Elements in Mathematics – Key Vocabulary

A comprehensive set of vocabulary flashcards covering fundamental definitions, laws, and concepts encountered in the lecture on Elements in Mathematics. Designed for quick review ahead of exams, they span algebraic laws, number theory, sequences and series, statistics, set theory, and more.

Even Symmetry

Property of a function where f(t) = f(–t); graph is symmetric about the y-axis.

Odd Symmetry

Property of a function where f(t) = –f(–t); graph is symmetric about the origin.

Significant Figures

Digits in a number that contribute to its precision, beginning with the first non-zero digit.

Napierian Logarithm

Another name for the natural logarithm, with base e ≈ 2.718.

Zeros of a Function

Roots of an equation where the function’s value equals zero.

Trivial Solution

Solution in which all variables equal zero.

Convergent Series

Infinite series whose partial sums approach a finite limit.

Divergent Series

Infinite series whose partial sums do not approach a finite limit.

Symmetric Axiom

Algebraic axiom stating that if a = b, then b = a.

Reflexive Axiom

Axiom stating any quantity equals itself, a = a.

Transitive Axiom

If a = b and b = c, then a = c.

Independent Events

Two events where occurrence of one does not affect the probability of the other.

Sinusoid

Curve whose equation’s second derivative equals the negative of the original function.

Law of Cosines

Relates sides of a triangle to the cosine of an included angle; used when all sides are known.

Determinant Zero Condition

A square matrix has determinant zero if two rows (or columns) are identical or proportional.

Quadrant Rule (ASTC)

Mnemonic: All Students Take Chemistry – signs of trig functions in quadrants I–IV.

Inverse of Cosecant

Sine function; sin θ is the reciprocal of csc θ.

Ogive

Graph of a cumulative frequency distribution.

Corollary

Statement that follows readily from a theorem with little or no additional proof.

Arithmetic Progression (AP)

Sequence whose successive terms differ by a constant difference.

Geometric Progression (GP)

Sequence whose successive terms have a constant ratio.

Harmonic Progression (HP)

Sequence whose reciprocals form an arithmetic progression.

Matrix

Rectangular array of numbers arranged in rows and columns.

Determinant

Single numerical value computed from a square matrix’s elements.

Binary Digits (Bits)

Base-2 numeral system digits (0 and 1); basis of the binary number system.

Rational Number

Number expressible as the quotient of two integers.

Irrational Number

Real number that cannot be expressed as a ratio of integers; decimal is non-terminating and non-repeating.

Transcendental Number

Real or complex number that is not a root of any non-zero polynomial equation with integer coefficients.

Roman Numeral MCMXCIV

Represents the decimal number 1994.

Like Terms

Algebraic terms that differ only in their numerical coefficients.

Lemma

Proved proposition used as a stepping-stone to a larger theorem.

Postulate

Statement accepted without proof as a basis for reasoning; synonymous with axiom in geometry.

Commutative Law of Multiplication

ab = ba; order of factors does not affect the product.

Monomial

Algebraic expression consisting of a single term.

Binomial

Algebraic expression consisting of exactly two terms.

Polynomial Degree

Highest sum of exponents in any term of the polynomial.

Proper Fraction

Fraction whose numerator’s absolute value is less than the denominator’s.

Improper Fraction

Fraction whose numerator’s absolute value is greater than or equal to the denominator’s.

Unit Fraction

Common fraction with numerator 1 and positive integer denominator.

Absolute Value

Non-negative distance of a real or complex number from zero.

Modulus (of a Complex Number)

Absolute value of a + bi, equal to √(a² + b²).

Variance

Mean of squared deviations from the mean; measure of spread.

Standard Deviation

Square root of variance; indicates dispersion around the mean.

Prime Number

Integer greater than 1 with exactly two positive divisors: 1 and itself.

Composite Number

Integer greater than 1 that has more than two positive divisors.

Perfect Number

Integer equal to the sum of all its proper divisors (e.g., 6).

Abundant Number

Integer whose sum of proper divisors exceeds the number itself.

Deficient Number

Integer whose sum of proper divisors is less than the number.

