Math Glossary
Plain-language definitions for the 71 terms behind WorkedMath's calculators, each with a worked example and a link to the calculator that puts it to use.
Absolute Value
Absolute value measures how far a number sits from zero on the number line, ignoring its sign — written with vertical bars, like |−5|. Since distance can't be negative, the absolute value of any number is always positive or zero, never negative.
Example
|−7| = 7 and |7| = 7, since both numbers sit exactly 7 units away from zero, just in opposite directions.
Try it: Absolute Value Calculator
Arc Length
Arc length measures the distance along a curved section of a circle's circumference, rather than the straight-line distance between its endpoints. It depends on both the circle's radius and the size of the angle (central angle) the arc spans, since a bigger circle or a wider angle both produce a longer arc.
Example
A circle with a radius of 10 and a central angle of 90° (a quarter of the full circle) has an arc length of (90/360) × 2π(10) = (1/4) × 20π = 5π ≈ 15.7.
Try it: Arc Length Calculator
Binary Number
Binary is a base-2 number system that represents every value using only the digits 0 and 1, unlike the base-10 (decimal) system most people use day to day. Each digit's position represents a power of 2 instead of a power of 10. Computers use binary internally because each digit maps directly onto a transistor's on/off state.
Example
The binary number 1011 equals 11 in decimal: (1×8) + (0×4) + (1×2) + (1×1) = 8 + 0 + 2 + 1 = 11.
Try it: Binary Calculator
Chi-Square Test
A chi-square test compares observed category counts against the counts you'd expect under some assumption, then measures how far apart they are. A small chi-square value means the data roughly matches expectations; a large one suggests something is genuinely different from the assumption. It's a common tool for goodness-of-fit tests, like checking whether a die is fair.
Example
Rolling a fair six-sided die 60 times, you'd expect 10 rolls of each number. If the actual counts were far from 10 each, the chi-square value would be large, suggesting the die isn't fair.
Try it: Chi-Square Calculator
Cone
A cone is a three-dimensional shape with a flat circular base that narrows smoothly to a single point called the apex. Its volume is (1/3)πr²h, exactly one-third the volume of a cylinder with the same base and height.
Example
A cone with a radius of 3 and a height of 6 has a volume of (1/3)π(3)²(6) = (1/3)π(54) = 18π ≈ 56.5.
Try it: Cone Calculator
Cube Root
A cube root is the value that, when multiplied by itself three times (cubed), produces the original number. It's written as ∛x, or x^(1/3). Unlike a square root, a cube root can be negative, since a negative number cubed stays negative.
Example
∛27 = 3, since 3 × 3 × 3 = 27. ∛(−8) = −2, since (−2) × (−2) × (−2) = −8.
Try it: Cube Root Calculator
Cylinder
A cylinder is a three-dimensional shape with two identical, parallel circular bases connected by a curved side surface, like a soup can. Its volume is πr²h — the area of one circular base multiplied by the height.
Example
A cylinder with a radius of 4 and a height of 10 has a volume of π(4)²(10) = π(160) ≈ 502.7.
Try it: Cylinder Calculator
Determinant
The determinant is a single number computed from a square matrix that reveals whether the matrix has an inverse (a determinant of 0 means it doesn't) and, geometrically, how much a transformation described by the matrix scales area or volume. It's a required step in finding a matrix's inverse.
Example
For the 2×2 matrix [[1, 2], [3, 4]], the determinant is (1×4) − (2×3) = 4 − 6 = −2.
Try it: Matrix Calculator
Discount
A discount is the amount subtracted from an item's original price, typically expressed as a percentage of that price. To find the final price, multiply the original price by (1 − the discount rate as a decimal).
Example
A $80 jacket with a 25% discount costs $80 × (1 − 0.25) = $80 × 0.75 = $60, a savings of $20.
Try it: Discount Calculator
Discriminant
The discriminant is the expression b² − 4ac, found under the square root in the quadratic formula. Its sign tells you what kind of solutions a quadratic equation has before you even solve it: positive means two real solutions, zero means one repeated real solution, and negative means two complex solutions.
