is positive in the first and fourth quadrants and negative in the second a... | 92 |
Query: Why is the cosine of an angle in the second quadrant always negative? | [
{
"doc_id": 1995,
"text": "\\begin{definition}[Definition:Cosine/Definition from Circle/Second Quadrant] Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P = \\tuple {x, y}$ be the point on $C$ in the second quadrant such that $\\theta$ is the angle made by $OP$ wit... | [
{
"doc_id": 3447,
"text": "['', 'Find the radian measure of the angle on the unit circle that corresponds to a cosine value of -1/2.\\nTo find the radian measure of the angle on the unit circle that corresponds to a cosine value of -1/2, we can use the properties of the unit circle and trigonometric functio... | 96 |
Query: If you have a point P in the second quadrant of a unit circle, how do you calculate its x-coordinate without directly measuring it? | [
{
"doc_id": 1995,
"text": "\\begin{definition}[Definition:Cosine/Definition from Circle/Second Quadrant] Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P = \\tuple {x, y}$ be the point on $C$ in the second quadrant such that $\\theta$ is the angle made by $OP$ wit... | [
{
"doc_id": 3551,
"text": "['', \"Find the values of sine, cosine, and tangent of an angle of 135 degrees by using the unit circle.\\nTo find the sine, cosine, and tangent of a 135-degree angle using the unit circle, we first need to convert the angle to radians. \\n\\n135 degrees * (π / 180) = 3π / 4 radia... | 92 |
Query: If I have a point on a unit circle in the second quadrant, how can I determine its x-coordinate by only knowing the angle formed with the positive x-axis? | [
{
"doc_id": 1995,
"text": "\\begin{definition}[Definition:Cosine/Definition from Circle/Second Quadrant] Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P = \\tuple {x, y}$ be the point on $C$ in the second quadrant such that $\\theta$ is the angle made by $OP$ wit... | [
{
"doc_id": 3551,
"text": "['', \"Find the values of sine, cosine, and tangent of an angle of 135 degrees by using the unit circle.\\nTo find the sine, cosine, and tangent of a 135-degree angle using the unit circle, we first need to convert the angle to radians. \\n\\n135 degrees * (π / 180) = 3π / 4 radia... | 92 |
Query: What is the formula for calculating the cosine of an angle when it forms a certain position on the Cartesian plane? | [
{
"doc_id": 1995,
"text": "\\begin{definition}[Definition:Cosine/Definition from Circle/Second Quadrant] Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P = \\tuple {x, y}$ be the point on $C$ in the second quadrant such that $\\theta$ is the angle made by $OP$ wit... | [
{
"doc_id": 445,
"text": "\\section{Tangent is Sine divided by Cosine} Tags: Trigonometry, Sine Function, Tangent Function, Cosine Function \\begin{theorem} Let $\\theta$ be an angle such that $\\cos \\theta \\ne 0$. Then: :$\\tan \\theta = \\dfrac {\\sin \\theta} {\\cos \\theta}$ where $\\tan$, $\\sin$ an... | 88 |
Query: How do I calculate the leftover value after dividing a polynomial by (x-a)? | [
{
"doc_id": 11032,
"text": "\\section{Little Bézout Theorem} Tags: Named Theorems, Algebra, Polynomial Theory \\begin{theorem} Let $\\map {P_n} x$ be a polynomial of degree $n$ in $x$. Let $a$ be a constant. Then the remainder of $\\map {P_n} x$ when divided by $x - a$ is equal to $\\map {P_n} a$. \\end{th... | [
{
"doc_id": 33520,
"text": "['', \"\\\\begin{definition}[Definition:Symmetric Function/Elementary]\\nLet $a, b \\\\in \\\\Z$ be integers such that $b \\\\ge a$.\\nLet $U$ be a set of $n = b - a + 1$ numbers $\\\\set {x_a, x_{a + 1}, \\\\ldots, x_b}$.\\nLet $m \\\\in \\\\Z_{>0}$ be a (strictly) positive inte... | 96 |
Query: Can you explain what it means when a polynomial's value at a specific point equals its remainder when divided by (x-a)? | [
{
"doc_id": 11032,
"text": "\\section{Little Bézout Theorem} Tags: Named Theorems, Algebra, Polynomial Theory \\begin{theorem} Let $\\map {P_n} x$ be a polynomial of degree $n$ in $x$. Let $a$ be a constant. Then the remainder of $\\map {P_n} x$ when divided by $x - a$ is equal to $\\map {P_n} a$. \\end{th... | [
