complex numbers in the real plane via the Argand diagram

Although the symbol manipulation surrounding the use of a symbol i which fulfils the arithmetical rule that i² = -1 is extremely convenient and widely used, complex numbers can be given a much more mundane explanation by making up special rules for performing arithmetic operations on pairs of numbers.

There are antecedents for such circumlocutions; for example it is possible to work with fractions without giving up integers. Simply keep the numerators and denominators separate, notice that to multiply fractions it suffices to multiply the numerators together, and then the denominators. Other rules of convenience, such as to omit common factors in numerator and denominator, keep the process from getting out of hand. The rule for adding fractions involves common denominators; to get reciprocals, exchange numerator and denominator. The point is that all the arithmetic operations for fractions can be covered, yet nothing ever actually gets broken into pieces.

Even the negative numbers succumb to this treatment: simply keep two accounts, one for what is on hand and one for what is owed, making opportune transfers from one to the other as occasion demands.

Apparently William Rowan Hamilton, who later invented quaternions, was the first to propose that x+iy be written instead as the pair (x,y) with the arithmetical rules for pairs

(a,b) + (c,d) = (a+c,b+d) (1)

$$(a,b)\times (c,d)$$ = (ac-bd,ad+bc). (2)

From this definition follows

(a,b) + (0,0) = (a,b) (3)

$$(a,b)\times (1,0)$$ = (a,b) (4)

$$(0,1)\times (0,1)$$ = (-1, 0), (5)

as do all the other things that ought to be checked, such as the associative, commutative, and distributive laws. The importance of doing things this way seems to be that all the manipulation is done with concrete, readily visible objects; there is no such thing as an ``inaginary'' number with some speculative properties.