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Leaving Cert Maths glossary

100 key terms in plain English, grouped by strand and labelled Higher or Ordinary. The same definitions the H1 Owl app teaches, written from the published syllabus.

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Algebra 20 terms

Algebraic fractionHigher: A quotient of polynomial expressions, combined using a common denominator just like numerical fractions. For example 1/(x - 1) + 1/(x + 1) = 2x/(x² - 1).
CoefficientOrdinary: The number written in front of a variable that multiplies it. In 7x the coefficient is 7.
Completing the squareHigher: Rewriting a quadratic in the form a(x + h)² + k, for example x² + 6x + 2 = (x + 3)² - 7. This reveals the maximum or minimum value and is the method behind the quadratic formula.
DiscriminantHigher: For ax² + bx + c = 0 the discriminant is b² - 4ac. A positive value gives two distinct real roots, zero gives a repeated real root, and a negative value gives a pair of complex conjugate roots.
Equating coefficientsHigher: If two polynomials are equal for all values of x, the coefficients of matching powers must be equal. This is how unknowns are found in identities such as x³ - 1 = (x - 1)(ax² + bx + c).
EquationOrdinary: A statement that two expressions have the same value, joined by an equals sign. You solve it to find the unknown, for example x + 4 = 9 gives x = 5.
Expanding BracketsOrdinary: Multiplying every term inside a bracket by the term outside so the brackets can be removed. Example: 3(x + 2) = 3x + 6.
ExpressionOrdinary: A group of numbers, letters and operation signs with no equals sign, such as 3x + 2y - 5.
Factor theoremHigher: A polynomial f(x) has (x - k) as a factor exactly when f(k) = 0. For f(x) = x³ - 7x + 6, f(1) = 0, so (x - 1) divides f(x) evenly.
FactorisingOrdinary: Writing an expression as a product of its factors, which is the reverse of expanding. Example: x² + 5x = x(x + 5).
IdentityHigher: An equation that is true for every allowed value of the variable, such as (x + y)² = x² + 2xy + y². This is stronger than an ordinary equation, which may hold only for particular values.
InequalityOrdinary: A statement that compares two amounts using <, >, ≤ or ≥ instead of an equals sign. Example: 2x + 1 < 9 is true for every x below 4.
Linear EquationOrdinary: An equation where the highest power of the variable is 1, such as 2x + 3 = 11. Its graph is a straight line.
Modulus inequalityHigher: An inequality involving absolute value. Since |x - a| measures the distance from x to a, the statement |x - a| < r is equivalent to a - r < x < a + r; squaring both sides is a common HL technique.
PolynomialHigher: An expression built from whole number powers of a variable with constant coefficients, such as 2x³ - x + 4. Its degree is the highest power that appears, so this example has degree 3.
Quadratic EquationOrdinary: An equation that can be written as ax² + bx + c = 0, where the highest power of x is 2. It can have up to two solutions.
Quadratic formulaHigher: The roots of ax² + bx + c = 0 with a ≠ 0 are x = (-b ± √(b² - 4ac)) / (2a). It works even when factorising is awkward or the roots are not rational.
Simultaneous equationsHigher: A set of equations that must all hold at once, solved by elimination or substitution. At HL this includes three linear equations in three unknowns and mixed systems such as x + y = 5 with x² + y² = 13.
Simultaneous EquationsOrdinary: Two equations with two unknowns that must both be true at the same time. Solving them gives the one pair of values that works in both.
VariableOrdinary: A letter such as x or y that stands for a number we do not know yet or that can change.

