>Introduction>What Is the Norm and the Argument of a Complex Number?

What Is the Norm and the Argument of a Complex Number?

Since Cartesian form and polar form are equivalent ways to write the same complex number $z$ , there are rules for how to convert between these two forms of representing complex numbers.

If you draw a complex number $z$ in the complex plane, you can use the Pythagorean Theorem and trigonometry to find the norm and the argument of $z$ .

In the complex plane, a complex number

z = a + b i

will form a right triangle with legs of lengths

a

and

b

. In this geometric picture, the norm of

z

—which is defined as the distance from

z

to the origin—is the length of the hypotenuse in the triangle. That means you can use the Pythagorean theorem to find the norm of complex numbers.

Formula

Norm

For all complex numbers $z = a + b i$ , you can find the norm $r$ of $z$ as

r = \sqrt{a^{2} + b^{2}} .

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You can find the argument $𝜃$ by using either sine, cosine or tangent:

Formula

Argument

For all complex numbers $z = a + b i$ with norm $r = \sqrt{a^{2} + b^{2}}$ , you can find the argument $𝜃$ using one of the following formulas:

\begin{array}{l} 𝜃 & = \cos^{- 1} (\frac{a}{r}), \\ 𝜃 & = \sin^{- 1} (\frac{b}{r}), \\ 𝜃 & = \arctan (\frac{b}{a}) . \end{array}

Every expression above yields two values for the argument $𝜃$ . You find the correct argument by using two of the formulas and choosing the repeated value. It is also possible to look at the arguments of $a$ and $b$ , or plot the number and choose the angle which lies in the correct quadrant.

The right triangle in the complex plane formed by a complex number $z$ can also be used to change from polar form $z = (r, 𝜃)$ to Cartesian form $z = a + b i$ .

Formula

From Polar Form to Cartesian Form

A complex number

z = (r, 𝜃)

can be written in Cartesian form

z = a + b i

, where

a = r \cos 𝜃

and

b = r \sin 𝜃 .

You can therefore write every complex number $z$ in the form

\begin{array}{l} z & = r \cos 𝜃 + i r \sin 𝜃 \\ = r (\cos 𝜃 + i \sin 𝜃) . \end{array}

This last form for complex numbers connects complex numbers with trigonometric functions. This is really important when you work with the exponential form of complex numbers.

Example 1

Find the norm $r$ and the argument $𝜃$ of the complex number $z = - 1 + \sqrt{3} i$

You can first use the Pythagorean theorem to find the norm of $z$ : $\begin{array}{l} r & = \sqrt{a^{2} + b^{2}} \\ = \sqrt{{(- 1)}^{2} + {(\sqrt{3})}^{2}} \\ = \sqrt{1 + 3} \\ = 2 . \end{array}$

The norm of $z$ is $r = 2$ . Second, you can use cosine to find the argument $𝜃$ : $\begin{matrix} 𝜃 = \cos^{- 1} (\frac{a}{r}) = \cos^{- 1} (- \frac{1}{2}) \\ ⇓ \\ 𝜃 = \frac{2 π}{3} or 𝜃 = \frac{4 π}{3} . \end{matrix}$

You can then use sine to check for the correct argument:

\begin{matrix} 𝜃 = \sin^{- 1} (\frac{b}{r}) = \sin^{- 1} (\frac{\sqrt{3}}{2}) \\ ⇓ \\ 𝜃 = \frac{π}{3} or 𝜃 = \frac{2 π}{3} . \end{matrix}

Since both expressions give $\frac{2 π}{3}$ as a value, that is the correct argument of $z$ .

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