Complex Numbers

A complex number is a number that combines a real part and an imaginary part, written in the form a + bi, where a and b are real numbers and i is the imaginary unit. The imaginary unit is defined by the property that its square equals minus one, which allows equations that have no real solution, such as the square root of a negative number, to be solved. Complex numbers extend the familiar real numbers and form a complete and consistent number system.

Complex numbers can be pictured as points or vectors on the complex plane, a two-dimensional diagram in which the horizontal axis represents the real part and the vertical axis represents the imaginary part. Identifying such points makes operations easy to visualise. Addition and subtraction combine the real and imaginary parts separately, while multiplication uses the rule that i squared equals minus one. Division can be carried out by multiplying by a suitable conjugate to remove the imaginary part from the denominator.

Every complex number also has a polar form, described by its modulus, the distance from the origin, and its argument, the angle it makes with the positive real axis. This representation makes multiplication and division especially simple, since the moduli multiply and the arguments add. De Moivre’s theorem builds on this idea to raise complex numbers to powers and to find their roots, including square roots. Complex numbers are widely used in mathematics, physics, and engineering.

Frequently asked questions

What is the imaginary unit?

It is the number, written i, whose square equals minus one. It lets equations like the square root of a negative number have solutions.

What is the complex plane?

It is a diagram in which a complex number is plotted as a point, with the real part on the horizontal axis and the imaginary part on the vertical axis.

What are modulus and argument?

The modulus is the distance of a complex number from the origin, and the argument is the angle it makes with the positive real axis in polar form.