A Darling 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).

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:

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

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.,

a2 + 4b3 + 4c3 +4a2 = 5a2 + 4(b3 + c3;
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 + 3i) + (17 - i) = 22 + 2i.

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.