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": 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: 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": 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 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": 35251,
"text": "['', '\\\\section{Area of Triangle in Terms of Two Sides and Angle}\\nTags: Triangles, Area of Triangle in Terms of Two Sides and Angle, Euclidean Geometry, Area Formulas, Areas of Triangles\\n\\n\\\\begin{theorem}\\nThe area of a triangle $ABC$ is given by:\\n:$\\\\displaystyle ... | 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": 37184,
"text": "['', 'A triangle has two sides of length 5 cm and 7 cm, and the included angle between them is 60 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)ab * sin(C)\\n\\nwhere a and b are the lengths... | 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": 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: 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": 35251,
"text": "['', '\\\\section{Area of Triangle in Terms of Two Sides and Angle}\\nTags: Triangles, Area of Triangle in Terms of Two Sides and Angle, Euclidean Geometry, Area Formulas, Areas of Triangles\\n\\n\\\\begin{theorem}\\nThe area of a triangle $ABC$ is given by:\\n:$\\\\displaystyle ... | 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": 29121,
"text": "['', '\\\\section{Measurements of Common Angles/Obtuse Angle}\\nTags: Angles\\n\\n\\\\begin{theorem}\\nAn obtuse angle measures $\\\\theta$, where:\\n: $90^\\\\circ < \\\\theta < 180^\\\\circ$\\nor:\\n: $\\\\dfrac \\\\pi 2 < \\\\theta < \\\\pi$\\n\\\\end{theorem}\\n\\n\\\\begin{p... | 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": 66228,
"text": "['', 'from math import log2\\n \"\"\"You are given a **binary string** `s`, and a **2D** integer array `queries` where `queries[i] = [firsti, secondi]`.\\n\\nFor the `ith` query, find the **shortest substring** of `s` whose **decimal value**, `val`, yields `secondi` when **bit... | 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": 3010,
"text": "['', \"There are 2 scales of temp A & B. It was given A 14 to 133, B 36 to 87. Find the temp when temp of A is equal to temp of B\\nAnswer Choices: (A) 52.5 (B) 53.5 (C) 51.5 (D) 54.5 (E) 55.5\\nLet's think about the multi-choice question.\\nActually, when corresponding ranges of ... | 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": 36547,
"text": "['', \"Suppose you have collected a sample of 40 students and calculated the mean height to be 168 cm with a standard deviation of 5 cm. Using the bootstrap method, estimate the standard error of the sample mean at a 95% confidence level.\\nThe bootstrap method is a resampling te... | 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": 56713,
"text": "['', '* [ Annual food expenditure vs. total consumer expenditure ](/grapher/food-expenditure-vs-gdp)\\n* [ Average income of large-scale food producers ](/grapher/income-large-scale-food-producers)\\n* [ Average income of small-scale food producers ](/grapher/income-small-scale-f... | 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": 36549,
"text": "['', \"Here's an example:\\n\\nSuppose we have a dataset with the following values: 10, 15, 18, 20, 22, 25, 30, 35, 40, 45. Find the mean, median and mode of this dataset.\\nTo find the mean, median, and mode of this dataset, we will follow these steps:\\n\\n1. Mean: Add up all t... | 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": 6506,
"text": "['', \"A thin metal plate is heated by a laser and the resulting temperature profile is measured. Given the temperature profile data, find the periodicities in the temperature variation using Fourier analysis. The temperature data over time is given by:\\n\\nTime (s) Temperatu... | 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": 13773,
"text": "['', 'Numbers can be positive, negative, or zero. Numbers can have up to 38 digits\\nof precision. Exceeding this results in an exception. If you need greater\\nprecision than 38 digits, you can use strings.\\n\\n * Positive range: 1E-130 to 9.99999999999999999999999999999999999... | 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": 36194,
"text": "['', 'Consider a set S = {2, 4, 6, 8, x, y} with distinct elements. If x and y are both prime numbers and 0 < x < 40 and 0 < y < 40, which of the following MUST be true?\\nI. The maximum possible range of the set is greater than 33.\\nII. The median can never be an even number.\\... | 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": 36965,
