Adding and multiplying mates together always produces a real number. In the
case of adding 1 + 2i and 1 −2i we get 2, and multiplying them we get 5. This
multiplication is more interesting. The answer 5 is the square of the ‘length’ of
the complex number 1 + 2i and this equals the length of its mate. Put the other
way, we could define the length of a complex number as:
Checking this for −3 + i, we find that length of
and so the length of.
The separation of the complex numbers from mysticism owes much to Sir
William Rowan Hamilton, Ireland’s premier mathematician in the 19th century.
He recognized that i wasn’t actually needed for the theory. It only acted as a
placeholder and could be thrown away. Hamilton considered a complex number
as an ‘ordered pair’ of real numbers (a, b), bringing out their two-dimensional
quality and making no appeal to the mystical √−1. Shorn of i, addition becomes
(2, 3) + (8, 4) = (10, 7)
and, a little less obviously, multiplication is
(2, 3) × (8, 4) = (4, 32)