Know the word. Understand the idea.
Clear definitions for 280 mathematical terms, each with an example and a common mistake to avoid. Explore algebra, geometry, trigonometry, statistics, and calculus at your own pace.
See these ideas in real-world lessonsShowing 280 of 280 terms.
Algebra
A symbol that represents a number that can vary or a value we have not yet found.
In C = 12 + 3n, n is the number of notebooks and C is the total cost in dollars.
A letter does not always mean an unknown to solve for; it can describe an input that changes.
Algebra
A value that stays fixed within the problem or relationship being studied.
In y = 4x + 9, the constant term is 9.
A constant is fixed for a particular model; a price can still change outside that model.
Algebra
A number or other factor multiplying a variable or a variable expression.
In 7x − 2, the coefficient of x is 7; in −x, it is −1.
Keep the sign with the coefficient. The coefficient in −5x is −5.
Algebra
A combination of numbers, variables, and operations that represents a value.
3n + 8 is an expression. When n = 4, its value is 20.
An expression alone does not state an equality; an equation does.
Algebra
A statement that two expressions have equal values.
Solving 4x + 6 = 30 gives x = 6, and 4(6) + 6 = 30 checks it.
Perform the same valid operation on both sides to preserve the equality.
Algebra
A comparison using less than, greater than, or their inclusive versions.
If each ticket costs $8 and you have $40, 8n ≤ 40 means n ≤ 5.
Reverse the inequality sign when multiplying or dividing both sides by a negative number.
Algebra
A rule that assigns exactly one output to each input in its domain.
The rule f(x) = x² gives one output for each real x, even though f(2) and f(−2) both equal 4.
Different inputs may share an output. One input cannot have two outputs in the same function.
Algebra
The set of allowed inputs for a function, including any restrictions imposed by the context.
For f(x) = 1/(x − 3), the real domain excludes 3.
Context can narrow an algebraic domain: the number of people cannot be −2 or 2.5.
Algebra
The set of output values a function actually produces over its domain.
For f(x) = x² with all real inputs, the range is all real numbers greater than or equal to 0.
The range is about outputs. It is different from the statistical range of a data set.
Algebra
The change in vertical coordinate divided by the change in horizontal coordinate between two points with different horizontal coordinates.
From (2, 5) to (6, 13), slope = (13 − 5)/(6 − 2) = 2.
Use the same point order in both subtractions. A vertical line has undefined slope, not zero slope.
Algebra
The point where a graph crosses the vertical axis, found by setting x equal to 0 when that input is allowed.
The graph of y = 3x + 12 has y-intercept (0, 12).
The constant term gives the y-coordinate, while the full intercept is a point.
Algebra
In school algebra, a function written as f(x) = mx + b whose graph is a straight line and whose rate of change is constant.
C(n) = 5n + 20 adds $5 for each extra item and starts at $20 when n = 0.
A graph can look nearly straight over a small interval without being a linear function.
Algebra
A symbol that indicates a power; for a positive integer exponent it counts repeated factors of the base.
2⁴ = 2 × 2 × 2 × 2 = 16; for nonzero a, a⁰ = 1.
2³ means 2 × 2 × 2, not 2 × 3.
Algebra
A function of the form f(x) = ax² + bx + c with a ≠ 0, whose graph is a parabola.
A(w) = 20w − w² is quadratic and has a maximum of 100 at w = 10.
The highest power must be 2 after simplification; x² − x² + 3x is not quadratic.
Algebra
A ratio expressed for one unit of a chosen quantity.
A 6 kg bag costing $15 has a unit price of $2.50 per kilogram.
Keep track of which unit is in the denominator: dollars per kilogram differs from kilograms per dollar.
Algebra
A ratio written per hundred, with a specified reference amount as its base.
15% of $80 is 0.15 × $80 = $12.
Percentage changes depend on their base; a 20% increase followed by a 20% decrease does not return to the starting value.
Geometry
An exact location with no length, width, or thickness in the mathematical model.
The point (3, 2) is three coordinate units right and two up from the origin.
A dot on paper has size, but the ideal mathematical point it represents does not.
Geometry
The part of a straight line between two endpoints, including both endpoints.
A segment from (0, 0) to (3, 4) has length 5 coordinate units.
A segment has finite length; a line extends without end in both directions.
Geometry
A measure of rotation between two rays that share an endpoint, usually stated in degrees or radians.
A right angle measures 90° or π/2 radians.
The lengths of the drawn rays do not determine the angle's measure.
Geometry
Distinct lines in the same plane that never intersect.
The lines y = 2x + 1 and y = 2x − 4 are parallel.
Equal slopes identify parallel nonvertical lines only when the lines are distinct; coincident lines are the same line.
Geometry
Lines that intersect at a right angle.
The lines y = 2x and y = −x/2 are perpendicular.
The negative-reciprocal slope rule needs special handling for horizontal and vertical lines.
Geometry
The total distance around the boundary of a two-dimensional shape.
A rectangle 8 m long and 3 m wide has perimeter 2(8 + 3) = 22 m.
Perimeter uses length units such as metres; area uses square units.
Geometry
The amount of a flat region covered, measured in square units.
A 4 m by 3 m rectangular floor has area 12 m².
Adding side lengths finds a perimeter, not an area.
Geometry
The amount of three-dimensional space occupied, measured in cubic units.
A rectangular box 2 m by 3 m by 4 m has volume 24 m³.
Surface area measures the outside covering and uses square units; volume uses cubic units.
Geometry
A segment from the centre of a circle to any point on the circle, or the length of that segment.
A circle with diameter 10 cm has radius 5 cm.
The radius is half the diameter, not half the circumference.
Geometry
A segment through the centre of a circle with both endpoints on the circle, or its length.
If r = 7 cm, then d = 2r = 14 cm.
A segment between two points on a circle is only a diameter when it passes through the centre.
Geometry
The distance around a circle, equal to 2πr or πd.
A wheel of radius 0.3 m has circumference 0.6π m, approximately 1.885 m.
πr² calculates the area enclosed by a circle, not its circumference.
Geometry
A relationship between shapes with equal corresponding angles and proportional corresponding lengths.
Triangles with side lengths 3, 4, 5 and 6, 8, 10 are similar with length scale factor 2.
Doubling every length multiplies area by 4, not by 2.
Geometry
In a right triangle, the squares of the two leg lengths sum to the square of the hypotenuse length: a² + b² = c².
Legs of 6 m and 8 m give a hypotenuse of √(36 + 64) = 10 m.
Use this formula for right triangles, and put the hypotenuse in the c position.
Geometry
The ratio of a new length to its corresponding original length when a figure is scaled uniformly.
Scaling a 4 cm segment to 10 cm gives a scale factor of 10/4 = 2.5.
Lengths scale by k, areas by k², and volumes by k³ for a positive scale factor k.
Trigonometry
A triangle with one 90° angle; its other two angles are acute and sum to 90°.
A triangle with angles 35°, 55°, and 90° is a right triangle.
A triangle cannot have two right angles in ordinary plane geometry.
Trigonometry
The side opposite the right angle in a right triangle; it is the longest side.
In a 5–12–13 right triangle, the hypotenuse is 13 units long.
The hypotenuse stays the same when you choose a different acute reference angle.
Trigonometry
For a chosen acute angle in a right triangle, the side across from that angle.
If an observer's angle of elevation is θ, the vertical rise above eye level is the opposite side.
Opposite depends on the reference angle; label the angle before choosing a ratio.
Trigonometry
For a chosen acute angle in a right triangle, the leg touching that angle; the hypotenuse is excluded.
For an angle of elevation measured from level ground, the horizontal distance is the adjacent leg.
Two sides touch an acute angle, but only the non-hypotenuse side is the adjacent leg in a trig ratio.
Trigonometry
For an acute angle in a right triangle, the ratio opposite/hypotenuse; on the unit circle, the y-coordinate at that angle.
If the opposite leg is 3 and the hypotenuse is 5, sin θ = 3/5 = 0.6.
Sine returns a ratio, not an angle. An inverse sine operation can recover an angle within its chosen range.
Trigonometry
For an acute angle in a right triangle, the ratio adjacent/hypotenuse; on the unit circle, the x-coordinate at that angle.
If the adjacent leg is 4 and the hypotenuse is 5, cos θ = 4/5 = 0.8.
cos(a + b) is generally not cos a + cos b.