{
"doc_id": 37222,
"text": "['', \"When positive integer k is divided by 5, the remainder is 2. When k is divided by 6, the remainder is 5. If k is less than 38, what is the remainder when k is divided by 7?\\nAnswer Choices: (A) 2 (B) 4 (C) 3 (D) 5 (E) 6\\nLet's reason about the multi-choice question.\\nCa... | 96 |
Query: What happens if you divide a polynomial by (x-a) when 'a' is a root of the polynomial? | [
{
"doc_id": 11032,
"text": "\\section{Little Bézout Theorem} Tags: Named Theorems, Algebra, Polynomial Theory \\begin{theorem} Let $\\map {P_n} x$ be a polynomial of degree $n$ in $x$. Let $a$ be a constant. Then the remainder of $\\map {P_n} x$ when divided by $x - a$ is equal to $\\map {P_n} a$. \\end{th... | [
{
"doc_id": 8,
"text": "\\section{Condition for Linear Divisor of Polynomial} Tags: Polynomial Theory \\begin{theorem} Let $\\map P x$ be a polynomial in $x$. Let $a$ be a constant. Then $x - a$ is a divisor of $\\map P x$ {{iff}} $a$ is a root of $P$. \\end{theorem} \\begin{proof} From the Little Bézout ... | 96 |
Query: I have a polynomial function and I want to quickly estimate its value without substituting x. Is there an easier way to find the remainder when dividing by (x-a)? | [
{
"doc_id": 11032,
"text": "\\section{Little Bézout Theorem} Tags: Named Theorems, Algebra, Polynomial Theory \\begin{theorem} Let $\\map {P_n} x$ be a polynomial of degree $n$ in $x$. Let $a$ be a constant. Then the remainder of $\\map {P_n} x$ when divided by $x - a$ is equal to $\\map {P_n} a$. \\end{th... | [
{
"doc_id": 38194,
"text": "['', 'Suppose you want to use RSA to encrypt a message where the plaintext is the number 42. The public key has the values n=323 and e=5. What is the ciphertext?\\nTo encrypt a message using RSA, you need to perform the following calculation:\\n\\nciphertext = (plaintext^e) mod n... | 96 |
Query: How can I express the greatest common factor of two numbers as a combination of those numbers? | [
{
"doc_id": 17074,
"text": "\\section{Bézout's Lemma} Tags: Number Theory, Bézout's Lemma, Greatest Common Divisor \\begin{theorem} Let $a, b \\in \\Z$ such that $a$ and $b$ are not both zero. Let $\\gcd \\set {a, b}$ be the greatest common divisor of $a$ and $b$. Then: :$\\exists x, y \\in \\Z: a x + b y ... | [
{
"doc_id": 4983,
"text": "['', \"If the GCD of two numbers (not equal, both integers, greater than 1) is 2, then which of the following can be true?\\nA. They are prime.\\nB. They are consecutive.\\nC. They do not have a common prime factor.\\nD. They have a common factor other than 2.\\nAnswer Choices: (A... | 96 |
Query: What method can I use to prove that any common divisor of two numbers must also divide their linear combination? | [
{
"doc_id": 17074,
"text": "\\section{Bézout's Lemma} Tags: Number Theory, Bézout's Lemma, Greatest Common Divisor \\begin{theorem} Let $a, b \\in \\Z$ such that $a$ and $b$ are not both zero. Let $\\gcd \\set {a, b}$ be the greatest common divisor of $a$ and $b$. Then: :$\\exists x, y \\in \\Z: a x + b y ... | [
{
"doc_id": 222,
"text": "\\section{Elements of Euclidean Domain have Greatest Common Divisor} Tags: Euclidean Domains, Ideal Theory \\begin{theorem} Let $\\struct {D, +, \\times}$ be a Euclidean domain. Then any two elements $a, b \\in D$ have a greatest common divisor $d$ such that: :$d \\divides a \\lan... | 96 |
Query: When dividing a polynomial by (x-a), how do I find the remainder? | [
{
"doc_id": 11032,
"text": "\\section{Little Bézout Theorem} Tags: Named Theorems, Algebra, Polynomial Theory \\begin{theorem} Let $\\map {P_n} x$ be a polynomial of degree $n$ in $x$. Let $a$ be a constant. Then the remainder of $\\map {P_n} x$ when divided by $x - a$ is equal to $\\map {P_n} a$. \\end{th... | [
{
"doc_id": 8,
"text": "\\section{Condition for Linear Divisor of Polynomial} Tags: Polynomial Theory \\begin{theorem} Let $\\map P x$ be a polynomial in $x$. Let $a$ be a constant. Then $x - a$ is a divisor of $\\map P x$ {{iff}} $a$ is a root of $P$. \\end{theorem} \\begin{proof} From the Little Bézout ... | 96 |
Query: How can I determine the value of a polynomial at a specific point using division? | [
