UPCAT Math

Complex Numbers

Numbers in the form a + bi, where i = √-1.

Real Numbers

Quantities on the number line, including all magnitudes.

Rational Numbers

Numbers expressible as a fraction a/b, b ≠ 0.

Irrational Numbers

Numbers that cannot be expressed as fractions.

Integers

Whole numbers including negatives, denoted as ℤ.

Whole Numbers

Natural numbers plus zero, denoted as 𝕎.

Natural Numbers

Counting numbers starting from 1, denoted as ℕ.

Even Numbers

Numbers divisible by 2, ending in 0, 2, 4, 6, 8.

Odd Numbers

Numbers not divisible by 2, ending in 1, 3, 5, 7, 9.

Terminating Decimals

Decimals that end after a finite number of digits.

Repeating Decimals

Non-terminating decimals that repeat a pattern.

Non-repeating Decimals

Decimals that do not repeat, like π and e.

Factors

Numbers that divide another number without a remainder.

Multiples

Products of a number and integers.

Commutative Property

Order of addition/multiplication does not change the result.

Associative Property

Grouping of numbers does not change the result.

Distributive Property

a(b + c) = ab + ac, distributes multiplication over addition.

Identity Property

Adding 0 or multiplying by 1 leaves numbers unchanged.

PEMDAS

Order of operations: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.

Imaginary Numbers

Square roots of negative numbers, represented as bi.

Zero

Considered both a real and imaginary number.

Cartesian Coordinate System

A plane defined by x and y axes.

Polynomial Equations

Equations involving variables raised to whole number powers.

Quadratic Equations

Polynomial equations of degree 2, in the form ax² + bx + c = 0.

Prime Number

Whole number greater than 1 with two factors.

Composite Number

Whole number greater than 1 with more than two factors.

Factoring

Rewriting a number as the product of its factors.

Prime Factorization

Factored form consisting only of prime numbers.

Greatest Common Factor (GCF)

Largest common factor of two numbers.

Least Common Multiple (LCM)

Smallest whole number multiple of two numbers.

Divisibility Rule of 1

Every number is divisible by 1.

Divisibility Rule of 2

Even numbers are divisible by 2.

Divisibility Rule of 3

Sum of digits divisible by 3 indicates divisibility.

Improper Fractions

Numerator greater than or equal to denominator.

Mixed Numbers

Combination of whole number and proper fraction.

Equivalent Fractions

Different fractions representing the same value.

Numerator

Top part of a fraction indicating parts taken.

Denominator

Bottom part of a fraction indicating equal parts.

Additive Identity

Adding zero to a number yields the same number.

Multiplicative Identity

Multiplying by one yields the same number.

Common Factors

Factors shared by two or more numbers.

Common Multiples

Multiples shared by two or more numbers.

GCD

Greatest common divisor of two numbers.

Divisibility Rule of 4

Last two digits divisible by 4 indicates divisibility.

Divisibility Rule of 5

Last digit is 0 or 5 for divisibility.

Divisibility Rule of 6

Divisible by both 2 and 3 indicates divisibility.

Divisibility Rule of 7

Remove last digit, double it, subtract repeatedly.

Divisibility Rule of 8

Last three digits divisible by 8 indicates divisibility.

Divisibility Rule of 9

Sum of digits divisible by 9 indicates divisibility.

Divisibility Rule of 10

Last digit is 0 indicates divisibility by 10.

Divisibility Rule of 11

Difference of sums of alternate digits divisible by 11.

Divisibility Rule of 12

Divisible by both 3 and 4 indicates divisibility.

Divisibility Rule of 13

Add four times last digit to remaining number.

Similar Fractions

Fractions with the same denominator.

Dissimilar Fractions

Fractions with different denominators.

Proper Fractions

Numerator is less than the denominator.

Mixed Number

Whole number combined with a proper fraction.

Multiplication of Fractions

Multiply numerators and denominators together.

Division of Fractions

Multiply by the reciprocal of the divisor.

Example of 534

Even number and sum 12 shows divisibility by 6.

Example of 462

Process shows divisibility by 7 through subtraction.

Example of 4983

Difference of sums shows divisibility by 11.

Example of 1157

Process shows divisibility by 13 through addition.

Original Price

Initial price before any discounts applied.

Sale Price

Final price after discounts are applied.

Discount Amount

Difference between original and sale price.

Principal (P)

Amount of money borrowed or loaned.

Rate of Interest (r%)

Percentage paid for borrowing money.

Percent to Decimal

Divide percent value by 100.

Decimal to Percent

Multiply decimal by 100, add percent symbol.

Fraction to Percent

Multiply fraction by 100, add percent symbol.

Percent to Fraction

Divide percent by 100, simplify if needed.

Improper Fraction

Numerator exceeds denominator in a fraction.

Ratio

Comparison of two quantities expressed as a fraction.

Proportion

Equality of two ratios, showing equivalence.

Terminating Decimal

Decimal that ends after a finite number of digits.

Repeating Decimal

Decimal with one or more repeating digits.

Base (B)

Amount being considered for percentage calculation.

Amount (A)

Total sum of interest and principal.

Time (t)

Duration for which money is borrowed.

Calculate Discount Percentage

Determine percentage reduction from original price.

Percentage

Part of a whole expressed in hundredths.

Example Calculation

$80 original price, $60 sale price, $20 discount.

Algebraic Methods

Techniques used to convert repeating decimals.

Parts Per Hundred

Concept underlying the definition of percentage.

Algebraic Expression

Mathematical phrase combining numbers and variables.

Ordered Pair

Two numbers representing a point in a plane.

Abscissa

The x-coordinate in an ordered pair.

Ordinate

The y-coordinate in an ordered pair.

Origin

Point where x-axis and y-axis intersect.

Polynomial Function

Function involving non-negative integer powers of x.

Quadratic Polynomial Function

Degree 2 polynomial: f(x) = ax² + bx + c.

Cubic Polynomial Function

Degree 3 polynomial: f(x) = ax³ + bx² + cx + d.

Linear Polynomial Function

Degree 1 polynomial: f(x) = ax + b.

Zero Polynomial Function

Polynomial of the form f(x) = 0.

Factoring Polynomials

Process of breaking down polynomials into factors.

Factoring by Grouping

Grouping terms to find common factors.

Trinomial

Polynomial with exactly three terms.

Degree of Polynomial

Highest power of the variable in a polynomial.

Constant Function

Function that always returns the same value.