# addition

Addition is associative – that is, a series of additions can be carried out in any order without affecting the result. This diagram shows the effects of successively weighing [A] 3, 4, and then 5 units of a substance on a spring balance and [B] weighing 4, 5, and then 3 units. In both cases the total weight – the sum or the additions – is 2 units. As in many other mathematical laws, this is applied common sense.

Addition is one of the four fundamental operations of arithmetic. It is denoted by the sign '+'.

The numbers that are added are called **terms**, **addends**, or **summands**. For example,
4 (an addend) + 3 (another addend) = 7 (sum).

## Addition of fractions

Fractions can be added directly only if they have equal denominators, e.g., 3/17 + 4/17 = 7/17.

If fractions with unequal denominators are to be added they must first be
brought to their **least common denominator**.

## Addition of literal numbers

Identically-named numbers can be added (and subtracted) by adding the coefficients:

2*a* + 3*b* + *a* + 4*b* = 3*a* + 7*b*

summands may be interchanged (commutative law):

*a* + *b* = *b* + *a*

With more than two summands, brackets may be inserted (associative law):

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

## Addition of directed numbers

No further rule is needed for the addition of positive numbers:

*a* + (+*b*) = *a* + *b*, 5 + (+3)
= 5 + 3 = 8.

to add one negative number to another, its absolute value is subtracted:

*a* + (–*b*) = *a* – *b*, (–*a*) + (–*b*) = –(*a* + *b*);

5 + (–3) = 5 – 3 = 2, (–5) + (–3) = –(5 + 3) = -8.

## Addition of powers and surds

Powers and surds must be treated like literal numbers for the purposes of addition, e.g.,

*a*^{ 2} + 4*b*^{ 3} + 4*c*^{ 3} +4*a*^{ 2} = 5*a *^{2} + 4(*b *^{3} + *c*^{ 3};

3√2 + 5√3 + √2 = 4√2 + 5√3

## Addition of complex numbers

In adding complex numbers the real and imaginary parts must be added separately, e.g.,

(5 + 3*i*) + (17 – *i*) = 22 + 2*i*.

## Addition in terms of sets

The addition of positive numbers can best be defined in terms of set
theory; if one considers set *A* to contain 4 elements, set *B* to contain 5 elements, then *A* U *B* (see union)
contains 9 elements; i.e., 4 + 5 = 9. The addition of negative numbers is
equivalent to subtraction in that *a* + (–*b*) is equivalent
to *a* – *b*.