Example
For x² − 4x + 4 = 0 (a = 1, b = −4, c = 4), the discriminant is (−4)² − 4(1)(4) = 16 − 16 = 0, so there's exactly one repeated solution.
Try it: Quadratic Formula Calculator
Distance Formula
The distance formula finds the straight-line distance between two points, (x₁,y₁) and (x₂,y₂), using √((x₂−x₁)² + (y₂−y₁)²). It's really just the Pythagorean theorem applied to a coordinate plane, treating the horizontal and vertical gaps between the points as the two legs of a right triangle.
Example
The distance between (0, 0) and (3, 4) is √((3−0)² + (4−0)²) = √(9+16) = √25 = 5.
Try it: Distance Formula Calculator
Domain and Range
The domain of a function is the complete set of input values (usually x) that the function accepts without breaking a math rule, like dividing by zero or taking the square root of a negative number. The range is the complete set of output values (usually y) the function can actually produce.
Example
For f(x) = √x, the domain is x ≥ 0 (you can't take the square root of a negative number), and the range is y ≥ 0 (a square root is never negative).
Try it: Domain and Range Calculator
Expanded Form
Expanded form breaks a number down into the sum of each digit multiplied by its place value (ones, tens, hundreds, and so on), making the value each digit actually represents explicit rather than implied by its position.
Example
3,482 in expanded form is 3,000 + 400 + 80 + 2.
Try it: Expanded Form Calculator
Exponent
An exponent (also called a power) tells you how many times to multiply a base number by itself. It's written as a small raised number, like the 3 in 2³. Exponents also extend to negative and fractional values: a negative exponent means take the reciprocal, and a fractional exponent means take a root.
Example
2³ means 2 × 2 × 2 = 8. 2⁻¹ means 1/2, and 2^(1/2) means the square root of 2.
Try it: Exponent Calculator
Exponential Growth
Exponential growth happens when a quantity grows by a constant percentage rate each time period, rather than by a constant amount. Because the growth compounds on an ever-larger base, exponential growth starts slowly but eventually increases far faster than linear (constant-amount) growth. It shows up in compound interest, population growth, and viral spread.
Example
$1,000 growing at 10% per year becomes $1,100 after year 1, $1,210 after year 2, and $1,331 after year 3 — the dollar amount added each year keeps increasing because it's 10% of a growing balance, not a flat $100.
Try it: Exponential Growth Calculator
Factor
A factor of a number is a whole number that divides into it with nothing left over. Every number has at least two factors, 1 and itself, and most have several more. Factors are the basis for finding a GCF, an LCM, or a prime factorization.
Example
The factors of 12 are 1, 2, 3, 4, 6, and 12, since each of those divides evenly into 12.
Try it: LCM Calculator – Least Common Multiple · GCF Calculator – Greatest Common Factor · Factor Calculator
Factorial
A factorial, written n!, is the product of every positive whole number from 1 up to n. Factorials grow extremely fast and are central to counting problems, since they represent the number of ways to arrange n distinct items in order.
Example
5! = 5 × 4 × 3 × 2 × 1 = 120.
Try it: Factorial Calculator
Factoring (Algebra)
Factoring rewrites an algebraic expression as a product of two or more simpler expressions, the reverse of expanding (multiplying) them back out. For quadratics, factoring means finding two binomials that multiply together to give the original expression, which is often the fastest way to solve the equation by setting each factor equal to zero.
Example
x² − 5x + 6 factors into (x − 2)(x − 3), since expanding (x − 2)(x − 3) gives back x² − 5x + 6. Setting each factor to 0 gives the solutions x = 2 and x = 3.
Try it: Factoring Calculator
GCF (Greatest Common Factor)
GCF stands for greatest common factor, sometimes called GCD (greatest common divisor). It's the largest number that divides evenly into every number in a set, with nothing left over. It's most often used to reduce a fraction to its simplest form.