"text": "['', \"Suppose we have a dataset containing 10 numbers: 3, 5, 7, 9, 11, 13, 15, 17, 19, 21. Calculate the median and mean of this dataset. Then, remove the outlier 21 from the dataset and recalculate the median and mean. Finally, explain which measure of central tendency (median ... | 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": 2922,
"text": "['', 'A school wants to divide 20 students into four groups of five students each for a class activity. How many different ways can the school create these four groups? (Hint: This problem involves combinatorial calculations using the concept of block designs.)\\nTo solve this pro... | 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": 37056,
"text": "['', \"A math teacher is investigating the effects of different teaching methods on exam scores. She divided her class into two groups: Group A received traditional teaching and Group B received problem-based teaching. She gave both groups the same exam and recorded their scores.... | 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": 6503,
"text": "['', \"The LCM of three numbers is four times their GCF. Which of the following must be true of the numbers?\\nI. At least one of the numbers is odd.\\nII. Two of the three numbers must be same.\\nIII. At least one number is the same as GCF.\\nAnswer Choices: (A) only (B) option 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": 6187,
"text": "['', 'A mathematical model predicts that the temperature of a certain region will follow a chaotic behavior with an average temperature of 25°C and a standard deviation of 5°C. A weather station records the temperature of the region at noon for 30 consecutive days. Calculate the p... | 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": 12779,
"text": "['', 'November 14, 2011 at 2:11 am\\n\\nIn the First Semester Scores worksheet, in cell F17, enter a formula to\\ncalculate the weighted average of the first student’s four exams. The formula\\nin cell F17 should use absolute references to the weights found in the range\\nC8:C11,... | 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": 55792,
"text": "['', 'Step 6: Multivariate Analysis\\nTo illustrate multivariate time-series analysis methods, a corresponding time-series of rainfall was created. The center of the postcodes in NSW from which CPV was reported during the study period was identified. This was achieved by joining ... | 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": 12873,
"text": "['', ' Retrieving Dates and Days of the Week ¶\\nGet the current day of the week as a number using the [ EXTRACT\\n>\\n> SELECT EXTRACT(\\'dayofweek\\',current_date());\\n> \\n>\\n> Copy\\nNote\\n * The ` dayofweek_iso ` part follows the ISO-8601 data elements and interc... | 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": 68409,
"text": "['', 'def get_months_of_year(year):\\n \"\"\"\\n Returns the number of months that have already passed in the given year.\\n\\n This is useful for calculating averages on the year view. For past years,\\n we should divide by 12, but for the current year, we should div... | 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": 16826,
"text": "['', ' Using ANY_VALUE with GROUP BY Statements ¶\\nANY_VALUE can simplify and optimize the performance of [ GROUP BY\\nthat the result of a query with a GROUP BY clause can only contain expressions\\nused in the GROUP BY clause itself, or results of aggregate functions. For\\n... | 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": 68912,
"text": "['', 'def f1_score(targets, predictions, average=\\'macro\\'):\\n r\"\"\"\\n Compute the F1 score (sometimes known as the balanced F-score or\\n F-measure). The F1 score is commonly interpreted as the average of\\n precision and recall. The score lies in the range [0,... | 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": 5816,
"text": "['', 'A math student is asked to graph the equation y = 2x + 5. What are the coordinates of the y-intercept and the x-intercept of the line?\\nTo find the y-intercept, we need to determine the value of y when x = 0. Plugging x = 0 into the equation:\\n\\ny = 2(0) + 5\\ny = 0 + 5\\... | 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": 6474,
"text": "['', 'A sector of a circle has a radius of 6 cm, and the central angle of the sector measures 45°. What is the perimeter of the sector?\\nTo find the perimeter of the sector, we need to calculate the length of the arc and then add the two radii.\\n\\nThe length of the arc can be f... | 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": 6705,
