It should not be a surprise that the first hints of the meaning of imaginary numbers came from the ancient Greeks. Diophantus discovered the following equation:
(a2 + b2)(c2 + d2) = (ac – bd)2 + (ad + bc)2
Just to foreshadow a little bit, this is like the formula for multiplying complex numbers:
(a + bi)(c + di) = (ac – bd) + (ad + bc)i
But Diophantus’s equation does a better job of revealing the mystery of imaginary numbers. We know from the Pythagorean formula that when you have a right triangle, a2 + b2 = c2. So the left of Diophantus’s equation represents two right triangles being multiplied together. The right side is also a triangle. So Diophantus found that two right triangles multiplied together equals another right triangle. That is a pretty amazing discovery!
Lets try it out. Our first triangle has sides of 3 and 4. The second triangle has sides of 5 and 12. Since they are right triangles we know by the Pythagorean theorem that the hypotenuses are 5 and 13 respectively. Plug that in and we get:
(32 + 42)(52 + 122) = 4225 = (3·5 – 4·12)2 + (3·12 + 4·5)2
Using the Pythagorean theorem we get:
52·132 = 4225 = 332 + 562
5·13 = 65
So the length of the hypotenuse of the new triangle is equal to the product of the hypotenuses of the original triangles. Pretty neat! But here is something even neater. Lets find the arc sin (AKA sin-1) for the triangles:
arc sin (3⁄5) = 36.87°
arc sin (5⁄13) = 22.62°
And we know that the legs of the product triangle are 33 and 56, and the hypotenuse is 65. So
arc sin ((56⁄65) ) = 59.49°
If you add up the angles of the original triangles, you get 59.49 degrees! Amazing! When you multiply two right triangles together, you get a new right triangle. The length of the hypotenuse is equal to the product of the lengths of the hypotenuses of the original triangles. And the main angle of the product angle is equal to the sum of the main angles of the original triangles! You can check this approximately on the figure above which is shown to scale (minus a small amount of error introduced making the figures – a limitations of Microsoft Paint).
Introducing The Complex Plane
Of course, we don’t really multiply triangles; we multiply vectors. And a vector is really just a point in the coordinate plane. Any point in the coordinate plane can be written as the pair (x,y), which determines a right triangle. One leg x, and the other leg is y. The hypotenuse is the line segment from the origin to the point (x,y). When you multiply two vectors together you multiply their lengths – which is given by the hypotenuse – and add their angles.
So what happens if you multiply the vector (0,1) by itself? The length is 1·1 = 1. And the angle is 90 degrees. So 90 + 90 = 180. So the product vector is of length 1 at 180 degrees – which means that the product vector is (-1,0). So this vector is the square root of negative one! We just found i!
We could write complex numbers as vectors, but we stick with the i notation. So we use i instead of (0,1). If we had the point (1,1) we would write it as 1 + i. More generally, we represent the y axis with i, so (a,b) becomes a + bi. The formula for multiplying two complex numbers is given by our Diophantus-like:
(a + bi)(c + di) = (ac – bd) + (ad + bc)i
This view also explains why a negative number squared is a positive. The angle of a negative number is 180 degrees. Square it and you get 180 + 180 = 360. We’ve come full circle and back to the positive direction. Similarly, the angle for a positive number is 0 degrees. So if you square a positive number the result is still zero degrees – it is still a positive number.
Let’s test this by using the formula for complex numbers to multipy the triangles (or vectors) in the example above: multiplying a triangle (or vector) of sides 3,4, and 5 by a triangle with sides of 5,12, and 13.
(4 + 3i)(12 + 5i) = 48 + 36i + 20i – 15 = 33 + 56i
Notice that I had to rearrange the terms compared to Diophantus’s equation. I had to make the real part 4 for the first triangle and 12 for the second. This is actually an advantage of complex numbers. They tell us which length belongs to which leg of the triangle. Diophantus’s equation does not do that.