Example
The GCF of 24 and 36 is 12, since 12 is the largest number that divides evenly into both.
Try it: LCM Calculator – Least Common Multiple · GCF Calculator – Greatest Common Factor · Factor Calculator
Golden Ratio
The golden ratio (denoted φ, or phi) is approximately 1.618033988749... — an irrational number where, if a line is split into two parts, the ratio of the whole line to the longer part equals the ratio of the longer part to the shorter part. It shows up in geometry, art, and famously (though often overstated) in nature.
Example
If a line 1.618 units long is split into a piece of length 1 and a piece of length 0.618, the ratio of the whole (1.618) to the longer piece (1) equals the ratio of the longer piece (1) to the shorter piece (0.618) — both equal about 1.618.
Try it: Golden Ratio Calculator
GPA (Grade Point Average)
GPA converts letter grades into numbers (typically A=4, B=3, C=2, D=1, F=0) and averages them, often weighted by how many credit hours each course is worth. A 4.0 GPA usually means straight A's, though some schools use a weighted scale above 4.0 for honors or AP courses.
Example
A student with a 4.0 in a 3-credit class and a 3.0 in a 4-credit class has a GPA of (4.0×3 + 3.0×4) / (3+4) = 24/7 ≈ 3.43, weighted by credit hours rather than a plain average of 3.5.
Hexagon
A hexagon is any two-dimensional shape with six straight sides and six angles. A regular hexagon has all six sides and angles equal, and its interior angles each measure 120°. Regular hexagons tile perfectly with no gaps, which is why they show up in honeycombs and some flooring patterns.
Example
A regular hexagon with a side length of 6 has an area of (3√3/2) × 6² ≈ 93.5.
Try it: Hexagon Calculator
Hypotenuse
The hypotenuse is the side of a right triangle that sits across from the 90° angle. It's always the longest of the triangle's three sides, and it's the side the Pythagorean theorem solves for when you know the other two.
Example
In a right triangle with legs 6 and 8, the hypotenuse is 10.
Try it: Pythagorean Theorem Calculator
Inverse Function
An inverse function undoes what the original function does: if a function f takes x to y, its inverse f⁻¹ takes y back to x. A function only has an inverse if it's one-to-one, meaning no two different inputs produce the same output. Graphically, a function and its inverse are mirror images across the line y = x.
Example
If f(x) = 2x + 3, its inverse is f⁻¹(x) = (x − 3)/2. Check: f(5) = 2(5)+3 = 13, and f⁻¹(13) = (13−3)/2 = 5, back where you started.
Try it: Inverse Function Calculator
Korean Age
Korean age (세는 나이) counts a newborn as already 1 year old at birth, then adds a year to everyone's age at once on January 1st, regardless of individual birth dates. This differs from international age (used officially in South Korea since 2023), which starts at 0 and only advances on a person's own birthday.
Example
A baby born on December 31 is already considered age 2 in the Korean system on January 1 of the following year — just one day after birth — since the birth year counts as year 1 and the new calendar year adds another.
Try it: Korean Age Calculator
Law of Cosines
The law of cosines generalizes the Pythagorean theorem to any triangle, not just right triangles: c² = a² + b² − 2ab×cos(C), where C is the angle opposite side c. It's used to find a missing side when you know two sides and the included angle, or to find an angle when you know all three sides.
Example
A triangle with sides a = 7, b = 10, and an included angle C = 60° has a third side c = √(7² + 10² − 2(7)(10)cos(60°)) = √(49 + 100 − 70) = √79 ≈ 8.89.
Try it: Law of Cosines Calculator
Law of Sines
The law of sines states that in any triangle, the ratio of a side's length to the sine of its opposite angle is the same for all three sides: a/sin(A) = b/sin(B) = c/sin(C). It's used to solve triangles that aren't right triangles, when you know either two angles and a side, or two sides and a non-included angle.