Trigonometry
For an acute angle in a right triangle, the ratio opposite/adjacent; more generally, sin θ/cos θ when cos θ ≠ 0.
A rise of 6 m over a horizontal run of 120 m gives tan θ = 0.05.
Tangent is undefined at 90° plus any integer multiple of 180°.
Trigonometry
An angle unit equal to one three-hundred-sixtieth of a complete turn.
A quarter-turn is 90°.
Check calculator mode: tan(30°) differs from tan(30 radians).
Trigonometry
An angle measure defined by arc length divided by radius; a full turn is 2π radians.
180° = π radians, so 60° = π/3 radians.
Standard calculus formulas such as the derivative of sin x assume x is measured in radians.
Trigonometry
The angle measured upward from a horizontal line of sight to an object above the observer.
Looking at a tree top 30° above eye-level horizontal gives an angle of elevation of 30°.
Measure from the horizontal, not from the vertical, and account for the observer's eye height.
Statistics
The arithmetic average: the sum of the data values divided by their count.
The mean of 4, 6, and 11 is 21/3 = 7.
An extreme value can move the mean considerably; it need not describe a typical observation well.
Statistics
The middle value of ordered numerical data, or the mean of the two middle values when the count is even.
For 2, 5, 8, 13, the median is (5 + 8)/2 = 6.5.
Sort the values first. Their original order does not define the middle.
Statistics
A value with the greatest frequency in a data set; there can be more than one mode.
In 1, 2, 2, 3, 3, 7, the values 2 and 3 are both modes.
Do not assume every data set has one unique mode. Conventions for all-distinct data vary.
Statistics
The difference between the largest and smallest numerical observations.
The range of 3, 8, 9, and 15 is 15 − 3 = 12.
The range uses just two observations and does not show how the rest of the data are distributed.
Statistics
The subset of a population from which observations are collected.
Surveying 80 randomly selected students from a school of 900 produces a sample of 80.
A large sample can still be biased if the selection process systematically excludes some groups.
Statistics
The full group of people, objects, or outcomes that a study aims to describe.
To study commute times for a school's students, the population is all students at that school.
The population is defined by the research question, not automatically by everyone in a city or country.
Statistics
A number from 0 to 1 that measures how likely an event is under a specified model.
For a fair six-sided die, P(rolling an even number) = 3/6 = 1/2.
Favourable outcomes divided by total outcomes works directly only when the elementary outcomes are equally likely.
Statistics
Events for which knowing that one occurred does not change the probability of the other; P(A and B) = P(A)P(B).
Two flips of a fair coin are independent in the usual model, so P(two heads) = 1/2 × 1/2 = 1/4.
Mutually exclusive events with positive probability are not independent: one occurring prevents the other.
Statistics
A measure of spread based on squared distances from the mean, expressed in the same units as the data.
The population standard deviation of 2 and 4 is 1: the mean is 3 and the average squared distance is 1.
Population standard deviation divides by N inside the square root; the usual sample formula divides by n − 1.
Statistics
An association between variables; Pearson's correlation coefficient specifically measures the direction and strength of a linear relationship.
If each observation satisfies y = 2x + 1 and x varies, Pearson's correlation is +1.
Correlation alone does not show causation, and a value near zero does not rule out a nonlinear relationship.
Calculus
The value a function approaches as its input approaches a specified value, when that approach has a consistent result.
As x approaches 2, (x² − 4)/(x − 2) approaches 4, even though the expression is undefined at x = 2.
A limit can exist even when the function is not defined at the point.
Calculus
A function is continuous at a point when its value is defined there and equals its limit at that point.
The polynomial f(x) = x² + 1 is continuous at every real input.
Having a limit is not enough; the function value must also exist and match it.
Calculus
The instantaneous rate of change of a function, defined by the limit of average rates of change when that limit exists.
If s(t) = t² metres with t in seconds, s′(3) = 6 metres per second.
A derivative has output units divided by input units; it is not generally the same quantity as the original function.
Calculus
The process of finding a derivative using its definition or valid derivative rules.
Differentiating f(x) = 3x² + 5 gives f′(x) = 6x.
A constant term differentiates to zero, not to itself.
Calculus
An interior point in a function's domain where the derivative is zero or does not exist.
For f(x) = x², x = 0 is critical because f′(0) = 0; for f(x) = |x|, the derivative is undefined at 0.
A critical point need not be a maximum or minimum: x³ has a critical point at 0 with neither.
Calculus
A function value that is at least as large as all nearby function values in its domain.
For f(x) = −x², f(0) = 0 is a local maximum.
A local maximum need not be the greatest value over the whole domain.
Calculus
The greatest value a function actually attains anywhere on its specified domain.
For f(x) = x² on [−1, 2], the absolute maximum is f(2) = 4.
Check domain endpoints as well as interior candidates; solving f′(x) = 0 alone can miss the answer.
Calculus
A function whose derivative is the given function on an interval.
x² + C is an antiderivative of 2x for any constant C.
Include the arbitrary constant when describing a family of antiderivatives.
Calculus
A limit of sums that measures signed accumulation over an interval, when the limit exists.
The integral of a constant speed of 3 m/s from 0 to 4 s gives a distance of 12 m.
Signed area can cancel across the horizontal axis; a definite integral is not always total geometric area.
Calculus
The derivative rule for a composition: differentiate the outer function at the inner value, then multiply by the derivative of the inner function.
For f(x) = (2x + 1)³, f′(x) = 3(2x + 1)² × 2 = 6(2x + 1)².
Do not forget the derivative of the inside function.
Algebra
The value a digit contributes because of its position in a numeral; adjacent places in base ten differ by a factor of ten.
In 4,572, the 5 contributes 500, while the 7 contributes 70.
A digit's face value stays the same, but its contribution changes when its position changes.
Algebra
One of the symbols used to write numbers in a numeral system; decimal notation uses 0 through 9.
The numeral 3,030 contains four digits, including two zeros that hold places.
A digit and a whole number are different ideas: 24 is a two-digit number, not one digit.
Algebra
A way of writing a numeral as a sum that shows the contribution of each place.
6,204 = 6,000 + 200 + 4, and 2.35 = 2 + 0.3 + 0.05.
The 5 in 2.35 contributes five hundredths, not five tenths.
Algebra
Replacing a number with a nearby value at a chosen precision, according to a stated rule.
Using ordinary nearest-tenth rounding, 7.26 becomes 7.3.
State the requested place. Rounding to the nearest ten differs from rounding to the nearest tenth.
Algebra
An approximate value obtained from limited information or simplified calculations, useful for judging scale and plausibility.
Estimating 19 × 31 by 20 × 30 gives 600, close to the exact result 589.
An estimate is not an exact answer; show an approximation sign or describe the rounding.
Algebra
A base-ten representation with places to the right of the decimal point for tenths, hundredths, and smaller powers of ten.
0.47 means 4 tenths plus 7 hundredths, or 47/100.
0.5 is greater than 0.47 even though 47 has more digits than 5.
Algebra
The expression above the fraction bar; in a part-of-a-whole model, it counts the equal parts being considered.
In 3/8, the numerator 3 counts three eighth-sized parts.
The numerator alone does not determine a fraction's size; the denominator also matters.
Algebra
The nonzero expression below the fraction bar; in a part-of-a-whole model, it describes how many equal parts make one whole.
The denominator of 5/12 is 12, so each part is one twelfth of the whole.
Fractions need the same whole before a visual comparison is meaningful, even when their denominators match.
Algebra
Fractions with the same value, even when written with different numerators and denominators.
3/4 = 6/8 because multiplying both numerator and denominator by 2 preserves the ratio.
Adding the same number to the numerator and denominator generally changes a fraction's value.
Algebra
The multiplicative inverse of a nonzero number: the number that multiplies it to give 1.
The reciprocal of 3/5 is 5/3, since (3/5)(5/3) = 1.
Zero has no reciprocal. A negative number's reciprocal remains negative.
Algebra
A notation combining a whole number and a proper fraction to express their sum.
2 3/4 = 2 + 3/4 = 11/4.
The space in a mixed number indicates addition, unlike the multiplication implied by 2x.
Algebra
For nonnegative numerator and positive denominator, a fraction whose numerator is at least as large as its denominator.
11/4 is improper and equals 2 3/4; 4/4 is also improper under this convention.
An improper fraction is a valid number, not a mathematical error.
