Engineering
Even engineers, a very practical breed, have found uses for complex numbers. When Michael
Faraday discovered alternating current in the 1830s, imaginary numbers gained a physical reality. In
this case the letter j is used to represent √–1 instead of i because i stands for electrical current.
The square root of –1
We have already seen that, restricted to the real number line,there is no square root of −1 as the square of any number cannot be negative.
If we continue to think of numbers only on the real number line, we might as
well give up, continue to call them imaginary numbers, go for a cup of tea with
the philosophers, and have nothing more to do with them. Or we could take the
bold step of accepting √−1 as a new entity, which we denote by i.
By this single mental act, imaginary numbers do exist. What they are we do
not know, but we believe in their existence. At least we know i^2 = −1. So in our
new system of numbers we have all our old friends like the real numbers 1, 2, 3,
4, π, e, and , with some new ones involving i such as 1 + 2i, −3 + i, 2 +
3 i, , , e + πi and so on.
This momentous step in mathematics was taken around the beginning of the
19th century, when we escaped from the one-dimensional number line into a
strange new two-dimensional number plane.
Adding and multiplying
Now that we have complex numbers in our mind, numbers with the form a +
bi, what can we do with them? Just like real numbers, they can be added and
multiplied together. We add them by adding their respective parts. So 2 + 3i
added to 8 + 4i gives (2 + 8) + (3 + 4)i with the result 10 + 7i.
Multiplication is almost as straightforward. If we want to multiply 2 + 3i by 8
- 4i we first multiply each pair of symbols together and add the resulting terms,
16, 8i, 24i and 12i^2 (in this last term, we replace i^2 by −1), together. The result
of the multiplication is therefore (16 – 12) + (8i + 24i) which is the complex