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# Complex Number

Complex numbers are numbers residing in a 2 dimensional plane and are more general than real numbers. Complex numbers are very convenient in many situations and sometimes allow solving problems at which using real numbers would fail.

Real numbers and imaginary numbers are both subsets of complex numbers.

A complex number can be written in the form:

a + b * i

where a and b are real numbers – representing the 2 dimensional coordinates – and i is so called imaginary unit.

The imaginary unit i is a special number which we define to have the property:

i^2 = -1

No real number has this property and so by defining this we invent a new number that exists outside the real number line. By having this new unit i we also get infinitely many new numbers of form a * i, e.g. 1i, 0.5i or -10i, which reside on a line separate from the real number line – we call these new numbers imaginary numbers.

Now since we have two number lines, real and imaginary, we can draw them perpendicular to each other with the intersection at 0, which is the only common number of both lines. This creates a 2 dimensional plane with real and imaginary dimensions. Numbers on this plane are called complex numbers and we can write them in the above mentioned form representing coordinates in the plane.

      imaginary
       numbers
          ^
          |      complex numbers
          |
          |
----------+----------> real numbers
          |0
          |
          |

for example:

• 3.5 is a real number and a complex number
• 2i is an imaginary number and a complex number
• 0 is both real and imaginary number and a complex number
• 2 + 6i is a complex numbers

# Credits and Notes

• Some text was made by Drummyfish, of the LRS Wiki, via Git, CC0.

This page was last edited on 2026-09-25 02:04:49.

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