{
"doc_id": 11032,
"text": "\\section{Little Bézout Theorem} Tags: Named Theorems, Algebra, Polynomial Theory \\begin{theorem} Let $\\map {P_n} x$ be a polynomial of degree $n$ in $x$. Let $a$ be a constant. Then the remainder of $\\map {P_n} x$ when divided by $x - a$ is equal to $\\map {P_n} a$. \\end{th... | [
{
"doc_id": 4810,
"text": "['', 'Consider a graph with 7 vertices. If the vertices can be colored with 3 different colors (red, blue, and green) such that no two adjacent vertices have the same color, how many distinct colorings are possible?\\nThis problem can be solved using the chromatic polynomial. The ... | 96 |
Query: If I divide a polynomial by (x-a) and set x=a, what will the result be? | [
{
"doc_id": 11032,
"text": "\\section{Little Bézout Theorem} Tags: Named Theorems, Algebra, Polynomial Theory \\begin{theorem} Let $\\map {P_n} x$ be a polynomial of degree $n$ in $x$. Let $a$ be a constant. Then the remainder of $\\map {P_n} x$ when divided by $x - a$ is equal to $\\map {P_n} a$. \\end{th... | [
{
"doc_id": 33520,
"text": "['', \"\\\\begin{definition}[Definition:Symmetric Function/Elementary]\\nLet $a, b \\\\in \\\\Z$ be integers such that $b \\\\ge a$.\\nLet $U$ be a set of $n = b - a + 1$ numbers $\\\\set {x_a, x_{a + 1}, \\\\ldots, x_b}$.\\nLet $m \\\\in \\\\Z_{>0}$ be a (strictly) positive inte... | 96 |
Query: Given a polynomial P(x) and a constant a, how do you determine the exact value of P(a)? | [
{
"doc_id": 11032,
"text": "\\section{Little Bézout Theorem} Tags: Named Theorems, Algebra, Polynomial Theory \\begin{theorem} Let $\\map {P_n} x$ be a polynomial of degree $n$ in $x$. Let $a$ be a constant. Then the remainder of $\\map {P_n} x$ when divided by $x - a$ is equal to $\\map {P_n} a$. \\end{th... | [
{
"doc_id": 4810,
"text": "['', 'Consider a graph with 7 vertices. If the vertices can be colored with 3 different colors (red, blue, and green) such that no two adjacent vertices have the same color, how many distinct colorings are possible?\\nThis problem can be solved using the chromatic polynomial. The ... | 96 |
Query: How do I find the remainder of a polynomial when divided by a linear factor? | [
{
"doc_id": 11032,
"text": "\\section{Little Bézout Theorem} Tags: Named Theorems, Algebra, Polynomial Theory \\begin{theorem} Let $\\map {P_n} x$ be a polynomial of degree $n$ in $x$. Let $a$ be a constant. Then the remainder of $\\map {P_n} x$ when divided by $x - a$ is equal to $\\map {P_n} a$. \\end{th... | [
{
"doc_id": 31531,
"text": "['', '\\\\section{Condition for Linear Divisor of Polynomial}\\nTags: Polynomial Theory\\n\\n\\\\begin{theorem}\\nLet $\\\\map P x$ be a polynomial in $x$.\\nLet $a$ be a constant.\\nThen $x - a$ is a divisor of $\\\\map P x$ {{iff}} $a$ is a root of $P$.\\n\\\\end{theorem}\\n\\n... | 96 |
Query: When dividing a polynomial by (x-a), what value does it leave as the remainder? | [
{
"doc_id": 11032,
"text": "\\section{Little Bézout Theorem} Tags: Named Theorems, Algebra, Polynomial Theory \\begin{theorem} Let $\\map {P_n} x$ be a polynomial of degree $n$ in $x$. Let $a$ be a constant. Then the remainder of $\\map {P_n} x$ when divided by $x - a$ is equal to $\\map {P_n} a$. \\end{th... | [
{
"doc_id": 5891,
"text": "['', \"x3−ax2+3x−b=0x3−ax2+3x−b=0 has one factor as (x - 2). When the equation is divided by (x + 1), it leaves a remainder -12. What are the values of ‘a’ and ‘b’?\\nAnswer Choices: (A) 2,2 (B) 2,6 (C) 2,1 (D) 2,9 (E) 2,7\\nLet's think about the multi-choice question.\\nx3−ax2+3x... | 96 |
Query: Imagine I have a polynomial function representing sales over time and I want to predict the exact sales value at a particular day. How would I use polynomial division to find this out? | [
{
"doc_id": 11032,
"text": "\\section{Little Bézout Theorem} Tags: Named Theorems, Algebra, Polynomial Theory \\begin{theorem} Let $\\map {P_n} x$ be a polynomial of degree $n$ in $x$. Let $a$ be a constant. Then the remainder of $\\map {P_n} x$ when divided by $x - a$ is equal to $\\map {P_n} a$. \\end{th... | [
{
"doc_id": 4957,