Algebra
A whole-number value, including negative whole numbers, zero, and positive whole numbers.
−4, 0, and 12 are integers; 3/2 is not.
Negative numbers can be integers. Integer does not mean positive only.
Algebra
For a real number, its distance from zero on the number line, always nonnegative.
|−7| = 7 and |7| = 7; the distance between −3 and 5 is |5 − (−3)| = 8.
Absolute value removes direction, so equal absolute values do not necessarily mean equal signed numbers.
Algebra
The number that adds to a given number to produce zero.
The additive inverse of −6 is 6 because −6 + 6 = 0.
The additive inverse changes the sign; the reciprocal instead produces 1 when multiplied.
Algebra
A comparison of two quantities by division, often written a:b or a/b with a nonzero second quantity.
A mixture with 2 cups of concentrate and 5 cups of water has concentrate-to-water ratio 2:5.
A part-to-part ratio of 2:5 makes the first part 2/7 of the total, not 2/5.
Algebra
An equation stating that two ratios have the same value.
3/5 = 12/20 is a proportion because both ratios equal 0.6.
Cross multiplication is valid only when the denominators are nonzero, and matching units must occupy matching positions.
Algebra
The convention for interpreting expressions: grouping first, then powers, then multiplication and division left to right, then addition and subtraction left to right.
18 − 6 ÷ 3 = 18 − 2 = 16.
Multiplication does not automatically precede division; operations at that level are evaluated left to right.
Algebra
A quantity multiplied by another to form a product; for positive integers, a factor divides the number without a remainder.
6 and 7 are factors of 42 because 6 × 7 = 42.
A factor is not the same as a multiple: 6 is a factor of 42, while 42 is a multiple of 6.
Algebra
A number obtained by multiplying a given number by an integer.
24 is a multiple of 6 because 24 = 6 × 4.
A number has many multiples. The smallest positive common multiple is a separate concept.
Algebra
A positive integer greater than 1 with exactly two positive factors: 1 and itself.
13 is prime because its only positive factors are 1 and 13.
1 is not prime, and 2 is prime even though it is even.
Algebra
A positive integer greater than 1 with a positive factor other than 1 and itself.
21 is composite because 21 = 3 × 7.
0 and 1 are neither prime nor composite under the usual positive-integer definitions.
Algebra
The largest positive integer that divides every integer in a given nonzero collection without a remainder.
The greatest common factor of 24 and 36 is 12.
A common factor can be correct without being the greatest common factor.
Algebra
The smallest positive integer that is a multiple of each given positive integer.
The least common multiple of 6 and 8 is 24.
Multiplying the numbers gives a common multiple, but not always the least one.
Algebra
The rule that multiplication distributes over addition or subtraction: a(b + c) = ab + ac.
7(10 + 3) = 70 + 21 = 91.
Multiply every term inside the grouping, including negative terms.
Algebra
Terms with exactly the same variable factors raised to the same powers, so their coefficients can be combined.
3x² + 5x² = 8x², while 3x² + 5x cannot combine into one term.
Matching the variable letter alone is insufficient; the exponents must also match.
Algebra
Replacing a variable or expression with an equal value or expression while preserving the surrounding operations.
If x = −2, then 3x² = 3(−2)² = 12.
Use parentheses around a substituted negative value, especially when powers are involved.
Algebra
An operation that reverses another on an appropriate domain, such as subtraction undoing addition.
To undo x + 9 = 14, subtract 9 from both sides and obtain x = 5.
Squaring is not one-to-one over all real numbers, so undoing a square can require both positive and negative possibilities.
Algebra
A value or collection of values that makes all requirements of an equation or system true.
x = 4 solves 3x − 1 = 11 because 3(4) − 1 = 11.
A value produced by algebra must still satisfy the original equation and any domain restrictions.
Algebra
The collection of every value that satisfies a stated equation, inequality, or system on its allowed domain.
Over the real numbers, the solution set of x² = 9 is {−3, 3}.
Finding one solution does not prove that all solutions have been found.
Algebra
Two or more equations whose shared unknowns must satisfy all equations at the same time.
x + y = 9 and x − y = 3 have the common solution x = 6, y = 3.
Solving each equation independently does not ensure a shared solution.
Algebra
A method that combines equations to remove one unknown, using valid equation operations.
Adding x + y = 9 to x − y = 3 removes y and gives 2x = 12.
When multiplying an equation before elimination, multiply both sides and every term.
Algebra
Two or more inequalities joined by a logical condition such as 'and' or 'or'.
2 ≤ x < 7 requires x to be at least 2 and less than 7.
'And' needs both conditions; 'or' allows either. Their solution sets can be very different.
Algebra
A way of describing intervals using endpoints, with brackets for included finite endpoints and parentheses for excluded endpoints.
[2, 7) means 2 ≤ x < 7.
Infinity is not an attained endpoint; use a parenthesis beside ∞ or −∞.
Algebra
Writing an output as f(x) to identify the function f and its input x.
If f(x) = 4x − 3, then f(5) = 17.
f(x) usually means the function's output, not f multiplied by x.
Algebra
A function that reverses a one-to-one function between its domain and range.
If f(x) = 3x + 2 over the reals, f⁻¹(y) = (y − 2)/3.
The inverse f⁻¹ is not the reciprocal 1/f. A domain restriction may be needed before an inverse exists.
Algebra
A sequence in which consecutive terms differ by a constant amount.
7, 11, 15, 19 is arithmetic with common difference 4.
Constant ratios describe geometric sequences, not arithmetic ones.
Algebra
The constant amount obtained by subtracting one term of an arithmetic sequence from the next.
For 20, 17, 14, 11, the common difference is −3.
Subtract earlier from later; reversing the order changes the sign.
Algebra
A sequence formed by multiplying each term by a fixed factor to obtain the next.
3, 6, 12, 24 is geometric with multiplier 2.
The terms themselves do not grow by a fixed addition unless the sequence is constant.
Algebra
The fixed multiplier between consecutive terms of a geometric sequence; it can be found by division when the earlier term is nonzero.
For 80, 40, 20, 10, the common ratio is 40/80 = 1/2.
Use next term divided by previous term, not the reverse.
Algebra
A relationship y = kx with constant k, so outputs scale in the same proportion as inputs.
At 3 litres per minute, V = 3t gives 15 litres after 5 minutes.
A fixed added fee creates y = kx + b, which is not direct variation when b is nonzero.
Algebra
A relationship y = k/x for nonzero x, so the product xy stays constant.
Sharing 60 markers equally gives m = 60/n markers per group when there are n groups.
Inverse variation is not the same as an inverse function or a relationship that merely decreases.
Algebra
The quantity repeatedly multiplied or raised to a specified exponent.
In (−3)⁴, the base is −3 and the result is 81.
Parentheses matter: −3² means −(3²) = −9, whereas (−3)² = 9.
Algebra
An exponent that indicates a reciprocal: a⁻ⁿ = 1/aⁿ for nonzero a and positive integer n.
2⁻³ = 1/8.
A negative exponent does not make the result negative; it changes which side of the fraction contains the power.
Algebra
A representation a × 10ⁿ where 1 ≤ |a| < 10 and n is an integer, for a nonzero real number.
0.00072 = 7.2 × 10⁻⁴.
A coefficient of 72 is not normalized scientific notation; adjust both coefficient and exponent together.
Algebra
A real function f(x) = abˣ with constant a ≠ 0, positive base b, and b ≠ 1.
f(n) = 5 × 2ⁿ gives 5, 10, 20, and 40 for inputs 0, 1, 2, and 3.
In an exponential function the variable is in the exponent; x² is instead a power function.
Algebra
A model in which a quantity is multiplied by the same factor between 0 and 1 during each equal input interval.
A value starting at 200 and retaining 80% each step is 200(0.8)ⁿ; after two steps it is 128.
Losing 20% each step differs from subtracting 20 units each step.
Algebra
The exponent needed for a positive base b other than 1 to produce a positive number: log_b(a) = c means bᶜ = a.
log₂(32) = 5 because 2⁵ = 32.
For real logarithms, arguments must be positive; log(a + b) is generally not log a + log b.
Algebra
A number whose square equals a given number; the symbol √a means the nonnegative root for real a ≥ 0.
√49 = 7, while the equation x² = 49 has two real solutions, 7 and −7.
The principal square-root symbol gives one nonnegative value; solving a squared equation may require ±.
