# Complex Numbers - Arithmetic, Graphical Representation, Uses Of Complex Numbers

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imaginary real

Complex numbers are numbers which can be put into the form a + bi, where a and b are real numbers and i^{2} = -1.

Typical complex numbers are 3 - i, 1/2 + 7i, and -6 - 2i. If one writes the real number 17 as 17 + 0i and the **imaginary number** -2.5i as 0 - 2.5i, they too can be considered complex numbers.

Complex numbers are so called because they are made up of two parts which cannot be combined. Even though the parts are joined by a plus sign, the **addition** cannot be performed. The expression must be left as an indicated sum.

Complex numbers are occasionally represented with ordered pairs, (z,b). Doing so shows the two-component nature of complex numbers but renders the operations with them somewhat obscure and hides the kind of numbers they are.

Inklings of the need for complex numbers were felt as early as the sixteenth century. Cardan, in about 1545, recognized that his method of solving **cubic equations** often led to solutions with the **square root** of **negative** numbers in them. It was not until the seventeenth and early eighteenth centuries that de Moivre, the Bernoullis, Euler, and others gave formal recognition to imaginary and complex numbers as legitimate numbers.

## Additional Topics

While the foregoing rules suffice for ordinary complex-number arithmetic, they must often be coupled with ingenuity for non-routine problems. An example of this can be seen in the problem of computing a square root of 3 - 4i. For two complex numbers to be equal, their real and imaginary parts must be equal
Solving these equations for a yields four roots, namely 2, -2, i, and -i. Discardi…

The mathematicians Wessel, Argand, and Gauss, separately devised a graphical method of representing
Figure 2. Illustration by Hans & Cassidy. Courtesy of Gale Group.
complex numbers. Real numbers have only a real component and can be represented as points on a one-dimensional number line. Complex numbers, on the other hand, with their real and imaginary components, require a two-dim…

Complex numbers are also used in the branch of mathematics known as "functions of complex variables." Such functions can often describe physical situations, as in electrical engineering and fluid dynamics, which real-valued functions cannot. …

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