"text": "['', \"A graph has 7 vertices and 9 edges. How many possible ways are there for this graph to be colored using 4 colors such that no two adjacent vertices have the same color?\\nTo solve this problem, we can use the chromatic polynomial. The chromatic polynomial P(G, k) of a graph... | 96 |
Query: If I have a polynomial and I want to quickly find its value at a specific point without using complex calculations, how can I do it using the division method? | [
{
"doc_id": 11032,
"text": "\\section{Little Bézout Theorem} Tags: Named Theorems, Algebra, Polynomial Theory \\begin{theorem} Let $\\map {P_n} x$ be a polynomial of degree $n$ in $x$. Let $a$ be a constant. Then the remainder of $\\map {P_n} x$ when divided by $x - a$ is equal to $\\map {P_n} a$. \\end{th... | [
{
"doc_id": 4957,
"text": "['', \"A graph has 7 vertices and 9 edges. How many possible ways are there for this graph to be colored using 4 colors such that no two adjacent vertices have the same color?\\nTo solve this problem, we can use the chromatic polynomial. The chromatic polynomial P(G, k) of a graph... | 88 |
Query: How do I determine if a specific number makes a polynomial zero? | [
{
"doc_id": 15965,
"text": "\\section{Condition for Linear Divisor of Polynomial} Tags: Polynomial Theory \\begin{theorem} Let $\\map P x$ be a polynomial in $x$. Let $a$ be a constant. Then $x - a$ is a divisor of $\\map P x$ {{iff}} $a$ is a root of $P$. \\end{theorem} \\begin{proof} From the Little Béz... | [
{
"doc_id": 4810,
"text": "['', 'Consider a graph with 7 vertices. If the vertices can be colored with 3 different colors (red, blue, and green) such that no two adjacent vertices have the same color, how many distinct colorings are possible?\\nThis problem can be solved using the chromatic polynomial. The ... | 96 |
Query: If a linear factor divides a polynomial evenly, what must be true about the input value? | [
{
"doc_id": 15965,
"text": "\\section{Condition for Linear Divisor of Polynomial} Tags: Polynomial Theory \\begin{theorem} Let $\\map P x$ be a polynomial in $x$. Let $a$ be a constant. Then $x - a$ is a divisor of $\\map P x$ {{iff}} $a$ is a root of $P$. \\end{theorem} \\begin{proof} From the Little Béz... | [
{
"doc_id": 37536,
"text": "['', \"Let's start with a problem on linear logic. Consider the following formulas:\\n\\nA & B, A -> B, A ⊸ B\\n\\nFor each of these formulas, suppose we know that A is true. Determine which of the following statements about B must be true and which must be false, and explain you... | 96 |
Query: Given a polynomial and a number, what test can I perform to see if the number is a root of the polynomial? | [
{
"doc_id": 15965,
"text": "\\section{Condition for Linear Divisor of Polynomial} Tags: Polynomial Theory \\begin{theorem} Let $\\map P x$ be a polynomial in $x$. Let $a$ be a constant. Then $x - a$ is a divisor of $\\map P x$ {{iff}} $a$ is a root of $P$. \\end{theorem} \\begin{proof} From the Little Béz... | [
{
"doc_id": 32797,
"text": "['', '\\\\section{Rational Number is Algebraic of Degree 1}\\nTags: Rational Numbers, Algebraic Numbers\\n\\n\\\\begin{theorem}\\nLet $r \\\\in \\\\Q$ be a rational number.\\nThen $r$ is an algebraic number of degree $1$.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nLet $r$ be expr... | 96 |
Query: Can I use a polynomial to find the value at which it equals zero, given a certain input? | [
{
"doc_id": 15965,
"text": "\\section{Condition for Linear Divisor of Polynomial} Tags: Polynomial Theory \\begin{theorem} Let $\\map P x$ be a polynomial in $x$. Let $a$ be a constant. Then $x - a$ is a divisor of $\\map P x$ {{iff}} $a$ is a root of $P$. \\end{theorem} \\begin{proof} From the Little Béz... | [
{
"doc_id": 3841,
"text": "['', \"A math student needs to find the value of 'y' for a given 'x' using interpolation method with the help of following data points:\\n\\n|x|1|2|3|4|\\n|--|-|-|-|-|\\n|y|3|2|1|4|\\n\\nUsing the Lagrange polynomial method, what is the value of 'y' when 'x' is equal to 2.5?\\nTo ... | 88 |
Query: If a polynomial equals zero at a specific value, how can it be factored? | [
{
"doc_id": 18781,