Algebra
A finite sum of terms with constant coefficients and variables raised to nonnegative integer powers.
4x³ − 2x + 7 is a polynomial of degree 3.
Expressions with a variable in a denominator or a negative power are not polynomials in that variable.
Algebra
Rewriting an expression as a product of factors without changing its value.
x² + 7x + 12 = (x + 3)(x + 4).
Factoring an expression does not by itself mean setting its factors equal to zero; that requires a product equal to zero.
Algebra
For real or complex numbers, a product equals zero only if at least one factor equals zero.
(x − 2)(x + 5) = 0 gives x = 2 or x = −5.
Do not apply the rule when the product equals a nonzero value. It also fails in some modular systems.
Algebra
The solutions of ax² + bx + c = 0, with a ≠ 0, are x = (−b ± √(b² − 4ac))/(2a), interpreted in the relevant number system.
For x² − 3x − 4 = 0, x = (3 ± 5)/2 gives 4 and −1.
The entire numerator is divided by 2a; use parentheses in a calculator.
Algebra
The value b² − 4ac for a real-coefficient quadratic, which determines its real-root pattern.
For x² − 6x + 9, the discriminant is 36 − 36 = 0, so there is one repeated real root.
A negative discriminant means no real roots, not no complex solutions.
Algebra
A rectangular array of entries organized into rows and columns; its dimensions specify rows first, then columns.
[[2, 3, 1], [4, 0, 5]] is a 2 × 3 matrix.
A matrix's dimensions are not the values of its entries; a 2 × 3 matrix need not contain a 2 or a 3.
Algebra
For an m × n matrix A and n × p matrix B, the m × p product AB has entries formed from row-by-column sums of products.
[[2, 3], [1, 4]] times [[5], [2]] equals [[16], [13]].
Ordinary matrix multiplication is not entry-by-entry, and AB need not equal BA.
Algebra
A number a + bi with real a and b, where i² = −1; it can be plotted as the point (a, b) in a plane.
(3 + 2i) + (1 − 5i) = 4 − 3i.
Combine real and imaginary components separately; 3 + 2i is not 5i.
Algebra
The number i satisfying i² = −1, used with real numbers to form complex numbers.
i³ = −i and i⁴ = 1.
i is not a variable with an arbitrary value, and i² is −1 rather than 1.
Algebra
Arithmetic that identifies integers having the same remainder modulo a fixed positive integer.
17 ≡ 5 (mod 12), so advancing 17 hours has the same clock position effect as advancing 5 hours.
Congruence modulo 12 does not mean ordinary numerical equality: 17 and 5 differ by 12.
Algebra
A set with a closed associative binary operation, an identity element, and an inverse for every element under that operation.
The residues {0, 1, 2, 3} with addition modulo 4 form a group; the inverse of 1 is 3.
A group need not use multiplication or be commutative; the operation and set must both be specified.
Geometry
Assigning a numerical value to a quantity by comparing it with a specified unit.
A length of 1.2 m also measures 120 cm.
A measurement needs a unit; more written decimals do not guarantee greater accuracy.
Geometry
Expressing the same quantity in another unit using an equality between units.
Because 1 m = 100 cm, 2 m² = 2 × 100² cm² = 20,000 cm².
Square the length conversion factor for area and cube it for volume.
Geometry
A closed plane figure bounded by a finite chain of straight segments; here the boundary does not cross itself.
A simple pentagon has five sides and five vertices.
A circle has a curved boundary, so it is not a polygon.
Geometry
A corner where sides of a polygon or edges of a solid meet; plural: vertices.
A rectangle has four vertices; a rectangular box has eight.
A vertex is a point, while an edge has length.
Geometry
A segment joining two nonadjacent vertices of a polygon.
A quadrilateral has two diagonals; a pentagon has five.
A segment joining adjacent vertices is a side, not a diagonal.
Geometry
An angle whose measure is greater than 0° and less than 90°.
An angle of 37° is acute.
The endpoints are excluded: 0° and 90° are not acute.
Geometry
An angle whose measure is greater than 90° and less than 180°.
An angle of 120° is obtuse.
An angle greater than 180° is reflex, not obtuse.
Geometry
Two angles whose measures add to 90°.
The complement of 28° is 62°.
Complementary angles need not be adjacent in a diagram.
Geometry
Two angles whose measures add to 180°.
An angle supplementary to 115° measures 65°.
Supplementary angles need not have equal measures.
Geometry
The interior angles of a nondegenerate triangle in Euclidean plane geometry add to 180°.
Angles 48° and 67° leave 65° for the third angle.
This rule does not apply unchanged to great-circle triangles on a sphere.
Geometry
A Euclidean triangle with three equal sides; all three interior angles measure 60°.
An equilateral triangle of side 7 cm has perimeter 21 cm.
An imprecise drawing alone does not establish equality of side lengths.
Geometry
A triangle with at least two equal side lengths; the angles opposite equal sides are equal.
Vertex angle 40° in an isosceles triangle leaves base angles of 70° each.
Under this inclusive definition, an equilateral triangle is also isosceles.
Geometry
A triangle whose three side lengths are all different.
Sides 4, 5, and 6 form a scalene triangle.
Unequal positive lengths form a triangle only when every pair sums to more than the remaining length.
Geometry
A simple four-sided polygon, with interior angles summing to 360° in Euclidean geometry.
Angles 80°, 100°, and 110° leave a fourth angle of 70°.
Four sides do not imply parallel opposite sides.
Geometry
A quadrilateral with both pairs of opposite sides parallel.
Base 9 cm and perpendicular height 4 cm give area 36 cm².
A sloping side is not generally the perpendicular height.
Geometry
Using the inclusive convention here, a quadrilateral with at least one pair of parallel sides.
Parallel bases 4 m and 10 m separated by height 3 m enclose 21 m².
Some courses require exactly one parallel pair; check the stated convention.
Geometry
A parallelogram with all four side lengths equal.
Perpendicular diagonals of 6 cm and 8 cm enclose area 6 × 8/2 = 24 cm².
Equal sides do not require right angles; a square is a special rhombus.
Geometry
Area found by combining simpler regions, subtracting holes, and avoiding overlap counts.
Removing a 2 m by 1 m corner from a 6 m by 4 m rectangle leaves 22 m².
Adding overlapping regions without subtracting the overlap counts some area twice.
Geometry
A region bounded by two radii and their connecting circular arc.
A 90° sector of radius 4 cm has area 4π cm².
A sector includes the centre; a segment cut off by a chord generally does not.
Geometry
Distance along a circle, equal to rθ for radius r and central angle θ in radians.
Radius 3 m and angle π/2 give arc length 3π/2 m.
Convert degrees before using rθ.
Geometry
A segment whose endpoints both lie on a circle.
A chord subtending 60° at the centre of a radius-5 circle has length 5.
A chord is straight; its corresponding arc is curved.
Geometry
The total area of the specified surfaces of a three-dimensional object.
A closed cube of side 3 cm has surface area 54 cm².
State whether a container is open or closed before counting faces.
Geometry
A solid with congruent parallel polygonal bases joined by parallelogram side faces.
Base area 12 cm² and perpendicular base separation 5 cm give volume 60 cm³.
For an oblique prism, use perpendicular base separation rather than a sloping edge.
Geometry
A solid with congruent parallel circular bases; a right cylinder has its axis perpendicular to the bases.
A right cylinder of radius 2 cm and height 5 cm has volume 20π cm³.
Halve a diameter before inserting it into V = πr²h.
Geometry
A plane arrangement of surface pieces that folds into a solid's surface without overlapping interiors.
A 2 by 3 by 4 box has two faces each of areas 6, 8, and 12.
Having the right number of connected faces does not guarantee a valid net.
Geometry
Having the same shape and size, so a rigid motion can carry one plane figure onto the other.
Two triangles each with sides 3, 4, and 5 are congruent by side-side-side.
Equal areas alone do not establish congruence.
Geometry
A transformation that multiplies directed position vectors from a fixed centre by a scale factor.
Factor 3 about the origin sends (2, −1) to (6, −3).
Factor 3 multiplies area by 9, not by 3.
Geometry
A transformation adding the same displacement vector to every point.
Translation by (4, −2) sends (1, 5) to (5, 3).
Apply both coordinate changes to every vertex.
Geometry
A rigid transformation sending each point to its mirror image across a line.
Reflection across the y-axis sends (3, −2) to (−3, −2).
