without complex numbers does not exist. However the number behaves exactly like any other number in algebra without any anomalies, allowing us to solve this problem.
The solutions are .
is an imaginary number. is a complex number.
Two complex numbers are added by .
Subtraction is obvious: .
Division can be worked out as an exercise. It requires as a common denominator. This is , (difference of two squares), and is .
In practice complex numbers allow one to simplify the mathematics of magnetism and angular momentum as well as completing the number system.
There is an apparent one to one correspondence between the Cartesian plane and the complex numbers, . This is called an Argand diagram. The correspondence is illusory however, because say for example you raise the square root of to a series of ascending powers. Rather than getting larger it goes round and round in circles around the origin. This is not a property of ordinary numbers and is one of the fundamental features of behaviour in the complex plane.
Plot on the same Argand diagram
(Answers -2 plus or minus 5i, 3/2 plus or minus 2i, i(-1 plus or minus root 2)
2 important equations to be familiar with, Euler's equation:
and de Moivre's theorem:
Euler's equation is obvious from looking at the Maclaurin expansion of .
To find the square root of we use de Moivre's theorem.
so de Moivre's theorem gives
Check this by squaring up to give .
The other root comes from:
de Moivre's theorem can be used to find the three cube roots of unity by
where can be .
Put , and
This is the difference of two squares so
Similarly any collection of th roots of 1 can be obtained this way.
Another example is to get the expressions for and without expanding .