"text": "['', 'A circular track has a diameter of 100 meters. What is the perimeter of the track?\\nTo find the perimeter (circumference) of a circular track, we use the formula:\\n\\nC = 2 * π * r\\n\\nwhere C is the circumference, r is the radius, and π (pi) is approximately 3.14159.\\n\... | 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": 6712,
"text": "['', \"A sector of a circle has a radius of 12 cm and an angle of 60°. What is the perimeter of the sector?\\nTo find the perimeter of the sector, we need to calculate the length of the arc and then add the two radii.\\n\\nFirst, let's find the length of the arc. The formula for t... | 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": 35397,
"text": "['', '\\\\section{Value of Radian in Degrees}\\nTags: Units of Measurement, Definitions: Geometry, Trigonometry, Definitions: Trigonometry, Angles, Definitions: Units of Measurement, Definitions: Angles\\n\\n\\\\begin{theorem}\\nThe value of a radian in degrees is given by:\\n:$1... | 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": 3246,
"text": "['', 'A math student needs to distribute 20 identical balls into 3 distinct boxes such that each box contains at least 2 balls. How many ways are there to do this?\\nSince each box must contain at least 2 balls, we can start by placing 2 balls in each box. This leaves us with 14 b... | 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": 3096,
"text": "['', 'Find the value of sin (45°) and cos (60°) using the unit circle method in trigonometry.\\nTo find the values of sin(45°) and cos(60°) using the unit circle method, we need to recall the coordinates of the points on the unit circle corresponding to these angles.\\n\\nThe unit... | 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": 4506,
"text": "['', 'A circle with radius 2cm is located at the origin in the xy-plane. Find the distance between the center of the circle and the point (3,4) in the plane using the contact geometry approach.\\nIn the contact geometry approach, we consider the distance between two points as the ... | 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": 8548,
"text": "['', ' * This parameter is only relevant for type \"circular\". It contains the radius of the circle. The center of the circle is located at the robot\\'s axis of rotation. ']",
"title": "",
"sim_score": 0.6746
},
{
"doc_id": 12247,
"text": "['', 'configuratio... | 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": 4782,
"text": "['', \"A precise numerical problem for a math student to solve on the topic of graphing trigonometric functions is: \\n\\nGraph the function y = 2sin(3x) in the interval [0, 2π]. Label the x and y-intercepts, the maximum and minimum points, and the points where the function inters... | 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": 37853,
"text": "['', 'A math student is asked to find the exact value of cos π/3 using the unit circle.\\nTo find the exact value of cos(π/3) using the unit circle, we need to recall that the unit circle has a radius of 1 and is centered at the origin (0,0) of the coordinate plane. The angle π/3... | 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": 29103,
"text": "['', \"\\\\begin{definition}[Definition:Cosine/Definition from Circle/Third Quadrant]\\nConsider a unit circle $C$ whose center is at the origin of a cartesian plane.\\n:500px\\nLet $P = \\\\tuple {x, y}$ be the point on $C$ in the third quadrant such that $\\\\theta$ is the angl... | 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": 4260,
"text": "['', 'A student is pushing a block with a force of 50 N at an angle of 30 degrees above the horizontal. The weight of the block is 100 N. Find the vertical component of the force acting on the block. (Assume that the force is acting on the center of the block)\\nTo find the vertic... | 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": 37853,
"text": "['', 'A math student is asked to find the exact value of cos π/3 using the unit circle.\\nTo find the exact value of cos(π/3) using the unit circle, we need to recall that the unit circle has a radius of 1 and is centered at the origin (0,0) of the coordinate plane. The angle π/3... | 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": 31448,
"text": "['', '\\\\section{Double Angle Formulas/Cosine/Corollary 1}\\nTags: Double Angle Formula for Cosine, Cosine Function\\n\\n\\\\begin{theorem}\\n: $\\\\cos \\\\left({2 \\\\theta}\\\\right) = 2 \\\\ \\\\cos^2 \\\\theta - 1$\\nwhere $\\\\cos$ denotes cosine.\\n\\\\end{theorem}\\n\\n\... | 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": 4306,