Reflecting across the y-axis changes the sign of x, not y.
Geometry
A rigid transformation turning points through a specified angle about a fixed centre.
A 90° counterclockwise turn about the origin sends (3, 2) to (−2, 3).
Specify centre and direction; reversing direction generally changes the image.
Geometry
The halfway point of a segment, found by averaging corresponding endpoint coordinates.
The midpoint of (2, 1) and (8, 5) is (5, 3).
Average x-coordinates together and y-coordinates together.
Geometry
In an orthonormal plane, point separation equals the square root of the sum of squared coordinate differences.
From (1, 2) to (4, 6), distance is √(3² + 4²) = 5.
Coordinate differences are displacement components, not the distance by themselves.
Geometry
Plane points whose distances to two fixed foci have a constant sum greater than the foci's separation.
For x²/25 + y²/9 = 1, semiaxes are 5 and 3 and foci are (±4, 0).
A denominator of 25 is a squared semiaxis, not a semiaxis of 25.
Geometry
A fixed point used in a distance-based definition of a conic such as an ellipse or parabola.
The parabola y² = 8x has focus (2, 0) and directrix x = −2.
A focus is not generally the centre or a point on the curve.
Geometry
The ratio e = c/a of focal distance from the centre to semimajor axis, with 0 ≤ e < 1.
An ellipse with a = 5 and c = 4 has eccentricity 0.8.
The full separation of the foci is 2c, not c.
Geometry
A curve made by intersecting a plane with a circular cone; nondegenerate examples include circles, ellipses, parabolas, and hyperbolas.
The circle x² + y² = 9 is a conic of radius 3.
A quadratic equation need not have real points; x² + y² = −1 has none.
Geometry
A directed displacement with magnitude and direction; components depend on chosen axes.
Vector (3, 4) has magnitude 5 in orthonormal coordinates.
Components can be negative even though magnitude is nonnegative.
Geometry
The sum of products of corresponding orthonormal components, also equal to the product of magnitudes times cosine of the included angle.
(2, 1) · (3, −4) = 6 − 4 = 2.
The dot product is a scalar, not a vector of component products.
Geometry
A path locally straight according to a surface's geometry; great circles are geodesics on an ideal sphere.
The equator is a spherical geodesic; the latitude circle at 45° north is not.
A long geodesic segment need not be globally shortest between its endpoints.
Geometry
For a simple triangle of shorter great-circle arcs on a sphere, its angle sum minus π radians; its area equals R² times this excess.
Three right angles give excess π/2 and area πR²/2.
Use radians for the excess in the area formula.
Trigonometry
Angle placement with vertex at the origin and initial ray along the positive x-axis.
A 90° angle in standard position ends on the positive y-axis.
An angle drawn elsewhere may be valid without being in standard position.
Trigonometry
The ray from which a directed angle begins its rotation.
The initial side of a standard-position angle is the positive x-axis.
A directed angle needs a starting ray as well as an ending ray.
Trigonometry
The ray where a directed angle's rotation ends.
Standard-position angles 45° and 405° share a terminal side.
An ending ray does not reveal how many full turns occurred.
Trigonometry
Standard-position angles sharing a terminal side, differing by an integer multiple of a full turn.
−30° and 330° differ by 360° and are coterminal.
The same terminal side does not require the same numerical angle measure.
Trigonometry
For a nonquadrantal angle, the acute angle between its terminal ray and the x-axis.
The reference angle of 150° is 30°.
Quadrant information still determines the signs of trigonometric values.
Trigonometry
An angle ending on a coordinate axis, with degree measure an integer multiple of 90°.
270° ends at unit-circle point (0, −1).
Some quadrantal angles have undefined tangent.
Trigonometry
The radius-1 circle centred at the origin, with angle-θ point (cos θ, sin θ).
At θ = π/3, the point is (1/2, √3/2).
Cosine is the x-coordinate and sine is the y-coordinate.
Trigonometry
Right triangles with 45°–45°–90° or 30°–60°–90° angle patterns and fixed side ratios.
Sides opposite 30°, 60°, and 90° have lengths k, k√3, and 2k.
The shortest side is opposite 30°, regardless of the drawing's orientation.
Trigonometry
A relationship expressing secant, cosecant, or cotangent as a reciprocal of cosine, sine, or tangent where defined.
If cos θ = 4/5, then sec θ = 5/4.
Reciprocal and inverse functions differ: sec θ is not arccos θ.
Trigonometry
The reciprocal of cosine, sec θ = 1/cos θ where cos θ ≠ 0.
sec 60° = 2.
Secant at 90° is undefined, not zero.
Trigonometry
The reciprocal of sine, csc θ = 1/sin θ where sin θ ≠ 0.
csc 30° = 2.
Cosecant pairs with sine, despite its similar spelling to cosine.
Trigonometry
The ratio cos θ/sin θ, defined where sin θ ≠ 0.
cot 45° = 1 and cot 90° = 0.
The form 1/tan θ requires tangent itself to exist; cos θ/sin θ is the broader definition.
Trigonometry
The function arcsin mapping [−1, 1] to the unique angle in [−π/2, π/2] with that sine.
arcsin(1/2) = π/6 = 30°.
The principal answer is not every solution of the original sine equation.
Trigonometry
The function arccos mapping [−1, 1] to the unique angle in [0, π] with that cosine.
arccos(−1/2) = 2π/3 = 120°.
The principal range differs from that of arcsin.
Trigonometry
The function arctan mapping a real ratio to the unique angle in (−π/2, π/2) with that tangent.
arctan(1) = π/4 = 45°.
arctan(y/x) alone does not identify an arbitrary point's correct quadrant.
Trigonometry
The single angle selected by an inverse trigonometric function's specified output interval.
arcsin(sin150°) = 30° because arcsin returns values from −90° to 90°.
An inverse trig function need not recover an original angle outside its principal interval.
Trigonometry
The equality sin²θ + cos²θ = 1, valid for every real angle θ.
If sin θ = 3/5, then cos²θ = 16/25.
A squared value determines magnitude, not sign; check the quadrant.
Trigonometry
A relationship between functions of complementary angles, such as sin θ = cos(π/2 − θ).
sin20° = cos70°.
Complementary angles sum to 90°, not 180°.
Trigonometry
A formula expressing a trigonometric function of a sum through functions of the component angles.
sin75° = sin45°cos30° + cos45°sin30° = (√6 + √2)/4.
Sine is not additive: sin(a + b) generally differs from sin a + sin b.
Trigonometry
An identity expressing a function of twice an angle through functions of the original angle.
sin60° = 2sin30°cos30° = √3/2.
sin(2θ) is generally not 2sinθ.
Trigonometry
An identity expressing squared sine or cosine of half an angle using cosine of the full angle.
sin²30° = (1 − cos60°)/2 = 1/4.
Taking a square root requires a sign consistent with the half-angle's quadrant.
Trigonometry
In a nondegenerate plane triangle, each side divided by the sine of its opposite angle has the same value.
If A = 30°, B = 60°, a = 5, then b = 5√3.
Match opposite pairs and check ambiguity when given two sides and a nonincluded angle.
Trigonometry
If C lies between sides a and b, the opposite side obeys c² = a² + b² − 2ab cos C.
For a = 3, b = 5, C = 60°, c = √19.
Use the included angle between the two multiplied sides.
Trigonometry
Two sides and a nonincluded angle may determine zero, one, or two plane triangles.
A = 30°, a = 7, b = 10 allow B ≈ 45.6° or 134.4°; both leave a positive third angle.
The first arcsin result can omit a second valid triangle.
Trigonometry
A triangle with no right angle; it may be acute or obtuse.
Angles 40°, 60°, and 80° form an acute oblique triangle.
Right-triangle side ratios do not apply directly to an arbitrary oblique triangle.
Trigonometry
The nonnegative distance |A| from midline to peak in A sin(Bx + C) + D or the corresponding cosine.
For y = 3sin x + 2, amplitude is 3 and range is [−1, 5].
Peak-to-peak distance is twice the amplitude.
Trigonometry
The smallest positive input shift repeating every value of a nonconstant periodic function.
sin(3t) has period 2π/3 when its argument is in radians.
A repeating shift may be a multiple of the smallest period.
Trigonometry
Cycles completed per unit time; a cycle duration T gives f = 1/T.
A period of 4 seconds gives 0.25 Hz.