"text": "\\section{Polynomial Factor Theorem/Corollary} Tags: Named Theorems, Polynomial Theory \\begin{theorem} Let $P \\left({x}\\right)$ be a polynomial in $x$ over the real numbers $\\R$ of degree $n$. Suppose there exists $\\xi \\in \\R: P \\left({\\xi}\\right) = 0$. Then $P \\left(... | [
{
"doc_id": 58,
"text": "\\section{Sum of Roots of Polynomial} Tags: Polynomial Equations, Sum of Roots of Polynomial \\begin{theorem} Let $P$ be the polynomial equation: : $a_n z^n + a_{n - 1} z^{n - 1} + \\cdots + a_1 z + a_0 = 0$ such that $a_n \\ne 0$. The sum of the roots of $P$ is $-\\dfrac {a_{n - 1... | 96 |
Query: Given a polynomial that has multiple roots, how do you express it as a product of factors? | [
{
"doc_id": 18781,
"text": "\\section{Polynomial Factor Theorem/Corollary} Tags: Named Theorems, Polynomial Theory \\begin{theorem} Let $P \\left({x}\\right)$ be a polynomial in $x$ over the real numbers $\\R$ of degree $n$. Suppose there exists $\\xi \\in \\R: P \\left({\\xi}\\right) = 0$. Then $P \\left(... | [
{
"doc_id": 1023,
"text": "\\begin{definition}[Definition:Cyclotomic Polynomial] Let $n \\ge 1$ be a natural number. The '''$n$th cyclotomic polynomial''' is the polynomial :$\\ds \\map {\\Phi_n} x = \\prod_\\zeta \\paren {x - \\zeta}$ where the product runs over all primitive complex $n$th roots of unity, ... | 96 |
Query: How can you prove that a polynomial of degree n can be written as a product of linear factors if it has n distinct roots? | [
{
"doc_id": 18781,
"text": "\\section{Polynomial Factor Theorem/Corollary} Tags: Named Theorems, Polynomial Theory \\begin{theorem} Let $P \\left({x}\\right)$ be a polynomial in $x$ over the real numbers $\\R$ of degree $n$. Suppose there exists $\\xi \\in \\R: P \\left({\\xi}\\right) = 0$. Then $P \\left(... | [
{
"doc_id": 33895,
"text": "['', \"\\\\section{Polynomial over Field has Finitely Many Roots}\\nTags: Polynomial Theory\\n\\n\\\\begin{theorem}\\nLet $F$ be a field.\\nLet $F \\\\left[{x}\\\\right]$ be the ring of polynomial functions in the indeterminate $x$.\\nIf $p \\\\in F \\\\left[{x}\\\\right]$ be non... | 96 |
Query: I have a polynomial equation and I know its roots. How can I write the polynomial in factored form without knowing the exact coefficients? | [
{
"doc_id": 18781,
"text": "\\section{Polynomial Factor Theorem/Corollary} Tags: Named Theorems, Polynomial Theory \\begin{theorem} Let $P \\left({x}\\right)$ be a polynomial in $x$ over the real numbers $\\R$ of degree $n$. Suppose there exists $\\xi \\in \\R: P \\left({\\xi}\\right) = 0$. Then $P \\left(... | [
{
"doc_id": 58,
"text": "\\section{Sum of Roots of Polynomial} Tags: Polynomial Equations, Sum of Roots of Polynomial \\begin{theorem} Let $P$ be the polynomial equation: : $a_n z^n + a_{n - 1} z^{n - 1} + \\cdots + a_1 z + a_0 = 0$ such that $a_n \\ne 0$. The sum of the roots of $P$ is $-\\dfrac {a_{n - 1... | 96 |
Query: Can you explain how to factor a polynomial when you know all its roots, but the polynomial itself is not given explicitly? | [
{
"doc_id": 18781,
"text": "\\section{Polynomial Factor Theorem/Corollary} Tags: Named Theorems, Polynomial Theory \\begin{theorem} Let $P \\left({x}\\right)$ be a polynomial in $x$ over the real numbers $\\R$ of degree $n$. Suppose there exists $\\xi \\in \\R: P \\left({\\xi}\\right) = 0$. Then $P \\left(... | [
{
"doc_id": 33895,
"text": "['', \"\\\\section{Polynomial over Field has Finitely Many Roots}\\nTags: Polynomial Theory\\n\\n\\\\begin{theorem}\\nLet $F$ be a field.\\nLet $F \\\\left[{x}\\\\right]$ be the ring of polynomial functions in the indeterminate $x$.\\nIf $p \\\\in F \\\\left[{x}\\\\right]$ be non... | 96 |
Query: When dividing a polynomial by a linear factor, under what condition does the result have no remainder? | [
{
"doc_id": 15965,
"text": "\\section{Condition for Linear Divisor of Polynomial} Tags: Polynomial Theory \\begin{theorem} Let $\\map P x$ be a polynomial in $x$. Let $a$ be a constant. Then $x - a$ is a divisor of $\\map P x$ {{iff}} $a$ is a root of $P$. \\end{theorem} \\begin{proof} From the Little Béz... | [
{
"doc_id": 5891,
