Arithmetic – Complete Reference Guide

Number Systems

Key Definitions

Natural Numbers: Counting numbers $\{1, 2, 3, \dots\}$

Whole Numbers: Natural + $\{0\}$

Integers: Whole + negatives $\{\dots, -2, -1, 0, 1, 2, \dots\}$

Digits: The symbols $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$

Place Value

$352.148$

$= 300 + 50 + 2 + 0.1 + 0.04 + 0.008$

Values maximize by x10 moving left

Values decrease by $\div 10$ moving right

Basic Operations & Properties

The Four Operations

Operation	Terms	Example
Addition (+)	Addend + Addend = Sum	$5 + 3 = 8$
Subtraction (-)	Minuend - Subtrahend = Difference	$10 - 4 = 6$
Multiplication (x)	Factor x Factor = Product	$6 \times 7 = 42$
Division (÷)	Dividend ÷ Divisor = Quotient	$20 \div 5 = 4$

Properties

Commutative: Order doesn't change result (Add/Mult only)
$a+b=b+a$ | $a \cdot b = b \cdot a$

Associative: Grouping doesn't change result (Add/Mult only)
$(a+b)+c = a+(b+c)$

Identity:
Add: $a+0=a$ | Mult: $a \cdot 1 = a$

Distributive: $a(b+c) = ab + ac$

Zero Property: $a \cdot 0 = 0$

Order of Operations (PEMDAS)

Rules

1. Parentheses / Grouping Symbols

2. Exponents / Roots

3. M/D Multiplication & Division (Left to Right)

4. A/S Addition & Subtraction (Left to Right)

Caution: Multiplication does NOT always come before Division. They are equal priority - do whichever comes first from left to right!

Factors & Multiples

Prime vs Composite

Prime: Only 2 factors (1 and itself). Ex: 2, 3, 5, 7, 11

Composite: More than 2 factors. Ex: 4, 6, 8, 9

1 is neither prime nor composite.

Factor Tree (Prime Factorization)

$12 = 2 \times 2 \times 3 = 2^2 \times 3$

GCF & LCM

GCF (Greatest Common Factor): Largest factor shared.

LCM (Least Common Multiple): Smallest multiple shared.

Fractions

Parts of a Fraction

$\frac{\text{Numerator}}{\text{Denominator}} = \frac{\text{Parts we have}}{\text{Total equal parts}}$

Types of Fractions

Proper: Num < Den ($\frac{2}{3}$)

Improper: Num ≥ Den ($\frac{5}{3}$)

Mixed Number: Whole + Fraction ($1\frac{2}{3}$)

Operations

Adding/Subtracting: MUST have common denominator.
$\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}$

Multiplying: Multiply straight across.
$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$

Dividing: "Keep, Change, Flip" (Multiply by reciprocal).
$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$

Mixed to Improper

$A\frac{b}{c} = \frac{(A \times c) + b}{c}$

Decimals & Percents

Conversions

Fraction	Decimal	Percent
1/2	0.5	50%
1/4	0.25	25%
3/4	0.75	75%
1/5	0.2	20%
1/10	0.1	10%
1/100	0.01	1%

Percent of a Number

Convert % to decimal, then multiply.

Ex: $20\%$ of $50 = 0.20 \times 50 = 10$

Rounding

Find the place value.

Look at digit to the right:

$\geq 5$: Round Up

$< 5$: Round Down (Keep)

Ratios, Rates & Proportions

Ratio

Comparison of two quantities such as a:b

Forms: $a \text{ to } b$, $a:b$, $\frac{a}{b}$

Unit Rate

A rate with a denominator of 1.

Ex: 60 miles per 1 hour (60 mph).

$\frac{\text{Total Quantity}}{\text{Total Units}} = \text{Unit Rate}$

Proportion

Equation stating two ratios are equal.

$\frac{a}{b} = \frac{c}{d}$

Cross Product Property: $a \cdot d = b \cdot c$

Use cross multiplication to solve for unknown variables in proportions!

Measurement & Geometry

Perimeter & Area

Perimeter: Distance around outside.

Square: $P = 4s$

Rectangle: $P = 2l + 2w$

Area: Space inside.

Square: $A = s^2$

Rectangle: $A = l \times w$

US Customary System

1 ft = 12 in

1 yd = 3 ft

1 lb = 16 oz

1 gal = 4 qt = 8 pt = 16 cups

Metric System

Based on powers of 10

Kilo (1000)

Centi (1/100)

Milli (1/1000)

ARITHMETIC Complete Reference