Complex Numbers
Imaginary numbers
Imaginary numbers are numbers defined in terms of their relation to $ \sqrt{-1} $, which is known as $ i $. Some equations which seemed impossible before are now completely doable. For example:
To solve this, you just work out the square root of 16 (4) and make it a coefficient of i:
Complex Numbers
Complex numbers are numbers with both a real and imaginary part. They can be squared and manipulated just like any other binomial - just keep in mind that while you are expanding these. They are expressed as:
In this equation:
- The real part is the part without an $ i $ coefficient:
x
. - The imaginary part is the part with an $ i $ coefficient:
y
.
Handy Expressions
Addition
Addition is the same as ever, the real part is the addition of the real parts while the imaginary part is the addition of the imaginary parts.
Subtraction
Subtraction, similarly, has the real part as the subtraction of the real parts and the imaginary part is the subtraction of the imaginary parts.
Multiplication
Division
Complex Conjugates
A complex conjugate to a complex number has the same real part but the opposite imaginary part to that number, as in, if:
Then its complex conjugate
The important equations formed from these properties are that:
Equating real and imaginary parts
If…
.. then due to the nature of complex numbers it is definite that: