Absolute Value of a Complex Number

For a complex number z = a + ib, the absolute or modulus value measures the linear distance from the origin (0, 0) to the point (a, b). But unlike the absolute value of real numbers, it is defined in the argand (complex) plane, as shown.

In the argand plane, the x-axis represents the real part, while the y-axis represents the complex number’s imaginary part.

If (a, b) is represented by the point P, then to find the absolute value |z|, we will join the point P with the origin O. Similarly, by joining P with A (a, 0), we get a right-angled triangle OPA at A. where the horizontal distance of P is ‘a’, and the vertical distance of P is ‘b.’

Now, applying the Pythagoras theorem to the triangle OPA, we get OP = |z| = ${\sqrt{a^{2}+b^{2}}}$, which represents the hypotenuse of the right-angled triangle.

Thus, the absolute value of z is given by the formula:

|z| = ${\sqrt{a^{2}+b^{2}}}$

Now, let us find the absolute value of a complex number z = 6 + 8i is ${\sqrt{6^{2}+8^{2}}}$ = ${\sqrt{100}}$ = 10.

In Unit Circle

Complex numbers can have an absolute value of 1. It is the same for -1, just as for the imaginary numbers i and -i. This is because all of them are one unit away from 0, either on the real number line or the imaginary axis. It includes all the complex numbers of absolute value 1. Thus, the equation of the unit circle is |z| = 1.

Properties

• The modulus of two complex numbers z and -z are equal, that is, |z| = |-z|.

If two complex numbers are z = 5 + 4i and -z = -5 – 4i.

Then, we have |z| = ${\sqrt{5^{2}+4^{2}}}$ and

|-z| = ${\sqrt{\left( -5\right) ^{2}+\left( -4\right) ^{2}}}$ = ${\sqrt{5^{2}+4^{2}}}$

• The modulus of the nth power of any complex number z equals the n^th power of the modulus of z, that is, |zⁿ| = |z|ⁿ

For example, if z = 5 + 2i, then, |z²| = |(5 + 2i)(5 + 2i)| = |21 + 20i|

= ${\sqrt{21^{2}+20^{2}}}$ = ${\sqrt{841}}$ = 29 and

|z|² = |5 + 2i|² = 5² + 2² = 29

• |z| = 0, if and only if z = 0.

If we consider z = 0 (= 0 + 0i), then |z| = ${\sqrt{0^{2}+0^{2}}}$ = 0.

• If any two complex numbers are z and w, then the mod of the product of complex numbers (${\left| z\cdot w\right|}$) equals the product of the mod of the complex numbers (${\left| z\right| \cdot \left| w\right|}$), that is, ${\left| z\cdot w\right|}$ = ${\left| z\right| \cdot \left| w\right|}$

Suppose two complex numbers are: z = 2 + 3i and w = 3 + 4i.

Now, ${z\cdot w}$ = (2 + 3i)(3 + 4i) = 6 + 8i + 9i + 12i² = -6 + 17i

${\left| z\cdot w\right|}$ = ${\sqrt{\left( -6\right) ^{2}+\left( 17\right) ^{2}}}$

= ${\sqrt{36+289}}$ = ${\sqrt{325}}$ = ${5\sqrt{13}}$

And, |z| = ${\sqrt{2^{2}+3^{2}}}$ = ${\sqrt{13}}$

|w| = ${\sqrt{3^{2}+4^{2}}}$ = ${\sqrt{25}}$ = 5

${\left| z\right| \cdot \left| w\right|}$ = ${5\sqrt{13}}$ = ${\left| z\cdot w\right|}$

•  The mod of the quotient of two complex numbers (${\left| \dfrac{z}{w}\right|}$) equals the quotient of the mod of the complex numbers (${\dfrac{\left| z\right| }{\left| w\right| }}$), that is, ${\left| \dfrac{z}{w}\right|}$ = ${\dfrac{\left| z\right| }{\left| w\right| }}$

For z = 2 + 3i and w = 3 + 4i,

${\left| \dfrac{z}{w}\right|}$

${\left| \dfrac{2+3i}{3+4i}\right|}$

Using The multiplicative property of modulus, we get,

${\left| \left( 2+3i\right) \dfrac{1}{\left( 3+4i\right) }\right|}$

= ${\left| 2+3i\right| \cdot \left| \dfrac{1}{3+4i}\right|}$

= ${\left| 2+3i\right| \cdot \dfrac{1}{\left| 3+4i\right| }}$

= ${\dfrac{\left| z\right| }{\left| w\right| }}$

• The modulus of a complex number equals the modulus of its conjugate number, that is, |z| = ${\left| \overline{z}\right|}$,

Let us consider z = 2 + i and its conjugate ${\overline{z}}$ = 2 – i.

Now, |z| = ${\sqrt{2^{2}+1^{2}}}$ = ${\sqrt{5}}$${\left| \overline{z}\right|}$ =  ${\sqrt{2^{2}+\left( -1\right) ^{2}}}$ = ${\sqrt{5}}$

Solved Examples