Example
In a triangle with angle A = 30°, angle B = 105°, and side a = 10, side b can be found with b = a × sin(B)/sin(A) = 10 × sin(105°)/sin(30°) ≈ 10 × 0.9659/0.5 ≈ 19.32.
Try it: Law of Sines Calculator
LCD (Least Common Denominator)
The least common denominator, or LCD, is the smallest number that works as a common denominator for a set of fractions. It's really just the LCM applied to the fractions' denominators. Once every fraction shares the LCD, they can be added or subtracted directly.
Example
The LCD of 1/4 and 1/6 is 12, since 12 is the smallest number both 4 and 6 divide into evenly.
Try it: LCM Calculator – Least Common Multiple · LCD Calculator – Least Common Denominator
LCM (Least Common Multiple)
LCM stands for least common multiple. It's the smallest positive number that every number in a set divides into evenly. It shows up constantly when adding or subtracting fractions with different denominators, since you need a shared denominator to work with.
Example
The LCM of 4 and 6 is 12, since 12 is the smallest number both 4 and 6 divide into evenly.
Try it: LCM Calculator – Least Common Multiple · GCF Calculator – Greatest Common Factor · LCD Calculator – Least Common Denominator
Linear Equation
A linear equation is any equation where every variable term is raised to the first power (no squares, roots, or products of variables), so its graph is always a straight line. It's typically written in slope-intercept form (y = mx + b), standard form (Ax + By = C), or point-slope form.
Example
y = 3x + 2 is a linear equation and graphs as a straight line. y = x² + 2 is not linear, since x is squared, and it graphs as a curve instead.
Try it: Linear Equation Calculator
Logarithm
A logarithm answers the question "what exponent do I need?" Specifically, log base b of x is the exponent you'd raise b to in order to get x. Logarithms are the inverse operation of exponentiation, the same way subtraction is the inverse of addition. Base 10 (common log) and base e (natural log) are the two most widely used.
Example
log₂(8) = 3, because 2³ = 8. In other words, you need to raise 2 to the 3rd power to get 8.
Try it: Log Calculator
Long Division
Long division is a written method for dividing one multi-digit number by another, working through the dividend one digit (or digit group) at a time: divide, multiply, subtract, and bring down the next digit, repeating until done. It's the standard hand method for division problems too large to do in one step mentally.
Example
847 ÷ 5 using long division: 8÷5=1 remainder 3, bring down 4 to make 34, 34÷5=6 remainder 4, bring down 7 to make 47, 47÷5=9 remainder 2. The result is 169 remainder 2.
Try it: Long Division Calculator
Markup
Markup is the amount added on top of an item's cost to determine its selling price, typically expressed as a percentage of that cost (not the final price, which is what makes it different from a profit margin). To find the selling price, multiply the cost by (1 + the markup rate as a decimal).
Example
An item that costs a store $40, marked up 50%, sells for $40 × (1 + 0.50) = $40 × 1.5 = $60.
Try it: Markup Calculator
Matrix
A matrix is a rectangular array of numbers arranged in rows and columns, used to represent and solve systems of linear equations, transform geometric shapes, and more. A matrix's size is described as rows × columns — a 2×3 matrix has 2 rows and 3 columns. Matrices can be added, subtracted, and multiplied following their own specific rules.
Example
A 2×2 matrix [[1, 2], [3, 4]] has two rows (1,2 and 3,4) and two columns (1,3 and 2,4).
Try it: Matrix Calculator
Mean, Median, and Mode
Mean is the arithmetic average — add up all the values and divide by how many there are. Median is the middle value when the data is sorted (or the average of the two middle values if there's an even count). Mode is the value that appears most often. Each measures "center" differently, and they can give very different answers on skewed data.
Example
For the data set 2, 3, 3, 5, 20: the mean is 6.6 (33 ÷ 5), the median is 3 (the middle value when sorted), and the mode is 3 (it appears twice). The single high value of 20 pulls the mean up but doesn't affect the median or mode.
