Branch of mathematics

Arithmetic

The four operations and the order they run in — the foundation of all of maths.

Topics in this branch

Basic
Addition
Addition combines two or more numbers into one — the sum. Learn the names of the parts, the laws of addition (commutativity, associativity, identity element) and how to add in columns with carrying.
Basic
Subtraction
Subtraction is the inverse of addition — take the subtrahend away from the minuend to get the difference. Learn the names, the properties, the link to addition and how to subtract in columns with borrowing.
Basic
Multiplication
Multiplication is repeated addition of the same number. Learn the names of the factors and the product, the laws of multiplication (commutativity, associativity, distributivity), the times tables and column multiplication.
Basic
Division
Division is the inverse of multiplication — divide the dividend by the divisor to get the quotient. Learn the names, the link to multiplication, division with a remainder and why you must never divide by zero.
Basic
Order of operations
When several operations meet in one expression, we evaluate them in a fixed order: brackets first, then powers, then multiplication and division, and finally addition and subtraction. Learn the rules and practise them on examples.

Branch formulas

Branch: Arithmetic

Addition

  • Sum

    a+b=ca + b = c

    addend + addend = sum

  • Commutativity

    a+b=b+aa + b = b + a

    the order of addends does not change the result

  • Associativity

    (a+b)+c=a+(b+c)(a + b) + c = a + (b + c)

    the grouping of addends does not change the result

  • Identity element

    a+0=aa + 0 = a

    zero does not change the number

Subtraction

  • Difference

    a−b=ca - b = c

    minuend − subtrahend = difference

  • Check

    a−b=c  ⟺  b+c=aa - b = c \iff b + c = a

    check subtraction with addition

  • Subtracting zero

    a−0=aa - 0 = a

    zero leaves the number unchanged

  • Difference of equals

    a−a=0a - a = 0

    a number minus itself is zero

Multiplication

  • Product

    a⋅b=ca \cdot b = c

    factor · factor = product

  • Commutativity

    a⋅b=b⋅aa \cdot b = b \cdot a

    the order of factors does not change the result

  • Associativity

    (a⋅b)⋅c=a⋅(b⋅c)(a \cdot b) \cdot c = a \cdot (b \cdot c)

    grouping of factors does not change the result

  • Distributivity

    a⋅(b+c)=a⋅b+a⋅ca \cdot (b + c) = a \cdot b + a \cdot c

    multiplication distributes over addition

  • Identity element

    a⋅1=aa \cdot 1 = a

    multiplying by one leaves the number unchanged

  • Multiplying by zero

    a⋅0=0a \cdot 0 = 0

    a product with zero is always zero

Division

  • Quotient

    a:b=ca : b = c

    dividend : divisor = quotient (b ≠ 0)

  • Check

    a:b=c  ⟺  c⋅b=aa : b = c \iff c \cdot b = a

    check division with multiplication

  • Division with remainder

    a=b⋅q+ra = b \cdot q + r

    the remainder satisfies 0 ≤ r < b

  • Dividing by one

    a:1=aa : 1 = a

    dividing by one leaves the number unchanged

  • Dividing by itself

    a:a=1a : a = 1

    for a ≠ 0

Order of operations

  • Multiplication before addition

    2+3⋅4=142 + 3 \cdot 4 = 14

    first 3 · 4 = 12, then 2 + 12

  • Brackets first

    (2+3)⋅4=20(2 + 3) \cdot 4 = 20

    brackets change the order: first 2 + 3

  • Power before multiplication

    23⋅2=162^3 \cdot 2 = 16

    first 2³ = 8, then 8 · 2

  • Left to right

    20−4−3=1320 - 4 - 3 = 13

    operations of the same rank are done left to right

Related articles

Plus, minus, times and division signs and digits floating in a geometric grid, joined by emerald lines of the order of operations, against a cosmic nebula in violet and magenta
Arithmetic

The grammar of arithmetic — why the order of operations is a convention

Nobody invented +, −, × and ÷ overnight, and the fight over 6 ÷ 2(1+2) still splits the internet. Here is why the notation of arithmetic is a human convention, the results are absolute truth, and division by zero simply cannot be defined.

10 min