Operations

An operation is a rule, or set of rules, for processing one or more objects.

A binary operation is one which combines two objects to make a third. Addition, subtraction, multiplication and division are all examples.

An operator is the symbol used to show which operation is to be done.

Addition (+)

Augend: the number to which another is added.

Addend: the number being added.

Summand: a quantity to be added to another.

Sum: the result of addition.

augend + addend = sum

Subtraction (−)

Minuend: the amount from which something is subtracted.

Subtrahend: the number taken away.

Difference: the result of subtraction.

minuend − subtrahend = difference

Multiplication (×)

Multiplicand: the amount to be multiplied.

Multiplier: the amount by which to multiply.

Product: the result of multiplication.

multiplicand × multiplier = product

Division (÷ or /)

Dividend: the amount to be divided.

Divisor: the amount by which to divide.

Quotient: the whole number result.

Remainder: what is left over.

dividend ÷ divisor = quotient + (remainder ÷ divisor)

The division algorithm

Commutative Property

A commutative operation is one in which the order does not matter.

Addition is commutative

Example

3 + 2 = 2 + 3

Subtraction is not commutative

Example

3 − 2 ≠ 2 − 3

Multiplication is commutative

Example

3 × 2 = 2 × 3

Division is not commutative

Example

3 ÷ 2 ≠ 2 ÷ 3

Associative Property

A binary operation is associative if grouping makes no difference.

Addition is associative

Example

3 + 2 + 5 = 3 + (2 + 5)

Subtraction is not associative

Example

5 − 3 − 2 ≠ 5 − (3 − 2)

Multiplication is associative

Example

3 × 2 × 5 = 2 × (3 × 5)

Division is not associative

Example

12 ÷ 3 ÷ 2 ≠ 12 ÷ (3 ÷ 2)

Distributive Property

The distributive property of multiplication over addition or subtraction:

\[ 4 \times (7 + 2) = 4 \times 7 + 4 \times 2 = 28 + 8 = 36 \] \[ \text{compare to} \] \[ 4 \times (7 + 2) = 4 \times 9 = 36 \]

In general:

\[ a \times (b + c) = a \times b + a \times c \]

Usually the multiplication sign is omitted:

\[ a (b + c) = a b + a c \]

Removing brackets

Identity

The identity for a binary operation is an object which, when combined with any other object, leaves it unchanged.

For addition: a + 0 = a

For multiplication: a × 1 = a

Addition

\[ \text{Commutative Law}\qquad a + b = b + a \] \[ \text{Associative Law}\qquad (a + b) + c = a + (b + c) \] \[ \text{Identity elements}\qquad a + 0 = a = 0 + a \]

Multiplication

\[ \text{Commutative Law}\qquad a \times b = b \times a \] \[ \text{Associative Law}\qquad (a \times b) \times c = a \times (b \times c) \] \[ \text{Identity elements}\qquad a \times 1 = a = 1 \times a \]

Subtraction

\[ \text{Commutative Law}\qquad a - b \ne b - a \] \[ \text{Associative Law}\qquad (a - b) - c \ne a - (b - c) \] \[ \text{Identity elements}\qquad a - 0 = a \ne 0 - a \]

Division

\[ \text{Commutative Law}\qquad a \div b \ne b \div a \] \[ \text{Associative Law}\qquad (a \div b) \div c \ne a \div (b \div c) \] \[ \text{Identity elements}\qquad a \div 1 = a \ne 1 \div a \]

Number Bases

Number bases are systems for writing numbers using different sets of digits and place values. In any base, each position in a numeral represents a power of the base, starting with the units place and increasing to the left.

Base ten uses powers of 10 , base two uses powers of 2, and base sixteen uses powers of 16. Although the symbols used may change from one base to another, the underlying quantity remains the same. Converting between bases involves expressing a number in terms of the powers of the new base and rewriting it using the appropriate digits.

Example

15₁₀ = 1111₂ = 30₅

(Read as: 15 in base ten is equal to 1111 in base 2 or 30 in base 5.)

The subscript shows the base.

They can also be written 15_dec = 1111_bin.