"text": "['', \"A student needs to find the third angle of a triangle which has sides of length 6, 8, and 10 using the law of cosines.\\nTo find the third angle of a triangle using the Law of Cosines, we first need to identify which angle we want to find. Let's call the angle opposite the ... | 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": 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: 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": 64513,
"text": "['', 'def algebra_simplify(alphabet_size=26,\\n min_depth=0,\\n max_depth=2,\\n nbr_cases=10000):\\n \"\"\"Generate the algebra simplify dataset.\\n\\n Each sample is a symbolic math expression involving unknown variab... | 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": 3695,
"text": "['', \"Find the exact value of sine and cosine of 300 degrees using the unit circle method.\\nTo find the exact value of sine and cosine of 300 degrees using the unit circle method, we first need to convert the angle from degrees to radians. \\n\\nTo convert from degrees to radian... | 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": 31448,
"text": "['', '\\\\section{Double Angle Formulas/Cosine/Corollary 1}\\nTags: Double Angle Formula for Cosine, Cosine Function\\n\\n\\\\begin{theorem}\\n: $\\\\cos \\\\left({2 \\\\theta}\\\\right) = 2 \\\\ \\\\cos^2 \\\\theta - 1$\\nwhere $\\\\cos$ denotes cosine.\\n\\\\end{theorem}\\n\\n\... | 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": 6653,
"text": "['', 'What is the value of cosine and sine for the angle θ = 60 degrees when measured in the standard position using the unit circle?\\nTo find the cosine and sine values for the angle θ = 60 degrees, we can use the unit circle. In the unit circle, the coordinates (x, y) of a poin... | 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": 2006,
"text": "\\begin{definition}[Definition:Cartesian Plane/Quadrants/Second] :400px Quadrant $\\text{II}: \\quad$ The area above the $x$-axis and to the left of the $y$-axis is called '''the second quadrant'''. That is, '''the second quadrant''' is where the $x$ coordinate of a point is negat... | 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": 35855,
"text": "['', \"\\\\begin{definition}[Definition:Cosine/Definition from Circle/Second Quadrant]\\nConsider a unit circle $C$ whose center is at the origin of a cartesian plane.\\n:500px\\nLet $P = \\\\tuple {x, y}$ be the point on $C$ in the second quadrant such that $\\\\theta$ is the an... | 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": 37959,
"text": "['', 'Suppose you want to estimate the value of π using Monte Carlo simulation. Use a program to randomly generate 10,000 points within a unit square (square with sides of length 1) centered at the origin. Count how many of these points are also inside the unit circle centered at... | 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": 35855,
"text": "['', \"\\\\begin{definition}[Definition:Cosine/Definition from Circle/Second Quadrant]\\nConsider a unit circle $C$ whose center is at the origin of a cartesian plane.\\n:500px\\nLet $P = \\\\tuple {x, y}$ be the point on $C$ in the second quadrant such that $\\\\theta$ is the an... | 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": 2132,
"text": "\\section{Cotangent is Cosine divided by Sine} Tags: Cotangent Function, Sine Function, Trigonometry, Cosine Function \\begin{theorem} Let $\\theta$ be an angle such that $\\sin \\theta \\ne 0$. Then: :$\\cot \\theta = \\dfrac {\\cos \\theta} {\\sin \\theta}$ where $\\cot$, $\\si... | 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": 35182,
"text": "['', '\\\\section{Multiplication using Parabola}\\nTags: Multiplication, Parabolas, Algebra, Quadratics\\n\\n\\\\begin{theorem}\\n:500pxrightthumb\\nLet the parabola $P$ defined as $y = x^2$ be plotted on the Cartesian plane.\\nLet $A = \\\\tuple {x_a, y_a}$ and $B = \\\\tuple {x... | 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": 3835,
"text": "['', \"Let's say there is a number which when divided by 3 leaves a remainder of 2, and when divided by 4 also leaves a remainder of 2. What is the smallest possible value of such a number?\\nLet's call the number x. According to the given information, we have the following two eq... | 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": 3835,