"text": "['', \"x3−ax2+3x−b=0x3−ax2+3x−b=0 has one factor as (x - 2). When the equation is divided by (x + 1), it leaves a remainder -12. What are the values of ‘a’ and ‘b’?\\nAnswer Choices: (A) 2,2 (B) 2,6 (C) 2,1 (D) 2,9 (E) 2,7\\nLet's think about the multi-choice question.\\nx3−ax2+3x... | 96 |
Query: How can I determine if a certain number makes a polynomial equal to zero? | [
{
"doc_id": 15965,
"text": "\\section{Condition for Linear Divisor of Polynomial} Tags: Polynomial Theory \\begin{theorem} Let $\\map P x$ be a polynomial in $x$. Let $a$ be a constant. Then $x - a$ is a divisor of $\\map P x$ {{iff}} $a$ is a root of $P$. \\end{theorem} \\begin{proof} From the Little Béz... | [
{
"doc_id": 4810,
"text": "['', 'Consider a graph with 7 vertices. If the vertices can be colored with 3 different colors (red, blue, and green) such that no two adjacent vertices have the same color, how many distinct colorings are possible?\\nThis problem can be solved using the chromatic polynomial. The ... | 92 |
Query: Suppose I have two integers, and I need to determine the largest number that can divide both without leaving a remainder. After finding this, how can I represent this number as a sum of the original integers multiplied by some coefficients? | [
{
"doc_id": 17074,
"text": "\\section{Bézout's Lemma} Tags: Number Theory, Bézout's Lemma, Greatest Common Divisor \\begin{theorem} Let $a, b \\in \\Z$ such that $a$ and $b$ are not both zero. Let $\\gcd \\set {a, b}$ be the greatest common divisor of $a$ and $b$. Then: :$\\exists x, y \\in \\Z: a x + b y ... | [
{
"doc_id": 1504,
"text": "\\section{Largest Number not Expressible as Sum of Multiples of Coprime Integers} Tags: Largest Number not Expressible as Sum of Multiples of Coprime Integers, Integer Combinations \\begin{theorem} Let $a, b$ be coprime integers, each greater than $1$. Then the largest number not... | 96 |
Query: I'm trying to figure out a way to summarize the common factors of two numbers in a unique way. Specifically, I want to know if I can write the largest common factor as a combination of the two original numbers. How would I go about doing this? | [
{
"doc_id": 17074,
"text": "\\section{Bézout's Lemma} Tags: Number Theory, Bézout's Lemma, Greatest Common Divisor \\begin{theorem} Let $a, b \\in \\Z$ such that $a$ and $b$ are not both zero. Let $\\gcd \\set {a, b}$ be the greatest common divisor of $a$ and $b$. Then: :$\\exists x, y \\in \\Z: a x + b y ... | [
{
"doc_id": 3073,
"text": "['', \"The function f is defined for all positive integers x by the following rule. f(x) is the number of positive integers each of which is less than x and has no positive factor in common with x other than 1. If p is any prime, number then f(p)=\\nAnswer Choices: (A) 2 (B) p-2 (... | 96 |
Query: What is a unique way to represent any number greater than 1 using only prime numbers? | [
{
"doc_id": 13145,
"text": "\\section{Fundamental Theorem of Arithmetic} Tags: Number Theory, Factorization, Fundamental Theorems, Prime Decompositions, Named Theorems, Prime Numbers \\begin{theorem} For every integer $n$ such that $n > 1$, $n$ can be expressed as the product of one or more primes, uniquel... | [
{
"doc_id": 2946,
"text": "['', 'Let\\'s consider the following statement: \"Every natural number greater than 1 is either a prime number or has a unique factorization into primes.\" Rewrite this statement as a predicate logic formula and use it to prove that 15 has a unique factorization into primes.\\nFir... | 96 |
Query: When trying to find a unique representation of a number using multiplication of primes, how do you ensure the order of the primes doesn't matter? | [
{
"doc_id": 13145,
"text": "\\section{Fundamental Theorem of Arithmetic} Tags: Number Theory, Factorization, Fundamental Theorems, Prime Decompositions, Named Theorems, Prime Numbers \\begin{theorem} For every integer $n$ such that $n > 1$, $n$ can be expressed as the product of one or more primes, uniquel... | [
{
"doc_id": 2946,
"text": "['', 'Let\\'s consider the following statement: \"Every natural number greater than 1 is either a prime number or has a unique factorization into primes.\" Rewrite this statement as a predicate logic formula and use it to prove that 15 has a unique factorization into primes.\\nFir... | 96 |
