Algebra

Algebra

The branch of mathematics that deals with the study of formal manipulations of equations involving symbols and numbers.

Number

An entity describing the quantity or position of a mathematical object or extensions of these concepts.

Cardinal Numbers

Describe the quantity or size of the collections of objects.

Ordinal Numbers

Refer to the position relative to an ordering (first, second, third, …).

Natural Numbers

Numbers considered as counting numbers (1, 2, 3, …).

Whole Numbers

Numbers from 0 to positive infinity, excluding fractions or decimals.

Integers

Whole numbers including negative numbers (negative integers, zero, and positive integers).

Rational Numbers

Numbers which can be expressed as a quotient of two integers (fractions); includes repeating and terminating decimals.

Irrational Numbers

Numbers which cannot be expressed as a fraction (e.g., √2, π, e).

Absolute Value

The numerical value of the number neglecting the sign; distance from zero on a number line.

Prime Numbers

Numbers greater than 1 that have exactly two distinct factors: 1 and itself.

Composite Numbers

Numbers greater than 1 that have more than two distinct factors.

Divisibility Test

Standard method used to identify if a number is prime or composite.

Simple Fraction

Numerator and denominator are both integers.

Proper Fraction

Numerator is smaller than the denominator.

Improper Fraction

Numerator is greater than the denominator.

Unit Fraction

Numerator is equal to one and denominator is any integer.

Mixed Number

Combination of an integer and a proper fraction.

Complex Fraction

Numerator and denominator are both fractions.

Similar Fraction

Two or more simple fractions that have the same denominator.

Reciprocal

A fraction that results in interchanging the numerator and denominator.

Zero Fraction

Fraction in which the numerator is 0; equal to zero.

Undefined Fraction

A fraction with a denominator of 0.

Indeterminate Fraction

A fraction which has no quantitative meaning (e.g., 0/0).

Least Common Multiple (LCM)

The smallest integer that is a multiple of each of the given numbers.

Least Common Denominator (LCD)

The smallest common multiple of the denominators of a set of fractions.

Greatest Common Factor (GCF)

The largest integer which is a factor of each of the given numbers.

Significant Figures

Digits that define the numerical value of a number, beginning with the first non-zero digit.

Rounding

Replacing a number by another of approximately the same value but with fewer digits.

Truncating

Shortening a number by dropping digits; always rounds down.

Dimensional Analysis

A method of setting up problems that involves converting between different units of measurement.

Exponent Law: Product

(am)(an) = am+n

Exponent Law: Quotient

am / an = am-n

Exponent Law: Power of a Power

(am)n = am*n

Exponent Law: Power of a Product

(ab)m = am * bm

Exponent Law: Power of a Quotient

(a/b)m = am / bm

Exponent Law: Fractional Exponent

am/n = n-root(am)

Exponent Law: Negative Exponent

a^-m = 1 / am

Exponent Law: Zero Exponent

a^0 = 1

Radical

An expression involving a root (index n, radicand a).

Surd

An irrational number expressed as a root of an integer.

Radical Law: Product

n-root(a) * n-root(b) = n-root(ab)

Radical Law: Quotient

n-root(a) / n-root(b) = n-root(a/b)

Radical Law: Root of a Root

m-root(n-root(a)) = mn-root(a)

Logarithm Definition

log_a(b) = x is equivalent to a^x = b.

Common Logarithm

Logarithm with base 10 (Briggsian logarithm).

Natural Logarithm

Logarithm with base e (Napierian logarithm).

Euler's Number (e)

Mathematical constant approximately 2.71828.

Log Law: Product

log(xy) = log x + log y

Log Law: Quotient

log(x/y) = log x - log y

Log Law: Power

log(x^n) = n * log x

Log Law: Change of Base

log_b(x) = log x / log b

System of Equations

A set of two or more equations involving the same variables solved simultaneously.

Substitution Method

Solving one equation for a variable and substituting it into the other.

Elimination Method

Adding or subtracting equations to eliminate one variable.

Quadratic Equation Standard Form

Ax^2 + Bx + C = 0

Quadratic Formula

x = [-B ± sqrt(B^2 - 4AC)] / 2A

Discriminant

B^2 - 4AC; determines the nature of roots.

Nature of Roots: B^2

4AC = 0 - Roots are real and equal.

Nature of Roots: B^2

4AC > 0 - Roots are real and unequal.

Nature of Roots: B^2

4AC < 0 - Roots are complex and unequal (no real roots).

Sum of Roots (Quadratic)

r1 + r2 = -B / A

Product of Roots (Quadratic)

r1 * r2 = C / A

Special Product: Difference of Two Squares

x^2 - y^2 = (x + y)(x - y)

Special Product: Square of Binomial

(x ± y)^2 = x^2 ± 2xy + y^2

Special Product: Cube of Binomial

(x ± y)^3 = x^3 ± 3x^2y + 3xy^2 ± y^3

Special Product: Sum/Difference of Two Cubes

x^3 ± y^3 = (x ± y)(x^2 ∓ xy + y^2)

Special Product: Square of Trinomial

(x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2xz + 2yz

Polynomial Function

A function involving only non-negative integer powers of x.

Remainder Theorem

If f(x) is divided by (x - k), the remainder is f(k).

Factor Theorem

If (x - k) is a factor of f(x), then f(k) = 0.

Binomial Theorem Expansion

(x + y)^n = sum from k=0 to n of [nCk * x^(n-k) * y^k]

Number of terms in (x + y)^n

n + 1

rth term of Binomial Expansion

nCr-1 * x^(n-r+1) * y^(r-1)

Sum of Coefficients in (Ax + By)^n

(A + B)^n

Sum of Exponents in (x^a + y^b)^n

n(n+1)/2 * (a+b) [Simplified: n(n+1) for linear]

Inverse Function

A function that "undoes" another, taking output y back to input x.

Mixture Problem Formula

V1C1 + V2C2 = VfCf

Direct Variation

y = kx

Inverse Variation

y = k/x

Joint Variation

y = kxz

Combined Variation

y = kz/x

Arithmetic Progression (AP)

A sequence where the difference between consecutive terms is constant.

nth term of AP

an = a1 + (n - 1)d

Arithmetic Mean

(a + b) / 2

Sum of Arithmetic Series

S = n/2 * (a1 + an) or S = n/2 * [2a1 + (n - 1)d]

Geometric Progression (GP)

A sequence where each term is found by multiplying the previous term by a constant ratio r.

nth term of GP

an = a1 * r^(n-1)

Geometric Mean

sqrt(ab)

Sum of Geometric Series

S = a1(1 - r^n) / (1 - r)

Sum of Infinite Geometric Series

S = a1 / (1 - r) where |r| < 1

Harmonic Progression

A sequence whose reciprocals form an arithmetic progression.

Clock Problem: Minute Hand vs Hour Hand

MH / HH = 12

Clock Problem: Angle between hands

theta = |30H - 5.5M|

Work Problem Formula

1/t1 + 1/t2 = 1/t_total

Complex Number Standard Form

z = a + bi

Imaginary Unit (i)

sqrt(-1); i^2 = -1, i^3 = -i, i^4 = 1

Permutation P(n, r)

n! / (n - r)!

Circular Permutation

(n - 1)!

Permutation with Identical Objects

n! / (r1! r2! … rk!)