"text": "['', \"Let's say there is a number which when divided by 3 leaves a remainder of 2, and when divided by 4 also leaves a remainder of 2. What is the smallest possible value of such a number?\\nLet's call the number x. According to the given information, we have the following two eq... | 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": 774,
"text": "\\begin{definition}[Definition:Greatest Common Divisor/Also known as] The '''greatest common divisor''' is often seen abbreviated as '''GCD''', '''gcd''' or '''g.c.d.''' Some sources write $\\gcd \\set {a, b}$ as $\\tuple {a, b}$, but this notation can cause confusion with ordered ... | 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": 35481,
"text": "['', \"\\\\section{Bézout's Lemma}\\nTags: Number Theory, Bézout's Lemma, Greatest Common Divisor\\n\\n\\\\begin{theorem}\\nLet $a, b \\\\in \\\\Z$ such that $a$ and $b$ are not both zero.\\nLet $\\\\gcd \\\\set {a, b}$ be the greatest common divisor of $a$ and $b$.\\nThen:\\n:$\... | 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": 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: 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": 30702,
"text": "['', '\\\\section{Little Bézout Theorem}\\nTags: Named Theorems, Algebra, Polynomial Theory\\n\\n\\\\begin{theorem}\\nLet $\\\\map {P_n} x$ be a polynomial of degree $n$ in $x$.\\nLet $a$ be a constant.\\nThen the remainder of $\\\\map {P_n} x$ when divided by $x - a$ is equal to... | 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": 34910,
"text": "['', '\\\\section{Polynomials Closed under Addition/Polynomial Forms}\\nTags: Polynomial Theory\\n\\n\\\\begin{theorem}\\nLet:\\n:$\\\\ds f = \\\\sum_{k \\\\mathop \\\\in Z} a_k \\\\mathbf X^k$\\n:$\\\\ds g = \\\\sum_{k \\\\mathop \\\\in Z} b_k \\\\mathbf X^k$\\nbe polynomials in... | 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": 32842,
"text": "['', \"\\\\begin{definition}[Definition:Constant Polynomial/Definition 1]\\nLet $R$ be a commutative ring with unity.\\nLet $P \\\\in R \\\\sqbrk x$ be a polynomial in one variable over $R$.\\nThe polynomial $P$ is a '''constant polynomial''' {{iff}} its coefficients of $x^k$ are... | 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": 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: 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": 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 ... | 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": 17297,
"text": "['', '[ RDocumentation ](/)\\n\\nMoon [ ](https://github.com/datacamp/rdocumentation-2.0) [ Learn R\\n](https://www.datacamp.com/learn/r)\\n\\nSearch all packages and functions\\n\\n[ base (version 3.6.2 ) ](/packages/base/versions/3.6.2)\\n\\n# polyroot: Find Zeros of a Rea... | 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": 70424,
"text": "['', 'def dimsplit(P):\\n \"\"\"\\n Segmentize a polynomial (on decomposed form) into it\\'s dimensions.\\n\\n In array missing values are padded with 1 to make dimsplit compatible with\\n ``poly.prod(Q, 0)``.\\n\\n\\n Args:\\n P (Poly):\\n Input ... | 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": 1486,
"text": "\\section{Product of Cyclotomic Polynomials} Tags: Cyclotomic Polynomials, Abstract Algebra \\begin{theorem} Let $n > 0$ be a (strictly) positive integer. Then: :$\\ds \\prod_{d \\mathop \\divides n} \\map {\\Phi_d} x = x^n - 1$ where: :$\\map {\\Phi_d} x$ denotes the $d$th cyclo... | 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": 32813,
"text": "['', '\\\\section{Polynomial with Algebraic Number as Root is Multiple of Minimal Polynomial}\\nTags: Minimal Polynomials\\n\\n\\\\begin{theorem}\\nLet $F$ be a field.\\nLet $\\\\map P x$ be a polynomial in $F$.\\nLet $z$ be a root of $\\\\map P x$.\\nThen $\\\\map P x$ is a mult... | 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": 62478,
"text": "['', 'In the special case _b_ 2 = 4 _ac_ where the quadratic has only one distinct\\nroot ( _i.e._ the discriminant is zero), the quadratic polynomial can be [\\nfactored ](/wiki/Factorization \"Factorization\") as']",
"title": "",
"sim_score": 0.6847
},
{
"doc_... | 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": 191,
"text": "\\section{Product of Roots of Polynomial} Tags: Polynomial Equations \\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 product of the roots of $P$ is $\\dfrac {\\paren {-1}^n a_0} {a_n}$. \... | 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": 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: 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": 34506,