Functions 20 terms

AntiderivativeHigher: A function whose derivative is the given function; the general antiderivative carries an arbitrary constant. For example ∫ x² dx = x³/3 + C, since d/dx (x³/3 + C) = x².
Axis of SymmetryOrdinary: A vertical line that splits the graph of a quadratic into two mirror image halves. It always passes through the turning point.
Bijective functionHigher: A function that is both injective (distinct inputs give distinct outputs) and surjective (every element of the codomain is reached). Exactly these functions have an inverse.
Chain ruleHigher: The rule for differentiating a composite function: if y depends on u and u on x, then dy/dx = (dy/du) × (du/dx). For example d/dx (sin(x²)) = 2x cos(x²).
Composite functionHigher: The result of applying one function and then another: f(g(x)) means g acts first, then f. Order matters; with f(x) = x² and g(x) = x + 1, f(g(2)) = 9 but g(f(2)) = 5.
Definite integralHigher: The integral of f(x) from x = a to x = b, giving the signed area between the curve and the x axis. It is evaluated as F(b) - F(a), where F is any antiderivative of f.
DerivativeHigher: The instantaneous rate of change of a function, defined as the limit of (f(x + h) - f(x))/h as h → 0. Geometrically it is the slope of the tangent; for example d/dx (x³) = 3x².
DomainOrdinary: The set of all input values that a function is allowed to take.
Exponential functionHigher: A function of the form f(x) = aˣ with a > 0, modelling growth when a > 1 and decay when 0 < a < 1. The base e ≈ 2.718 is special because d/dx (eˣ) = eˣ.
Exponential FunctionOrdinary: A function where the variable is in the power, such as f(x) = 3 × 2ˣ. Its graph climbs or drops faster and faster as x grows.
FunctionOrdinary: A rule that turns each input into exactly one output. Example: f(x) = 2x + 1 turns the input 3 into the output 7.
Inverse functionHigher: The function f⁻¹ that undoes f, so f⁻¹(f(x)) = x for every x in the domain. It exists only when f is bijective, and its graph is the reflection of the graph of f in the line y = x.
LimitHigher: The value a function approaches as its input approaches a given point, written lim f(x). A key HL example is lim (sin x)/x = 1 as x → 0, and the derivative itself is defined as a limit.
Point of inflectionHigher: A point on a curve where the concavity switches between concave up and concave down. There f″(x) = 0 and f″ changes sign; the origin on y = x³ is the standard example.
Quadratic FunctionOrdinary: A function of the form f(x) = ax² + bx + c. Its graph is a U shaped curve called a parabola, which opens upwards when a is positive.
RangeOrdinary: The set of all output values that a function actually produces from its domain.
Root of a FunctionOrdinary: An input value that makes the function equal to zero, so f(x) = 0. On a graph, the roots are where the curve crosses the x axis.
SlopeOrdinary: A measure of how steep a line is, found by dividing the rise by the run between two points. A slope of 3 means the line goes up 3 units for every 1 unit across.
Turning PointOrdinary: The point where a graph changes direction, from going down to going up or the other way round. For a quadratic it is the maximum or minimum point of the curve.
y-interceptOrdinary: The point where a graph crosses the y axis. You find it by putting x = 0 into the function.