Query: If I have a large number and want to prove its prime factorization is unique, how would I go about doing that? | [
{
"doc_id": 13145,
"text": "\\section{Fundamental Theorem of Arithmetic} Tags: Number Theory, Factorization, Fundamental Theorems, Prime Decompositions, Named Theorems, Prime Numbers \\begin{theorem} For every integer $n$ such that $n > 1$, $n$ can be expressed as the product of one or more primes, uniquel... | [
{
"doc_id": 2946,
"text": "['', 'Let\\'s consider the following statement: \"Every natural number greater than 1 is either a prime number or has a unique factorization into primes.\" Rewrite this statement as a predicate logic formula and use it to prove that 15 has a unique factorization into primes.\\nFir... | 96 |
Query: Can you give an example of how to break down a number into its prime factors and explain why the factors are unique? | [
{
"doc_id": 13145,
"text": "\\section{Fundamental Theorem of Arithmetic} Tags: Number Theory, Factorization, Fundamental Theorems, Prime Decompositions, Named Theorems, Prime Numbers \\begin{theorem} For every integer $n$ such that $n > 1$, $n$ can be expressed as the product of one or more primes, uniquel... | [
{
"doc_id": 65082,
"text": "['', 'def stoneGame(stones):\\n \"\"\"You are given a positive integer `primeFactors`. You are asked to construct a positive integer `n` that satisfies the following conditions:\\n\\n* The number of prime factors of `n` (not necessarily distinct) is **at most** `primeFactors... | 96 |
Query: How can understanding the unique prime factorization of numbers help in solving complex mathematical problems? | [
{
"doc_id": 13145,
"text": "\\section{Fundamental Theorem of Arithmetic} Tags: Number Theory, Factorization, Fundamental Theorems, Prime Decompositions, Named Theorems, Prime Numbers \\begin{theorem} For every integer $n$ such that $n > 1$, $n$ can be expressed as the product of one or more primes, uniquel... | [
{
"doc_id": 2946,
"text": "['', 'Let\\'s consider the following statement: \"Every natural number greater than 1 is either a prime number or has a unique factorization into primes.\" Rewrite this statement as a predicate logic formula and use it to prove that 15 has a unique factorization into primes.\\nFir... | 88 |
Query: How can I break down a large number into its smallest unique prime components? | [
{
"doc_id": 13145,
"text": "\\section{Fundamental Theorem of Arithmetic} Tags: Number Theory, Factorization, Fundamental Theorems, Prime Decompositions, Named Theorems, Prime Numbers \\begin{theorem} For every integer $n$ such that $n > 1$, $n$ can be expressed as the product of one or more primes, uniquel... | [
{
"doc_id": 4929,
"text": "['', 'Let P be the set of all prime numbers, and let S be the set of all positive integers. Define a second-order formula for the set Q which contains all positive integers n which satisfy the following conditions:\\n\\n1) n is divisible by some prime p in P, and\\n2) for every pr... | 96 |
Query: What if I have a number that is already prime, does it still need to be broken down further? | [
{
"doc_id": 13145,
"text": "\\section{Fundamental Theorem of Arithmetic} Tags: Number Theory, Factorization, Fundamental Theorems, Prime Decompositions, Named Theorems, Prime Numbers \\begin{theorem} For every integer $n$ such that $n > 1$, $n$ can be expressed as the product of one or more primes, uniquel... | [
{
"doc_id": 36297,
"text": "['', \"A manufacturing plant produces three types of machines: type A, type B, and type C. The percentage of machines produced daily is as follows: 60% of type A, 30% of type B, and 10% of type C. The probability of a type A machine breaking down on any given day is 0.05, for a t... | 96 |
Query: How can I break down any large number into its smallest unique building blocks? | [
{
"doc_id": 13145,
"text": "\\section{Fundamental Theorem of Arithmetic} Tags: Number Theory, Factorization, Fundamental Theorems, Prime Decompositions, Named Theorems, Prime Numbers \\begin{theorem} For every integer $n$ such that $n > 1$, $n$ can be expressed as the product of one or more primes, uniquel... | [