Try it: Mean, Median, Mode Calculator
Midpoint
The midpoint of two points is the point that sits exactly halfway between them, found by averaging their x-coordinates and averaging their y-coordinates separately. It's the center point of the line segment connecting the two original points.
Example
The midpoint of (2, 4) and (8, 10) is ((2+8)/2, (4+10)/2) = (5, 7).
Try it: Midpoint Calculator
Mixed Number
A mixed number combines a whole number and a proper fraction (a fraction less than 1) into a single value, like 2¾. It represents the same value as an improper fraction (one where the numerator is larger than the denominator) — 2¾ and 11/4 mean exactly the same amount, just written differently.
Example
The improper fraction 11/4 converts to the mixed number 2¾, since 11 ÷ 4 = 2 remainder 3, giving a whole number of 2 and a remaining fraction of 3/4.
Try it: Mixed Number Calculator
Percentile
A percentile shows where a value stands compared to the rest of a data set. If a score is in the 90th percentile, that means it beat about 90% of the other scores in the set. It's a way of ranking, not a percentage of correct answers.
Example
Scoring in the 95th percentile on a standardized test means you scored higher than about 95% of the other test-takers, whatever your raw score happened to be.
Try it: Standard Deviation Calculator · Percentile Calculator
Perfect Square
A perfect square is a number you get by multiplying a whole number by itself. Its square root is always a whole number too, with no radical needed to express it exactly.
Example
16 is a perfect square, since 4 × 4 = 16. 18 is not, since no whole number squared equals 18.
Try it: Factor Calculator · Square Root Calculator
Perimeter
Perimeter is the total length of the boundary around a flat (two-dimensional) shape — add up the lengths of every side. For a circle, the equivalent measurement is called the circumference instead, calculated differently since a circle has no straight sides.
Example
A rectangle with a length of 8 and a width of 5 has a perimeter of 2(8) + 2(5) = 16 + 10 = 26.
Try it: Perimeter Calculator
Permutation
A permutation counts how many different ordered arrangements are possible when choosing r items from a set of n, where the order of selection matters. It's closely related to a combination, which counts the same kind of selection but ignores order. Permutations are always greater than or equal to the corresponding combination, since each combination can usually be arranged in more than one order.
Example
Choosing and ordering 2 winners (1st and 2nd place) from 5 runners gives 5 × 4 = 20 permutations, since there are 5 choices for 1st place and 4 remaining choices for 2nd.
Try it: Combination Calculator (nCr)
Place Value
Place value is the idea that the same digit is worth different amounts depending on its position in a number. Each column one step to the left of a digit is worth 10 times more; each column one step to the right (including past the decimal point, into tenths, hundredths, and so on) is worth 10 times less. It's distinct from a digit's face value, which is just the digit itself.
Example
In 526, the 5 has a face value of 5 but a place value of 500, because it sits in the hundreds column.
Try it: Expanded Form Calculator · Place Value Chart – Interactive, Ones to Hundred Millions & Decimals
Point-Slope Form
Point-slope form writes a line's equation using its slope (m) and the coordinates of one known point on the line (x₁, y₁): y − y₁ = m(x − x₁). It's especially useful when you know a point and a slope but don't yet know the y-intercept needed for slope-intercept form.
Example
A line through the point (2, 5) with a slope of 3 is written in point-slope form as y − 5 = 3(x − 2), which simplifies to y = 3x − 1.
Try it: Point-Slope Form Calculator
Polynomial
A polynomial is an expression built from one or more terms, each a coefficient multiplied by a variable raised to a non-negative whole-number power, combined with addition or subtraction. Its degree is the highest exponent that appears. Polynomials cannot include variables in a denominator, negative exponents, or exponents that are fractions or variables themselves.
Example
3x² + 5x − 7 is a polynomial of degree 2 (a quadratic), with terms 3x², 5x, and −7.