Frequency and period are reciprocals, not interchangeable labels.
Trigonometry
The phase-change rate in radians per unit time for uniform oscillation, ω = 2πf.
A 2 Hz sinusoid has angular frequency 4π rad/s.
Divide the coefficient in sin(ωt) by 2π to obtain cycle frequency.
Trigonometry
The horizontal displacement h in A sin(B(x − h)) + D, with B nonzero.
sin(2(x − π/4)) shifts sin(2x) right by π/4.
Factor B first; sin(2x − π/2) shifts by π/4, not π/2.
Trigonometry
The horizontal line halfway between a sinusoid's maximum and minimum.
A wave ranging from 2 to 10 has midline y = 6.
A shifted sinusoid need not oscillate around zero.
Trigonometry
A scaled and shifted sine or cosine function with a linear argument.
4cos(πt) + 1 has amplitude 4 and period 2.
A periodic square wave is not itself a sinusoid.
Trigonometry
Plane coordinates giving a directed radius r and angle θ from a reference axis.
The polar pair (2, π/3) represents (1, √3) in Cartesian coordinates.
Coordinates are not unique; adding 2π to the angle preserves the point.
Trigonometry
The origin from which radial distances are measured in a polar system.
Every pair (0, θ) describes the pole.
The angle at the pole is not uniquely determined.
Trigonometry
The reference ray for polar angle measurement, usually the positive Cartesian x-axis.
Polar coordinates (3, 0) lie 3 units along the positive polar axis.
Changing the reference ray changes coordinates without moving the physical point.
Trigonometry
Equations making coordinates functions of a shared parameter, describing position and traversal together.
x = 2cos t, y = sin t trace x²/4 + y² = 1 for 0 ≤ t ≤ 2π.
Eliminating the parameter can lose direction, speed, or domain information.
Trigonometry
A sinusoidal component whose frequency is a positive integer multiple of a chosen fundamental frequency.
For a 2 Hz fundamental, the third harmonic has frequency 6 Hz.
Harmonic number describes frequency, not amplitude.
Trigonometry
The reciprocal of a signal's smallest positive repetition period, when such a period exists.
sin(2πt) + 0.2sin(6πt) has fundamental frequency 1 Hz.
The largest-amplitude component need not by itself identify the fundamental period.
Trigonometry
Forming a combined mathematical signal by adding component values at each input.
Components 2 and −0.5 combine to give 1.5 at that instant.
Add signed instantaneous values, not just component amplitudes.
Trigonometry
A finite sum of a constant and sine/cosine terms at integer-multiple frequencies relative to a chosen fundamental.
F(t) = sin t + sin3t/3 gives F(π/2) = 2/3.
A finite sum is not automatically an exact representation of a target waveform.
Calculus
The output change divided by the input change over a nonzero interval: [f(b) − f(a)]/(b − a). It describes the slope of a secant line.
For s(t) = t², the average velocity from t = 2 to t = 4 is (16 − 4)/2 = 6 distance units per time unit.
An interval average need not equal the rate at either endpoint.
Calculus
The limiting average rate over intervals shrinking toward one input, provided that limit exists. It describes local change rather than change over a finite interval.
For s(t) = t² + 2t metres, the instantaneous velocity at t = 2 seconds is 2(2) + 2 = 6 m/s.
Dividing the function value by the input usually does not give its instantaneous rate.
Calculus
The expression [f(x + h) − f(x)]/h for h ≠ 0. Its limit as h approaches zero defines the derivative when the limit exists.
For f(x) = x², the quotient simplifies to 2x + h, whose limit is 2x.
Do not substitute h = 0 before simplifying or evaluating the limit.
Calculus
A straight line passing through two distinct points on a curve. When their input coordinates differ, its slope gives an average rate of change.
The secant through (1, 1) and (3, 9) on y = x² has slope 4.
A secant slope uses two points; a tangent slope is obtained from local limiting behavior.
Calculus
For a differentiable function at x = a, the line y = f(a) + f′(a)(x − a) matching its value and first-order change at that input.
For f(x) = x² at a = 2, the tangent is y = 4 + 4(x − 2).
A tangent can cross its curve; touching without crossing is not its general definition.
Calculus
The value approached when inputs approach a point from only the left or only the right. A finite two-sided limit requires both one-sided limits to agree.
For f(x) = |x|/x, the left-hand limit at 0 is −1 and the right-hand limit is 1.
Finding just one side does not establish a two-sided limit.
Calculus
A description of function values growing without bound positively or negatively as inputs approach a point. Infinity is not a finite real function value.
As x approaches 0 from either side, 1/x² increases without bound, written as a limit of +∞.
The two sides of 1/x at zero have different signs and do not share the same infinite limit.
Calculus
The value a function approaches as its input grows without bound in a specified direction. A finite result can identify a horizontal asymptote.
As x tends to +∞, (3x + 1)/(x + 2) tends to 3.
Large inputs and inputs approaching a finite singularity are different limit situations.
Calculus
A point where a finite two-sided limit exists but the function is undefined or has a different value there. Redefining that single value can restore continuity.
(x² − 9)/(x − 3) has limit 6 at 3; assigning the value 6 fills its hole.
Cancellation simplifies values away from the excluded point; it does not automatically redefine the original function.
Calculus
A discontinuity where the two finite one-sided limits exist but differ. No single value at the point can make both approaches agree.
A function equal to 0 for x < 0 and 1 for x ≥ 0 jumps at zero.
Changing only the point value cannot repair a jump.
Calculus
The existence of a finite derivative at a point. For ordinary real functions at an interior point, differentiability implies continuity there.
|x| is continuous at 0 but is not differentiable there because its one-sided slopes are −1 and 1.
Continuity alone does not guarantee differentiability.
Calculus
The derivative rule d(xⁿ)/dx = nxⁿ⁻¹ wherever the real-valued power and derivative are defined. Domain restrictions matter for negative or fractional powers.
For x > 0, d(√x)/dx = 1/(2√x).
Reducing the exponent is only half the rule: multiply by the original exponent too.
Calculus
For differentiable functions u and v, the derivative of their product is u′v + uv′. Both changing factors contribute to the rate.
d(x²eˣ)/dx = 2xeˣ + x²eˣ.
The derivative of a product is generally not the product of the derivatives.
Calculus
For differentiable u and v with v ≠ 0, the derivative of u/v is (u′v − uv′)/v².
d[x/(x + 1)]/dx = 1/(x + 1)² for x ≠ −1.
Reversing the numerator subtraction reverses the sign of the answer.
Calculus
Differentiating an equation involving x and y while treating y as a differentiable function of x locally. Derivatives of y expressions need the chain rule.
From x² + y² = 25, 2x + 2yy′ = 0, so y′ = −x/y when y ≠ 0.
The derivative of y² with respect to x is 2yy′, not just 2y.
Calculus
The derivative of the first derivative, describing how a rate of change itself changes. Position differentiated twice gives acceleration.
If s(t) = 10t − t² metres, then s″(t) = −2 m/s².
Negative acceleration does not always mean decreasing speed; the velocity direction matters.
Calculus
A description of how slopes change across an interval. Where a second derivative exists, positive values indicate concave up behavior and negative values indicate concave down behavior.
For f(x) = x², f″ = 2, so its tangent slopes increase everywhere.
Increasing and concave up are different: a decreasing curve can still have increasing slopes.
Calculus
A point on a continuous graph where concavity changes across the point. A zero second derivative is a candidate, not by itself a conclusion.
x³ changes concavity at x = 0, whereas x⁴ does not despite f″(0) = 0.
Check a change in concavity instead of declaring every zero of f″ an inflection point.
Calculus
The local approximation L(x) = f(a) + f′(a)(x − a), obtained from the tangent line to a differentiable function at a.
For √x near 4, L(x) = 2 + (x − 4)/4; thus √4.04 is approximately 2.01.
Accuracy is local and depends on curvature; the approximation is not an identity.
Calculus
For a differentiable function, the first-order change dy = f′(x) dx associated with an input increment dx. It approximates the actual output change for a small increment.
For y = x² at x = 3 and dx = 0.1, dy = 0.6; the actual change is 0.61.
The differential need not equal the finite change Δy.
Calculus
A sum of function values at chosen sample points multiplied by subinterval widths. Under suitable conditions, refining the partition makes these sums approach the definite integral.
For f(x) = x on [0, 2], two unit-width right-endpoint rectangles give 1 + 2 = 3, while the integral is 2.