{
"doc_id": 3655,
"text": "['', 'Here is a precise numerical problem for a math student to solve related to Catalan numbers:\\n\\nAlice wants to build a tower using blocks of two different colors: red and blue. She has 8 red blocks and 5 blue blocks. If she wants to build a tower that has exactly 6 blocks a... | 96 |
Query: If a number is greater than 1, is there only one way to write it as a product of primes? | [
{
"doc_id": 13145,
"text": "\\section{Fundamental Theorem of Arithmetic} Tags: Number Theory, Factorization, Fundamental Theorems, Prime Decompositions, Named Theorems, Prime Numbers \\begin{theorem} For every integer $n$ such that $n > 1$, $n$ can be expressed as the product of one or more primes, uniquel... | [
{
"doc_id": 2946,
"text": "['', 'Let\\'s consider the following statement: \"Every natural number greater than 1 is either a prime number or has a unique factorization into primes.\" Rewrite this statement as a predicate logic formula and use it to prove that 15 has a unique factorization into primes.\\nFir... | 96 |
Query: Can you explain in simple terms why breaking down a number into its prime components is important? | [
{
"doc_id": 13145,
"text": "\\section{Fundamental Theorem of Arithmetic} Tags: Number Theory, Factorization, Fundamental Theorems, Prime Decompositions, Named Theorems, Prime Numbers \\begin{theorem} For every integer $n$ such that $n > 1$, $n$ can be expressed as the product of one or more primes, uniquel... | [
{
"doc_id": 74265,
"text": "['', 'In cooperative settings, awareness is a term used to denote \"knowledge created\\nthrough the interaction of an agent and its environment — in simple terms\\n\\'knowing what is going on\\'\". [26] In this setting, awareness is meant to\\nconvey how individuals monitor and... | 84 |
Query: How do I prove a number can only be written as a unique product of primes? | [
{
"doc_id": 13145,
"text": "\\section{Fundamental Theorem of Arithmetic} Tags: Number Theory, Factorization, Fundamental Theorems, Prime Decompositions, Named Theorems, Prime Numbers \\begin{theorem} For every integer $n$ such that $n > 1$, $n$ can be expressed as the product of one or more primes, uniquel... | [
{
"doc_id": 2946,
"text": "['', 'Let\\'s consider the following statement: \"Every natural number greater than 1 is either a prime number or has a unique factorization into primes.\" Rewrite this statement as a predicate logic formula and use it to prove that 15 has a unique factorization into primes.\\nFir... | 96 |
Query: What's the rule for integers where the prime factors are unique, regardless of their sequence? | [
{
"doc_id": 13145,
"text": "\\section{Fundamental Theorem of Arithmetic} Tags: Number Theory, Factorization, Fundamental Theorems, Prime Decompositions, Named Theorems, Prime Numbers \\begin{theorem} For every integer $n$ such that $n > 1$, $n$ can be expressed as the product of one or more primes, uniquel... | [
{
"doc_id": 4929,
"text": "['', 'Let P be the set of all prime numbers, and let S be the set of all positive integers. Define a second-order formula for the set Q which contains all positive integers n which satisfy the following conditions:\\n\\n1) n is divisible by some prime p in P, and\\n2) for every pr... | 96 |
Query: In practical scenarios, like cryptography, why is it important to know that the prime factorization of a number is unique? | [
{
"doc_id": 13145,
"text": "\\section{Fundamental Theorem of Arithmetic} Tags: Number Theory, Factorization, Fundamental Theorems, Prime Decompositions, Named Theorems, Prime Numbers \\begin{theorem} For every integer $n$ such that $n > 1$, $n$ can be expressed as the product of one or more primes, uniquel... | [
{
"doc_id": 2946,
"text": "['', 'Let\\'s consider the following statement: \"Every natural number greater than 1 is either a prime number or has a unique factorization into primes.\" Rewrite this statement as a predicate logic formula and use it to prove that 15 has a unique factorization into primes.\\nFir... | 96 |
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