The imaginary unit is defined by the equation . Equivalently, , allowing numbers with negative square roots to be represented within the complex number system.
This is HL-only content from AHL 1.12, Introduction to complex numbers. It may be tested in any GDC-active Mathematics: Applications and Interpretation HL paper.
Why Is Needed?
No real number has a square equal to : a positive number squared is positive, and a negative number squared is also positive. Defining extends the real numbers so that equations such as
can be solved. Rearranging gives
so
These solutions are not real numbers. They belong to the complex numbers, whose Cartesian form is
where is the real part and is the coefficient of the imaginary part. For example, in , the real part is and the imaginary part is .
Working with in Cartesian Form
Use whenever a product contains . For example,
The final answer is written in Cartesian form . Real terms and imaginary terms are collected separately.
Complex roots of real-coefficient quadratics occur in conjugate pairs. Therefore, if is a root, is also a root. On a GDC, complex mode can check calculations, but the written solution should still show the mathematical setup rather than calculator syntax.
Powers of
Powers of repeat in a cycle of four:
| Power | Value |
|---|---|
For example, since leaves remainder when divided by ,
A common misconception is that “imaginary” means mathematically unreal or invalid. In fact, complex numbers are a consistent extension of the real number system and are widely used in electrical circuits, waves, and rotations.
Exam Technique
For a calculate or solve question, show the step using before giving the final answer in the form . Use , not the engineering symbol , and check whether both roots of a quadratic are required.