Amicable Numbers

Pair of integers each equal to the sum of the other’s proper divisors (smallest pair 220 & 284).

Twin Primes

Pair of prime numbers differing by 2 (e.g., 11 and 13).

Goldbach Conjecture

Every even integer greater than 2 can be expressed as the sum of two primes.

Mean Proportional

Geometric mean; number x such that a:x = x:b, hence x = √(ab).

Inequality

Mathematical statement that two expressions are not equal, using >, <, ≥, or ≤.

Conditional Inequality

Inequality true only for certain values of the variable(s).

Cramer's Rule

Method using determinants to solve systems of linear equations.

Radical Symbol (√ )

Symbol indicating the principal nth root of a quantity.

Radicand

Quantity under a radical sign whose root is being taken.

Index (of Radical)

Small number written above and to the left of the radical sign indicating the root’s order.

Surd

Irrational root expressed in radical form (e.g., √3).

Pure Surd

Surd containing no rational term, e.g., √2.

Mixed Surd

Surd that includes at least one rational term combined with irrational terms.

Quadratic Discriminant

Value B² – 4AC determining nature of roots of Ax² + Bx + C = 0.

Sequence

Ordered list of numbers following a rule.

Series

Sum of the terms of a sequence.

Fibonacci Numbers

Sequence where each term equals the sum of the two preceding terms (1, 1, 2, 3, 5…).

Triangular Numbers

Figurate numbers representable as dots forming an equilateral triangle (1, 3, 6, 10…).

Square Numbers

Integers that are squares of natural numbers (1, 4, 9, 16…).

Pentagonal Numbers

Figurate numbers forming pentagons (1, 5, 12, 22, 35…).

Pascal’s Triangle

Triangular arrangement of binomial coefficients.

Binomial Expansion

Expansion of (x + y)ⁿ; exponents of x decrease while those of y increase.

Permutation

Arrangement of objects in a specific order; order matters.

Combination

Selection of objects without regard to order.

Probability

Ratio of successful outcomes to total possible outcomes, between 0 and 1.

Range (Statistics)

Difference between the largest and smallest values in a data set.

Mean (Average)

Sum of data values divided by their number.

Median

Middle value of an ordered data set; 50 % below and above.

Mode

Most frequently occurring value in a data set.

Absolute Error

Magnitude of the difference between an approximate value and the true value.

Relative Error

Absolute error divided by the true value; expresses error as a fraction or percent.

Significant Digit

Digit that carries meaning contributing to measurement precision.

Leading Digit

First non-zero digit in a number from the left.

Function

Relation assigning exactly one output (y) to each input (x).

Scientific Notation

Expression of a number as m × 10ⁿ with 1 ≤ m < 10.

Factorial (n!)

Product of all positive integers from 1 to n inclusive.

Absolute Convergence

Series converges even when each term is replaced by its absolute value.

Conditional Convergence

Series converges but diverges when terms are replaced by their absolute values.

Least Common Denominator (LCD)

Least common multiple of denominators of several fractions.

Power Set

Set of all subsets of a given set, including the empty set and the set itself.

Venn Diagram

Pictorial representation of set relationships introduced by John Venn.

Equality Symbol (=)

Introduced by Robert Recorde in 1557 to denote equivalence of quantities.

Euler’s Number (e)

Mathematical constant approximately 2.718 281 828, base of natural logarithms.

Mersenne Numbers

Numbers of the form 2ᵖ – 1; prime values are called Mersenne primes.

Fermat Numbers

Numbers of the form 2^(2ⁿ) + 1.

Q.E.D.

Initialism for Latin “quod erat demonstrandum,” meaning ‘which was to be demonstrated,’ placed at the end of proofs.

Mathematical Induction

Proof technique establishing validity of a statement for all natural numbers.

Additive Identity

Number 0, since a + 0 = a.

Multiplicative Identity

Number 1, since a × 1 = a.

Additive Inverse

Number that when added to a given number yields zero (–a for a).

Multiplicative Inverse

Number that when multiplied by a given non-zero number yields 1 (1/a for a).