Number 20 terms

AmortisationHigher: Repaying a loan by equal instalments, each covering the interest due plus part of the principal. With rate i per period over t periods, the instalment is A = P i (1 + i)ᵗ / ((1 + i)ᵗ - 1).
Argand diagramHigher: A plane for plotting complex numbers, with the real part on the horizontal axis and the imaginary part on the vertical axis. The number 3 + 4i corresponds to the point (3, 4).
Arithmetic sequenceHigher: A sequence with a constant difference d between consecutive terms, so the nth term is a + (n - 1)d. The sequence 5, 8, 11, 14, ... has a = 5 and d = 3.
Compound InterestOrdinary: Interest worked out on the original amount plus any interest already added. The final value after t years is F = P(1 + i)ᵗ.
ConjugateHigher: For a complex number z = a + bi, the conjugate is a - bi. Multiplying them gives the real number a² + b², and nonreal roots of a polynomial with real coefficients always occur in conjugate pairs.
De Moivre's theoremHigher: The result (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ, proved by induction for natural n at HL. It gives an efficient route to powers and to the n roots of a complex number.
Geometric seriesHigher: The sum of terms in which each term is the previous one multiplied by a fixed ratio r. The first n terms add to Sₙ = a(1 - rⁿ)/(1 - r) for r ≠ 1.
Highest Common FactorOrdinary: The largest number that divides evenly into two or more given numbers. Example: the HCF of 12 and 18 is 6.
IntegerOrdinary: Any whole number, whether positive, negative or zero, such as -3, 0 or 7.
Lowest Common MultipleOrdinary: The smallest number that two or more given numbers all divide into evenly. Example: the LCM of 4 and 6 is 12.
ModulusHigher: The size of a complex number z = a + bi, given by |z| = √(a² + b²), which is its distance from the origin on the Argand diagram. For example |3 + 4i| = 5.
Natural NumbersOrdinary: The positive counting numbers 1, 2, 3, 4 and so on.
PercentageOrdinary: A way of writing a part of an amount as so many out of 100. Example: 25% of 80 is 20.
Prime NumberOrdinary: A natural number greater than 1 whose only factors are 1 and itself. The first few primes are 2, 3, 5, 7 and 11.
Proof by inductionHigher: A proof method for statements about natural numbers: verify a base case, assume the statement for n = k, then deduce it for n = k + 1. HL uses it for series sums, divisibility and inequalities.
RatioOrdinary: A comparison of two or more quantities in a set order. Example: sharing 20 euro in the ratio 3:2 gives 12 euro and 8 euro.
Rational NumberOrdinary: A number that can be written as a fraction of two integers, such as 3/4 or -5. As a decimal it either stops or repeats a pattern.
Scientific NotationOrdinary: Writing a number in the form a × 10ⁿ, where a is at least 1 but less than 10 and n is an integer. Example: 4500 = 4.5 × 10³.
Sum to infinityHigher: The limiting value of a geometric series when |r| < 1, given by S∞ = a/(1 - r). It converts recurring decimals to fractions, for example 0.3 + 0.03 + 0.003 + ... = 1/3.
SurdHigher: An irrational root such as √2 kept in exact form rather than as a decimal. Denominators are rationalised using a conjugate, for example 1/(√5 - 2) = √5 + 2.

Geometry and Trigonometry 20 terms

CentroidHigher: The point where the three medians of a triangle meet; it divides each median in the ratio 2:1 measured from the vertex. Its coordinates are the averages of the coordinates of the vertices.
CircumferenceOrdinary: The distance all the way around a circle, given by 2πr where r is the radius.
Compound angle formulaHigher: An expansion of a trigonometric function of a sum or difference, such as sin(A + B) = sin A cos B + cos A sin B. Setting B = A produces double angle results like cos 2A = 1 - 2 sin²A.
Congruent TrianglesOrdinary: Triangles that are exactly the same shape and size, so their matching sides and matching angles are equal.
CosineOrdinary: For an angle in a right angled triangle, the length of the adjacent side divided by the length of the hypotenuse.
Cosine ruleHigher: In any triangle, a² = b² + c² - 2bc cos A, linking three sides and one angle. When A = 90° the formula reduces to the theorem of Pythagoras.
EnlargementHigher: A transformation that scales a figure from a centre by a factor k, keeping the shape similar to the original. Lengths are multiplied by k and areas by k².
HypotenuseOrdinary: The longest side of a right angled triangle, always found opposite the right angle.
MidpointOrdinary: The point exactly halfway between two points. On the coordinate plane you find it by averaging the x values and averaging the y values of the two points.
PerpendicularOrdinary: Describes two lines that meet at a right angle of 90°.
Perpendicular distanceHigher: The shortest distance from a point (x₁, y₁) to the line ax + by + c = 0, equal to |ax₁ + by₁ + c| / √(a² + b²). It is the standard test for a line being tangent to a circle.
RadianHigher: The angle at the centre of a circle for which the arc length equals the radius, so π radians = 180°. With θ in radians the arc length is s = rθ.
SectorHigher: The region of a circle bounded by two radii and the arc between them. With the angle θ in radians, its area is (1/2) r² θ and its arc length is rθ.
Similar TrianglesOrdinary: Triangles with the same shape but possibly different sizes. Their matching angles are equal and their matching sides are in the same ratio.
SineOrdinary: For an angle in a right angled triangle, the length of the opposite side divided by the length of the hypotenuse.
Sine ruleHigher: In any triangle, a/sin A = b/sin B = c/sin C. It solves triangles given two angles and a side, and leads to the ambiguous case when two sides and a non included angle are known.
TangentOrdinary: For an angle in a right angled triangle, the length of the opposite side divided by the length of the adjacent side.
Theorem of PythagorasOrdinary: In a right angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides, so a² + b² = c².
Trigonometric identityHigher: A relation between trigonometric functions valid for every angle, such as cos²θ + sin²θ = 1 or tan θ = sin θ / cos θ. Identities are used to simplify expressions and solve equations.
Unit circleHigher: The circle of radius 1 centred at the origin. The point reached after turning through angle θ is (cos θ, sin θ), which extends the trigonometric ratios to angles of any size.