Try it: Factoring Calculator · Polynomial Calculator
Prime Number
A prime number is a whole number greater than 1 that can only be divided evenly by 1 and itself. Any number with more than two factors is called composite instead. Primes are the building blocks of every whole number, since every number greater than 1 can be broken down into a unique product of primes.
Example
7 is prime, since only 1 and 7 divide into it evenly. 8 is not prime, since 1, 2, 4, and 8 all divide into it.
Try it: GCF Calculator – Greatest Common Factor · Factor Calculator
Proportion
A proportion is an equation showing that two ratios are equal, usually written as a/b = c/d. Proportions are solved by cross-multiplying: a×d = b×c. They're the basis for scaling recipes, maps, and unit conversions, whenever one known ratio needs to be applied to a new quantity.
Example
In the proportion 2/3 = x/12, cross-multiplying gives 2×12 = 3×x, so 24 = 3x, and x = 8.
Try it: Proportion Calculator
Pythagorean Theorem
The Pythagorean theorem describes a relationship that holds in every right triangle: the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides. It only applies to right triangles, not triangles in general.
Example
A right triangle with legs 3 and 4 has a hypotenuse of 5, since 3² + 4² = 9 + 16 = 25 = 5².
Try it: Pythagorean Theorem Calculator
Quadratic Equation
A quadratic equation is any equation that can be written as ax² + bx + c = 0, where a, b, and c are numbers and a isn't 0. Graphed, it forms a parabola. It can have two real solutions, one repeated real solution, or two complex solutions, depending on the discriminant.
Example
x² − 5x + 6 = 0 is a quadratic equation. It factors to (x − 2)(x − 3) = 0, giving solutions x = 2 and x = 3.
Try it: Quadratic Formula Calculator
Quartile
Quartiles divide a sorted data set into four equal-sized groups. Q1 (the first quartile) marks the boundary below which 25% of the data falls, Q2 is the median (50%), and Q3 marks the 75% boundary. The gap between Q1 and Q3, called the interquartile range, is a common way to measure spread while ignoring extreme outliers.
Example
For the sorted data set 1, 3, 5, 7, 9, 11, 13: Q2 (the median) is 7, Q1 is 3, and Q3 is 11.
Try it: Quartile Calculator · Five Number Summary Calculator
Ratio
A ratio compares two (or more) quantities, showing their relative sizes, often written as a:b or a/b. Ratios can be simplified the same way fractions can, by dividing both sides by their greatest common factor, and they stay meaningful even when the actual quantities scale up or down together.
Example
A recipe using 2 cups of flour to 1 cup of sugar has a ratio of 2:1. Doubling the recipe to 4 cups of flour and 2 cups of sugar keeps the same 2:1 ratio.
Try it: Ratio Calculator
Rationalizing the Denominator
Rationalizing the denominator means rewriting a fraction so that its bottom no longer contains a radical (square root), typically by multiplying both the numerator and denominator by a value that clears the root out of the denominator. The value of the fraction doesn't change, only its form, since multiplying by a fraction equal to 1 never changes the result.
Example
1/√2 is rationalized by multiplying top and bottom by √2, giving (1×√2)/(√2×√2) = √2/2.
Remainder
A remainder is the amount left over when one whole number doesn't divide evenly into another. It's always smaller than the number you're dividing by (the divisor) — if it weren't, you could fit at least one more whole group in.
Example
17 ÷ 5 = 3 remainder 2, since 5 fits into 17 three whole times (15), leaving 2 left over.
Try it: Remainder Calculator
Roman Numerals
Roman numerals represent numbers using combinations of seven letters: I (1), V (5), X (10), L (50), C (100), D (500), and M (1000). Letters are generally added together left to right, except when a smaller value appears before a larger one, which means subtract instead — like IV meaning 4 (5 − 1), not 6.
Example
1994 is written in Roman numerals as MCMXCIV: M (1000) + CM (900) + XC (90) + IV (4) = 1994.
Try it: Roman Numeral Converter
Sales Tax
Sales tax is a percentage of a purchase price added on top at checkout, set by state and local governments and varying widely by location. To find the total price, multiply the pre-tax price by (1 + the tax rate as a decimal).
