Imaginary unit

Sponsorship the way you would do it

In mathematics, the imaginary unit i allows the real number system \R to be extended to the complex number system C. Its precise definition is dependent upon the particular method of extension.

The primary motivation for this extension is the fact that not every polynomial equation f(x) = 0 has a solution in the real numbers. In particular, the equation x² + 1 = 0 has no real solution. However, if we allow complex numbers as solutions, then this equation, and indeed every polynomial equation f(x) = 0 does have a solution. (See algebraic closure and fundamental theorem of algebra.)

Table of contents

1 Definition
2 Warning
3 Powers of i
4 i and Euler's Formula
5 Alternate notation
6 Also see:

Definition

By definition, the imaginary unit i is a solution of the equation

x² = −1

Remark: after i has been defined in this way, one can define multiplication of a real number and i. For instance, one can define −i to be the result of multiplying −1 and i. It follows that −i is also a solution of the equation ((−i)² = (−1)²(i)² = −1). This is in accordance with the fact that the complex roots of a real polynomial always occur in conjugate pairs.

(The existence of two distinct roots may lead one to feel the above definition is not well-defined. The moral of the story is that you can define i by the above equation, but you must fix this i as the "positive" solution and not confuse it with −i. See complex number, where C is defined as R[x]/(x² + 1). In essence, this amounts to defining C to be "polynomials" in i with the rule that i² = − 1. See also complex conjugation and field automorphism.)

Warning

The imaginary unit should not be written down or treated as √(−1). This notation is reserved either for the principal square root function, which is only defined for real x ≥ 0, or for the principal branch of the complex square root function. Attempting to apply the calculation rules of the principal (real) square root function to manipulate the principal branch of the complex square root function will produce false results:

which is absurd!

The calculation rule

is only valid for real, non-negative numbers a and b.

For a more thorough discussion of this phenomenon, see square root and branch (complex analysis).

Powers of i

The powers of i repeat in a cycle:

This can be expressed with the following pattern where n is any integer:

i and Euler's Formula

Taking Euler's formula e^ix = cos(x) + isin(x), and substituting π/2 in for x, we get

If both sides are raised to the power of i, remembering that i² = −1, we obtain the remarkable identity

It should be noted that iⁱ actually has infinitely many values among the complex numbers, and this particular value is just one of them. (See exponential function.)

Alternate notation

In electrical engineering and related fields, the imaginary unit is often written as j to avoid confusion with a changing current, traditionally denoted by i.