Hi Keith.

Think of an imaginary number plotted on a cartesian grid such that the real part is the x axis and the imaginary part is the y axis. Then you'd plot 3+4i as the point (3,4).

The distance from the origin to the point is the absolute value of that complex number. The distance formula says the distance from the original to any point (x,y) is sqrt(x^{2} + y^{2}), so the absolute value of 3+4i = sqrt(3^{2} + 4^{2}) = 5.

Notice that when you have a complex number with no imaginary part (such as 4+0i), the absolute value is the positive square root of the real part squared - in other words, the absolute value of the real part. So this "distance" method is just an extension of what you already use for real numbers.

Hope this explanation helps,

Stephen La Rocque.

If you think of the 'length' of a + bi as the distance from the origin to the point (a,b) then all you need is Pythagoras' Theorem; in your example you need the distance from (0,0) to (3,4) which is exactly what you said by Pythagoras. (What we are doing here is thinking of a + bi in the 'complex' plane where the usual Y-axis is now thought of as the i-axis.)

Penny