Arithmetic

Arithmetic (from Ancient Greek ἀριθμός (arithmós) ‘number’, and τική (tikḗ) ‘art, craft’) is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers — addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th century, Italian mathematician Giuseppe Peano formalized arithmetic with his Peano axioms (disputed), which are highly important to the field of mathematical logic today.
— Wikipedia

Addition (+)

Addition commutative (move) property: $5 + 3 =$ $3 + 5$ , order doesn’t matter.
Addition associative Property: $(2 + 3) + 4 =$ $2 + (3 + 4)$ , grouping doesn’t matter, and it’s useless in this example.
Addition identity element: $x + 0 =$ $x$ , zero is like adding nothing.

Multiplication (×)

Multiplication commutative Property: $6 \times 2 = 2 \times 6$
Multiplication associative Property: $(2 \times 3) \times 4 = 2 \times (3 \times 4)$
Multiplication identity Element: $x \times 1 = x$
Multiplication zero Property: Any number $\times 0 = 0$
Multiplication distributive property: $a \times (b + c) = (a \times b) + (a \times c)$
- Analogy: Like giving the same gift to each person at a party
- Example: $3 \times (2 + 4) = 3 \times 6 = 18$ OR $(3 \times 2) + (3 \times 4) = 6 + 12 = 18$

Subtraction (−)

Subtraction is not commutative: $8 - 3 \neq = 3 - 8$ (order matters!) TODO: rewrite
Subtraction identity: $x - 0 = x$ the same number

Division (÷)

Division splits quantities into equal parts or finds how many times one number fits into another, like dividing a pizza equally among friends.

Division is not commutative: $12 \div 3 \neq = 3 \div 12$
Division identity: $x \div 1 =$ the same number
Division zero rule: “Cannot” divide by zero (undefined)

Order of Operations, PEMDAS

Think of this as a recipe - you must follow the steps in order, here is the order: TODO: rewrite

Parentheses/(Brackets) - Handle what’s in containers first
Exponents/(Orders) - Handle powers and roots
MD - Multiplication and Division (left to right) - These are equals
AS - Addition and Subtraction (left to right) - These are equals
Memory aid: “Please Excuse My Dear Aunt Sally”

Modular Arithmetic

Operations performed on remainders when numbers are divided by a fixed modulus.

Simplified: Think of modular arithmetic like a clock. When it’s 11 o’clock and you add 3 hours, you get 2 o’clock (not 14 o’clock). The clock “wraps around” at 12.

Example: $17 \equiv 2 (mod 5)$ means 17 divided by 5 leaves remainder 2.

Exponent Rules

Think of exponents as “how many times you multiply”:

Same base, add exponents when multiplying: $x^{3} \times x^{2}$ = $x^{5}$ (like stacking building blocks)
Same base, subtract exponents when dividing: $x^{5} \div x^{2}$ = $x^{3}$ (like removing blocks)
Power of a power, multiply exponents: $(x^{2})^{3}$ = $x^{6}$ (like boxes within boxes)

Logarithmic Properties

Inverse exponential function manipulation rules. Logarithms are like “undo buttons” for exponents. If exponents ask “what do you get?”, logarithms ask “what power do you need?”. $lo g_{2} 1024$ = $2$ .

Key Rules:

$lo g (a \times b)$ = $lo g (a) + lo g (b)$ (multiplication becomes addition)
$lo g (\frac{a}{b})$ = $lo g (a) - lo g (b)$ (division becomes subtraction)

Complex Number Arithmetic

Operations on numbers containing real and imaginary components. Think of complex numbers like coordinates on a map - they have two parts: horizontal (real) and vertical (imaginary). Just like you can add two locations by combining their coordinates.

TODO: verify in wolfram alpha Example: $(3 + 2 i) + (1 + 4 i)$ = $(3 + 1) + (2 + 4) i = 4 + 6 i$

Factorial Properties

Product sequences of consecutive positive integers. Factorials are like a multiplication countdown. $5! = 5 \times 4 \times 3 \times 2 \times 1$ = == $120$ ==. It’s like asking “if you have 5 different books, in how many ways can you arrange them on a shelf?”