"text": "['', '\\\\section{Differentiation on Polynomials is Linear Operator}\\nTags: Polynomial Theory, Real Polynomial Functions, Differential Calculus, Linear Operators, Linear Algebra\\n\\n\\\\begin{theorem}\\nLet $\\\\map P \\\\R$ be the vector space of all polynomial functions on th... | 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": 63190,
"text": "['', 'The solution for this equation is the values of x, which are also called\\nzeros. Zeros of the polynomial are the solution for which the equation is\\nsatisfied. In the case of quadratics, there are two roots or zeros of the\\nequation. And if we put the values of roots or ... | 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": 37431,
"text": "['', \"Prove that the sum of two even numbers is always even.\\nLet's consider two even numbers, say 'a' and 'b'. By definition, an even number can be represented as 2n, where n is an integer.\\n\\nSo, we can write 'a' and 'b' as:\\na = 2n (where n is an integer)\\nb = 2m (where ... | 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": 32723,
"text": "['', '\\\\section{Prime Decomposition of Integer is Unique}\\nTags: Prime Decompositions\\n\\n\\\\begin{theorem}\\nLet $n$ be an integer such that $n > 1$.\\nThen the prime decomposition of $n$ is unique.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Integer is Expressible as P... | 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": 37378,
"text": "['', \"If the GCD of two distinct numbers (not equal, both integers, greater than 1) is 3, then which of the following can be true?\\nA. They are prime.\\nB. They are not even.\\nC. They do not have a common prime factor.\\nD. They do not have a common factor other than 3.\\nAnsw... | 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": 282,
"text": "\\section{Definition:Prime Decomposition} Tags: Definitions: Prime Decompositions, Definitions: Prime Decomposition, Number Theory, Definitions: Prime Numbers, Definitions: Number Theory, Definitions \\begin{theorem} Let <math>n > 1 \\in \\mathbb{Z}</math>. Then <math>n</math> has... | 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": 30137,
"text": "['', '\\\\section{Prime Number Formed by Concatenating Consecutive Integers down to 1}\\nTags: Prime Numbers\\n\\n\\\\begin{theorem}\\nLet $N$ be an integer whose decimal representation consists of the concatenation of all the integers from a given $n$ in descending order down to... | 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": 36593,
"text": "['', \"Suppose Alice has three boxes containing 7, 9, and 11 marbles respectively. What is the smallest number of marbles that she needs to transfer among the boxes so that each box would have the same number of marbles? Use the Chinese remainder theorem to solve the problem.\\nT... | 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": 29260,
"text": "['', '\\\\section{Fundamental Theorem of Arithmetic}\\nTags: Number Theory, Factorization, Fundamental Theorems, Prime Decompositions, Named Theorems, Prime Numbers\\n\\n\\\\begin{theorem}\\nFor every integer $n$ such that $n > 1$, $n$ can be expressed as the product of one or mo... | 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": 2781,
"text": "['', \"Let G be a group of order p, where p is a prime number. If the group G has exactly 5 distinct subgroups, what is the value of p?\\nLet's consider the properties of a group of prime order p. Since p is prime, the only divisors of p are 1 and p itself. By Lagrange's theorem, ... | 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": 32723,
"text": "['', '\\\\section{Prime Decomposition of Integer is Unique}\\nTags: Prime Decompositions\\n\\n\\\\begin{theorem}\\nLet $n$ be an integer such that $n > 1$.\\nThen the prime decomposition of $n$ is unique.\\n\\\\end{theorem}\\n\\n\\\\begin{proof}\\nFrom Integer is Expressible as P... | 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": 37378,
"text": "['', \"If the GCD of two distinct numbers (not equal, both integers, greater than 1) is 3, then which of the following can be true?\\nA. They are prime.\\nB. They are not even.\\nC. They do not have a common prime factor.\\nD. They do not have a common factor other than 3.\\nAnsw... | 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": 282,
"text": "\\section{Definition:Prime Decomposition} Tags: Definitions: Prime Decompositions, Definitions: Prime Decomposition, Number Theory, Definitions: Prime Numbers, Definitions: Number Theory, Definitions \\begin{theorem} Let <math>n > 1 \\in \\mathbb{Z}</math>. Then <math>n</math> has... | 96 |
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