A finite sum is generally an approximation, and the choice of sample points matters.
Calculus
An ordered finite set of points dividing an interval into subintervals. Their widths need not be equal when constructing integration sums.
The points 0, 1, and 3 partition [0, 3] into widths 1 and 2.
Do not multiply every height by the same width unless the partition is uniform.
Calculus
The function being integrated. Together with the differential and bounds, it determines the quantity and units being accumulated.
In ∫₀⁴(2 + 3t) dt, the integrand is 2 + 3t.
The integrand is not the resulting antiderivative or the final integral value.
Calculus
The family of antiderivatives of a function on an interval, customarily written with an arbitrary additive constant.
∫3x² dx = x³ + C.
An indefinite integral is a family of functions; a definite integral has specified bounds and gives a number when it exists.
Calculus
A theorem linking differentiation and integration: a continuous integrand is recovered by differentiating its accumulation function, and a definite integral can be evaluated using an antiderivative.
For A(x) = ∫₀ˣ(1 + t) dt, A′(x) = 1 + x; ∫₀²(1 + t) dt = 4.
A variable upper bound other than x requires the chain rule.
Calculus
A change of variable that reverses the chain rule, transforming an integral together with its differential and, for a definite integral, its bounds.
With u = x², ∫₀¹2x cos(x²) dx = ∫₀¹cos u du = sin 1, using radians.
Changing the integrand without changing the differential leaves an inconsistent integral.
Calculus
The integration rule ∫u dv = uv − ∫v du, derived from the product rule. A useful choice makes the remaining integral easier.
∫x eˣ dx = xeˣ − eˣ + C.
Track the minus sign and verify the result by differentiation.
Calculus
An integral defined through a limit because an interval is unbounded or the integrand is unbounded. It converges only when the required limits are finite.
∫₁^∞ x⁻² dx = 1, but ∫₁^∞ x⁻¹ dx diverges.
Treating infinity as an ordinary endpoint can hide divergence.
Calculus
An infinite sum whose sequence of partial sums approaches a finite limit. The individual terms tending to zero is necessary but not sufficient.
Σ from n = 1 to ∞ of 1/2ⁿ converges to 1, while Σ1/n diverges.
Do not infer convergence merely because the terms approach zero.
Calculus
A series a + ar + ar² + … with constant ratio r. For |r| < 1 its sum is a/(1 − r).
3 + 1.5 + 0.75 + … sums to 3/(1 − 0.5) = 6.
The infinite-sum formula requires |r| < 1; a finite geometric sum has a different formula.
Calculus
The power series built from a function's derivatives at a center: Σf⁽ⁿ⁾(a)(x − a)ⁿ/n!. Equality to the function requires the remainder to approach zero.
At a = 0, eˣ = 1 + x + x²/2! + … for every real x.
Having derivatives of every order does not alone guarantee equality with the Taylor series.
Calculus
The distance R from a power series center within which the series converges absolutely. Outside that distance it diverges; endpoints require separate checks.
Σxⁿ has radius 1 and converges to 1/(1 − x) for |x| < 1.
The radius does not tell you whether either boundary point is included.
Calculus
An upper bound on the error after truncating a Taylor polynomial. If |f⁽ⁿ⁺¹⁾| ≤ M between the center and input, the error is at most M|x − a|ⁿ⁺¹/(n + 1)!.
Approximating e^0.1 by 1 + 0.1 + 0.1²/2 has error below 0.000185 because e^t < 1.106 on [0, 0.1].
Use a derivative bound over the whole connecting interval, not just at the center.
Calculus
An equation relating an unknown function to one or more of its derivatives. Solutions are functions satisfying the relation over a stated domain.
M′ = 0.2M is solved by M(t) = Ce^0.2t for constant C.
A derivative equation usually does not determine one unique solution without additional conditions.
Calculus
A specified function value, or derivative value, at a starting input used to select a solution of a differential equation.
M′ = 0.2M with M(0) = 50 selects M(t) = 50e^0.2t.
Substitute the condition into the full solution, including any arbitrary constant.
Calculus
A first-order equation expressible as y′ = g(x)h(y), so variables can be separated where division by h(y) is valid. Equilibrium solutions must be checked separately.
For y′ = 2xy and y ≠ 0, dy/y = 2x dx gives y = Ceˣ²; y = 0 is also a solution.
Dividing by a function of y can discard constant solutions where that function is zero.
Calculus
The rate of change of a multivariable function with respect to one input while its other inputs are held fixed.
For T(x, y) = 20 + x² + 2y², Tₓ = 2x and Tᵧ = 4y.
Hold the other independent variables fixed; do not differentiate all inputs as if they depended on x.
Calculus
The vector of a scalar function's partial derivatives. For a differentiable function in Euclidean coordinates, it points toward the fastest local increase when nonzero.
For T = 20 + x² + 2y², ∇T(1, 2) = (2, 8).
The gradient is a vector, not just its magnitude; its direction is undefined when it is the zero vector.
Calculus
The rate of change of a scalar function per unit distance in a specified unit-vector direction. For a differentiable function it equals ∇f · u.
With gradient (2, 8) and u = (3/5, 4/5), the directional derivative is 38/5 = 7.6.
Normalize the direction vector first when the requested rate is per unit distance.
Calculus
An integral along a curve. A vector line integral ∫F · dr accumulates the component of a vector field along directed motion, such as work done by force.
For F = (2x, y) and r(t) = (t, 2t), 0 ≤ t ≤ 1, work = ∫₀¹6t dt = 3.
The path direction matters for vector line integrals; reversing it changes the sign.
Statistics
A recorded measurement or set of attributes for one unit in a study. The observational unit and measurement rules should be defined before analysis.
One bus's recorded delay of 3 minutes is an observation when each arriving bus is the unit.
Several measurements from one unit may be related and should not automatically be counted as independent units.
Statistics
A variable whose values identify groups or labels rather than measured numerical magnitudes. Categories may be unordered or have a meaningful order.
Transport mode can take the categories walk, bus, bike, and car.
Numerical codes assigned to categories do not automatically make arithmetic averages meaningful.
Statistics
A variable recording a numerical amount for which numerical differences have meaning. It may be a discrete count or a continuous measurement.
The number of visitors is a discrete count; a measured wait time is usually modeled continuously.
A number printed on an identifier is not necessarily a quantitative measurement.
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The number of observations in a stated value, category, or interval. Frequencies across nonoverlapping exhaustive categories add to the sample size.
If 3 of 8 buses have a two-minute delay, the frequency of that delay is 3.
A frequency is a count, whereas a relative frequency is a proportion.
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The fraction of observed cases in a category, calculated as its frequency divided by the total number of observations.
Three occurrences among eight observations have relative frequency 3/8 = 0.375.
An observed relative frequency estimates a model probability only with appropriate sampling and assumptions.
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An average calculated as Σwᵢxᵢ/Σwᵢ with nonnegative weights and positive total weight. It accounts for differing frequencies or importance assigned by the model.
Combining 2 scores of 80 and 3 scores of 90 gives (160 + 270)/5 = 86.
Averaging group means without accounting for group sizes can give the wrong overall mean.
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A cutoff describing a position in an ordered distribution, commonly associated with a specified percentage of values at or below it. Sample interpolation conventions can differ.
If a reported 90th-percentile wait is 12 minutes, it describes an upper cutoff for roughly 90% of waits under the stated method.
A percentile is a position in a distribution, not a percentage increase in the variable.
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One of the three cutoffs dividing an ordered distribution into four parts: the 25th percentile, median, and 75th percentile. Sample conventions should be stated.
Using medians of halves for 1, 2, 3, 4, 5, 6, 7, 8 gives Q₁ = 2.5 and Q₃ = 6.5.
Different accepted sample quartile conventions can produce different cutoffs in small data sets.
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The difference Q₃ − Q₁, measuring the spread of the middle portion of an ordered distribution. It is less sensitive to extremes than the full range.
If Q₁ = 2.5 and Q₃ = 6.5 minutes, the IQR is 4 minutes.
Do not subtract the minimum from the maximum; that gives the range.
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An observation unusually far from the main pattern according to a stated context or screening rule. It may represent an error or a valid unusual event.
With Q₁ = 10 and Q₃ = 14, the 1.5-IQR upper fence is 20, so 22 is flagged by that rule.
Being flagged does not justify deleting an observation without investigating its meaning.
