number 4 + 32i.
(2 + 3i) × (8 + 4i) = (2 × 8) + (2 × 4i) + (3i × 8) + (3i × 4i)
With complex numbers, all the ordinary rules of arithmetic are satisfied.
Subtraction and division are always possible (except by the complex number 0 +
0 i, but this was not allowed for zero in real numbers either). In fact the complex
numbers enjoy all the properties of the real numbers save one. We cannot split
them into positive ones and negative ones as we could with the real numbers.
The Argand diagram
The two-dimensionality of complex numbers is clearly seen by representing
them on a diagram. The complex numbers −3 + i and 1 + 2i can be drawn on
what we call an Argand diagram: This way of picturing complex numbers was
named after Jean Robert Argand, a Swiss mathematician, though others had a
similar notion at around the same time.
Every complex number has a ‘mate’ officially called its ‘conjugate’. The mate of
1 + 2i is 1 − 2i found by reversing the sign in front of the second component.
The mate of 1 − 2i, by the same token, is 1 + 2i, so that is true mateship.