Example
A $50 purchase with an 8% sales tax costs $50 × (1 + 0.08) = $50 × 1.08 = $54, meaning $4 in tax.
Try it: Sales Tax Calculator
Sample Size
Sample size is how many people, items, or observations a survey or experiment needs to draw a conclusion with a chosen level of confidence and margin of error. Larger populations, higher confidence levels, and smaller margins of error all push the required sample size up. Using too small a sample risks a result that doesn't reflect reality; too large wastes time and money.
Example
Surveying a population of 10,000 people with a 95% confidence level and a 5% margin of error requires a sample of about 370 people — sample size requirements level off quickly, so a population of 50,000 would need a similar sample.
Try it: Sample Size Calculator
Scale Factor
A scale factor is the number you multiply every length in a shape by to make it proportionally larger or smaller, while keeping its overall shape identical. A scale factor greater than 1 enlarges a shape; a scale factor between 0 and 1 shrinks it.
Example
A model car built at a 1:24 scale factor means every real measurement is divided by 24 — a real car 192 inches long becomes an 8-inch model, since 192 ÷ 24 = 8.
Try it: Scale Factor Calculator
Scientific Notation
Scientific notation expresses a number as a value between 1 and 10 multiplied by 10 raised to some power. It makes extremely large or small numbers easier to read and compare, since the exponent immediately shows the number's scale rather than requiring you to count digits or zeros.
Example
602,000,000,000,000,000,000,000 (Avogadro's number, roughly) is written in scientific notation as 6.02 × 10²³.
Try it: Scientific Notation Calculator
Simple Interest
Simple interest is calculated as a fixed percentage of the original principal amount for each period, using the formula I = P × r × t (principal × rate × time). Unlike compound interest, it never earns interest on previously accumulated interest, so it grows at a constant, linear rate rather than accelerating.
Example
A $1,000 loan at 5% simple annual interest for 3 years accrues 1,000 × 0.05 × 3 = $150 in interest, regardless of when during those 3 years the interest is calculated.
Try it: Simple Interest Calculator
Slope
Slope measures how much a line rises or falls vertically for every unit it moves horizontally, calculated as the change in y divided by the change in x between two points. A positive slope rises left to right, a negative slope falls, a slope of 0 is flat, and an undefined slope is a vertical line.
Example
The line through (1, 2) and (4, 11) has a slope of (11−2)/(4−1) = 9/3 = 3 — for every 1 unit the line moves right, it rises 3 units.
Slope-Intercept Form
Slope-intercept form writes a straight line's equation as y = mx + b, where m is the line's slope and b is the y-coordinate where it crosses the y-axis (where x = 0). It's the most common way to write a linear equation because both key features of the line, its steepness and its starting point, are visible directly in the equation.
Example
y = 2x + 3 has a slope of 2 and crosses the y-axis at (0, 3). At x = 0, y = 2(0) + 3 = 3, confirming the y-intercept.
Try it: Slope-Intercept Form Calculator
Sphere
A sphere is a three-dimensional shape where every point on its surface sits the same distance (the radius) from a single center point — the 3D equivalent of a circle. Its volume is (4/3)πr³ and its surface area is 4πr².
Example
A sphere with a radius of 3 has a volume of (4/3)π(3)³ = (4/3)π(27) = 36π ≈ 113.1, and a surface area of 4π(3)² = 36π ≈ 113.1.
Try it: Sphere Calculator
Standard Deviation
Standard deviation measures how far, on average, the values in a data set sit from the mean. A small standard deviation means the numbers cluster tightly around the average; a large one means they're spread out. It's the most common way statisticians describe variability, from test scores to manufacturing tolerances.
Example
Two classes can both average 80 on a test, but one class has scores tightly bunched near 80 while the other ranges from 40 to 100. The second class has a much larger standard deviation.