Statistics
A measure of spread using squared deviations from the mean. Population variance averages those squared deviations; the usual sample estimator divides their sum by n − 1.
For the complete population 8, 10, 12, variance is (4 + 0 + 4)/3 = 8/3.
Variance has squared measurement units and is not the same quantity as standard deviation.
Statistics
A numerical function of a random outcome, equipped with a probability distribution. Its value is uncertain before the outcome is observed.
The number of false alarms among five independent trials is a random variable taking values 0 through 5.
A random variable is the rule assigning values, not just one observed result.
Statistics
The set of possible elementary outcomes for a random experiment. Probabilities require a model on these outcomes, which need not be equally likely.
Two ordered coin flips have sample space HH, HT, TH, TT.
The categories zero, one, and two heads are not equally likely for two fair independent flips.
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The probability of an event after restricting attention to another event known to occur: P(A|B) = P(A∩B)/P(B), provided P(B) > 0.
For a fair die, P(even | greater than 4) = 1/2 because only 5 and 6 remain possible.
P(A|B) generally differs from P(B|A).
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Events that cannot occur together in one trial, so their intersection is empty and the probability of that intersection is zero.
A single die roll cannot be both 2 and 5.
Mutually exclusive events with positive probabilities are not independent.
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The event that the original event does not occur. Its probability is one minus the original event's probability.
If P(no alarm) = 0.9⁵, then P(at least one alarm) = 1 − 0.9⁵.
The complement of at least one is none, not exactly one.
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A probability rule that updates an event's probability using evidence: P(A|B) = P(B|A)P(A)/P(B), when P(B) > 0.
With defect rate 0.02, flag sensitivity 0.9, and false-flag rate 0.05, P(defect|flag) = 0.018/0.067 ≈ 0.269.
Ignoring the base rate can greatly overstate how convincing a positive flag is.
Statistics
The probability or density of observed data regarded as a function of a model parameter. It compares how compatible parameter values are with the same observed data.
For 3 successes in 5 independent Bernoulli trials, the binomial likelihood is proportional to p³(1 − p)².
A likelihood is not automatically a probability distribution over parameter values.
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A probability assigned before incorporating the particular new evidence under consideration. In a Bayesian update it combines with the evidence likelihood.
A manufacturing defect probability of 0.02 can be the prior before a screening flag is observed.
A prior should have an explicit basis; it is not the test's sensitivity.
Statistics
A probability after updating prior information with specified evidence using a probability model. It depends on both the prior and the evidence mechanism.
In the defect-screen example, the posterior defect probability after a flag is 18/67 ≈ 26.9%.
Do not confuse a Bayesian posterior probability with a frequentist confidence level.
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The probability-weighted average of a random variable when the defining sum or integral exists. It describes a model average, not a guaranteed individual outcome.
For demand values 0, 1, 2 with probabilities 0.5, 0.3, 0.2, expected demand is 0.7 units.
An expected count can be fractional even though every observed count is a whole number.
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A random experiment with two designated outcomes, success and failure, whose probabilities are p and 1 − p. Success is simply the event being counted.
A sensor either gives a false alarm or does not on one defined test, with false-alarm probability 0.1.
Repeated Bernoulli trials are not necessarily independent or identically distributed.
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The distribution of a success count in a fixed number n of independent Bernoulli trials with the same success probability p.
For n = 5 and p = 0.1, P(X = 2) = 10(0.1)²(0.9)³ = 0.0729.
Changing probabilities, dependence, or stopping when a target count is reached can invalidate the binomial model.
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A count distribution with probabilities P(X = k) = e⁻λ λᵏ/k! for k = 0, 1, … and mean λ > 0. A homogeneous Poisson process is one model generating such interval counts.
With mean count λ = 2, P(X = 0) = e⁻² ≈ 0.1353.
Match λ to the interval length; two arrivals per hour means an expected one in half an hour.
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A continuous, symmetric bell-shaped probability distribution specified by its mean μ and positive standard deviation σ. Interval probabilities are areas under its density.
For X normal with mean 500 and SD 4, P(492 ≤ X ≤ 508) is approximately 0.9545.
Not every symmetric or real-world measurement distribution is normal.
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A standardized value z = (x − μ)/σ indicating the signed distance from a mean in standard-deviation units, with σ > 0.
A fill weight of 494 g when μ = 500 g and σ = 4 g has z = −1.5.
A z-score alone does not imply a normal probability model.
Statistics
The probability distribution of a statistic across repeated samples generated by the same sampling procedure. It differs from the distribution of individual observations.
For independent normal observations with SD 12 and n = 36, sample means have SD 12/√36 = 2.
A histogram of the raw observations is not a sampling distribution of their mean.
Statistics
The standard deviation of a statistic's sampling distribution, or an estimate of it. It describes sampling variability in the statistic.
For an independent sample mean with known population SD 12 and sample size 36, SE = 12/√36 = 2.
Standard deviation describes observation spread; standard error describes estimator variability.
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Under conditions such as independent identically distributed observations with finite nonzero variance, the standardized sample mean approaches a standard normal distribution as sample size grows.
Even with nonnormal observations, sufficiently large independent sample means may be approximately normal when the theorem's conditions and approximation are appropriate.
It does not make the original observations normal, and no universal sample size works for every population.
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An interval produced by a sampling procedure designed to cover a fixed population parameter with a stated long-run frequency under its assumptions.
With mean 82, known SD 12, and n = 36 normal independent observations, a 95% z-interval is 82 ± 1.96(2), or [78.08, 85.92].
A frequentist 95% interval does not assign 95% posterior probability to the fixed parameter after the interval is observed.
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The half-width in a symmetric estimate-plus-or-minus interval, usually a critical value multiplied by a standard error. It reflects the stated sampling model.
For a 95% z-interval with SE = 2, the margin is 1.96 × 2 = 3.92.
A margin of error does not automatically account for selection bias or inaccurate measurements.
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The statistical claim used to generate the reference distribution for a hypothesis test, often a specified parameter value or absence of a modeled effect.
H₀: μ = 100 specifies the mean used to calculate a test statistic's null distribution.
Failing to reject a null hypothesis does not prove it true.
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The competing statistical claim defining departures of interest from the null hypothesis. Its direction determines whether a test is one-sided or two-sided.
H₁: μ ≠ 100 calls for a two-sided test, while H₁: μ > 100 is one-sided.
Choose the direction before seeing the data rather than switching to obtain a smaller p-value.
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Assuming the null model and testing assumptions hold, the probability of a test statistic at least as extreme as the observed one in the directions specified by the alternative.
For a two-sided standard normal test with z = 2, p = 2P(Z ≥ 2) ≈ 0.0455.
A p-value is not the probability that the null hypothesis is true or that results occurred by chance.
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A threshold α chosen for a test's rejection rule, controlling its Type I error probability at the stated level under appropriate assumptions.
For a valid test at α = 0.05, reject when the p-value is at most 0.05.
Statistical significance does not measure effect size or practical importance.
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Rejecting the null hypothesis when it is true. A test's significance level controls this error probability under its null model.
A correctly calibrated α = 0.05 test falsely rejects about 5% of repeated true-null cases in the long run.
The significance level is not the probability that one particular rejection is wrong.
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Failing to reject a false null hypothesis. Its probability depends on the particular alternative, sample size, variability, and decision rule.
A small noisy sample can miss a real increase in a process mean.
There is no single Type II error probability until a specific alternative is specified.
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A method choosing model coefficients to minimize the sum of squared residuals. In simple linear regression it fits a line predicting a response from one predictor.
For (1, 2), (2, 3), (3, 7), the least-squares line is ŷ = −1 + 2.5x.
A fitted association does not establish a causal effect of changing x.
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The observed response minus the response predicted by a fitted model. Residual patterns help assess whether the model misses systematic structure.
At x = 2, the line ŷ = −1 + 2.5x predicts 4; an observed value 3 gives residual −1.
Do not reverse observed minus predicted; that reverses the residual sign.
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A stochastic sequence where the conditional distribution of the next state depends on the current state rather than the entire past. A time-homogeneous model uses fixed transition probabilities.
With transitions dry→wet 0.2 and wet→dry 0.3, a dry starting day gives a two-step wet probability of 0.8(0.2) + 0.2(0.7) = 0.3.
State probabilities are updated using a consistent row or column convention; rows and columns cannot be mixed.
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