Statistics and Probability 20 terms

Bernoulli trialHigher: A single experiment with exactly two outcomes, success with probability p and failure with probability 1 - p. Repeating n independent trials gives the binomial model P(X = r) = ⁿCᵣ pʳ (1 - p)ⁿ⁻ʳ.
Central Limit TheoremHigher: As the sample size n grows, the distribution of the sample mean approaches a normal distribution with mean μ and standard error σ/√n, regardless of the shape of the parent population.
Conditional probabilityHigher: The probability of A given that B has occurred, P(A|B) = P(A ∩ B)/P(B). Events A and B are independent exactly when P(A|B) = P(A).
Confidence intervalHigher: A range of plausible values for a population parameter built from a sample. A 95% interval for a proportion is p ± 1.96√(p(1 - p)/n), where p is the sample proportion and n the sample size.
CorrelationOrdinary: A measure of how strongly two sets of data are related on a scatter plot. Points lying close to a straight line that rises show strong positive correlation.
Correlation coefficientHigher: A number between -1 and 1 measuring the strength and direction of a linear relationship between paired data. Values near ±1 indicate a strong linear trend, and correlation alone never proves causation.
Expected valueHigher: The long run average of a random variable, E(X) = Σ x · P(X = x). A game is fair when the expected winnings equal zero.
HistogramOrdinary: A chart that displays grouped numerical data using bars with no gaps between them. The height of each bar shows how many values fall in that group.
Interquartile RangeOrdinary: The spread of the middle half of the data, found by subtracting the lower quartile from the upper quartile. Extreme values do not affect it.
MeanOrdinary: The average found by adding all the values and dividing by how many there are. Example: the mean of 2, 4 and 9 is 15 ÷ 3 = 5.
MedianOrdinary: The middle value when the data are arranged in order. If there are two middle values, take the average of the two.
ModeOrdinary: The value that appears most often in a set of data. A data set can have more than one mode or none at all.
Normal distributionHigher: A bell shaped distribution symmetric about its mean μ, with roughly 95% of values within 1.96 standard deviations of μ. Values are compared to tables after standardising to z scores.
P-valueHigher: The probability, assuming the null hypothesis is true, of observing a result at least as extreme as the one obtained. A p-value below 0.05 leads to rejecting H₀ at the 5% significance level.
ProbabilityOrdinary: A number between 0 and 1 that measures how likely an event is, where 0 means impossible and 1 means certain. Example: a fair coin lands on heads with probability 1/2.
Relative FrequencyOrdinary: An estimate of probability taken from an experiment, found by dividing the number of times an event happened by the total number of trials.
Sample SpaceOrdinary: The set of every possible outcome of an experiment. Example: for one roll of a die it is {1, 2, 3, 4, 5, 6}.
Standard deviationHigher: A measure of spread equal to the square root of the average squared deviation from the mean, σ = √(Σ(x - μ)²/n). It carries the same units as the data itself.
Standard DeviationOrdinary: A measure of how spread out the data values are around the mean. A small standard deviation means most values sit close to the mean.
Z-scoreHigher: The standardised value z = (x - μ)/σ, counting how many standard deviations an observation lies from the mean. A score of z = 1.5 sits 1.5 standard deviations above average.

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