Try it: Standard Deviation Calculator · Percentile Calculator
Surface Area
Surface area is the total area of every face or curved surface on the outside of a three-dimensional shape, as if you unfolded it flat and measured all the pieces. Different shapes use different formulas — a cube adds up six equal squares, while a sphere or cylinder involves curved surfaces calculated with π.
Example
A cube with 4-inch sides has 6 faces, each 4×4=16 square inches, for a total surface area of 6×16 = 96 square inches.
Try it: Surface Area Calculator
Synthetic Division
Synthetic division is a streamlined way to divide a polynomial by a binomial of the form (x − c), using only the coefficients instead of writing out the full long-division process. It's faster than polynomial long division and is especially useful for testing whether a given value is a root.
Example
Dividing x² − 5x + 6 by (x − 2) using synthetic division gives a quotient of x − 3 with a remainder of 0, confirming that x = 2 is a root of the original polynomial.
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System of Equations
A system of equations is a group of two or more equations that all use the same variables, where a solution must satisfy every equation at once. Graphically, solving a system of two linear equations means finding the point (or points) where their lines intersect. Systems can have exactly one solution, no solution (parallel lines), or infinitely many solutions (the same line written two ways).
Example
The system x + y = 10 and x − y = 2 has the solution x = 6, y = 4, since 6 + 4 = 10 and 6 − 4 = 2 both hold true.
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T-Test
A one-sample t-test compares the average of a sample to a known or hypothesized value, accounting for the sample's size and variability. It answers whether an observed difference is likely real or just due to random chance. It's one of the most widely used tests in science and business for comparing averages when the true population variance isn't known.
Example
A factory expects light bulbs to last 1,000 hours on average. Testing a sample of 20 bulbs, a t-test can determine whether the sample's average lifespan is significantly different from 1,000 hours or just randomly varying.
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Trapezoid
A trapezoid is a quadrilateral (a four-sided shape) with exactly one pair of parallel sides, usually called the bases. The other two sides, called legs, don't have to be parallel or equal in length. Its area is the average of the two parallel sides, multiplied by the height between them.
Example
A trapezoid with parallel sides of 8 and 14, and a height of 5 between them, has an area of ((8 + 14) / 2) × 5 = 55.
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Variance
Variance measures how spread out a data set is by averaging the squared distance of each value from the mean. Squaring makes every difference positive and weights larger deviations more heavily. Its square root is the standard deviation, which is usually easier to interpret since it's back in the same units as the original data.
Example
For the data set 2, 4, 6: the mean is 4. The squared differences are (2−4)²=4, (4−4)²=0, (6−4)²=4. The population variance is the average of those: (4+0+4)/3 = 2.67.
Vertex Form
Vertex form writes a quadratic equation as y = a(x − h)² + k, where (h, k) is the vertex (the parabola's highest or lowest point) and a controls how wide or narrow it opens. It's the easiest form to graph from directly, since the vertex is visible without any extra calculation.
Example
y = 2(x − 3)² + 1 has its vertex at (3, 1). At x = 3, the squared term becomes 0, leaving y = 1, the parabola's minimum value since a = 2 is positive.
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Weighted Average
A weighted average multiplies each value by its own weight (its relative importance), adds those products together, and divides by the total weight — instead of treating every value equally like a regular average does. It's used whenever some data points should count more, like a course grade where the final exam is worth more than a homework assignment.
Example
A grade of 90 weighted 20% and a grade of 70 weighted 80% gives a weighted average of (90×0.2) + (70×0.8) = 18 + 56 = 74, not the plain average of 80.
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Z-Score
A z-score converts a raw value into a standardized measure of how far it sits from the mean, expressed in standard deviations. A z-score of 0 means the value equals the mean; positive means above it, negative means below it. Z-scores make it possible to compare values from different data sets that use different scales.
Example
A test has a mean of 70 and a standard deviation of 10. A score of 85 has a z-score of (85−70)/10 = 1.5, meaning it's 1.5 standard deviations above average.
Try it: Z-Score Calculator