Modulus of Complex Number - Properties, Formula, Graph and Examples

Modulus of Complex Number - Properties, Formula, Graph and Examples

Edited By Komal Miglani | Updated on Jul 02, 2025 07:36 PM IST

Modulus of complex number means the distance of the complex number from the origin of the argand plane. It also helps in finding the distance between two points of complex numbers. The modulus of a complex number is also called the absolute value of the complex number.

In this article, we will cover the concept of the modulus of a complex number. This concept falls under the broader category of complex numbers. It is essential not only for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), as well as other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of seventeen questions have been asked on this concept, including two in 2013, two in 2014, one in 2015, one in 2019, three in 2020, three in 2021, three in 2022, and two in 2023.

This Story also Contains
  1. What is Modulus of Complex Number?
  2. Modulus of Complex Number Formula
  3. Modulus of a complex number using graph
  4. Summary
  5. Solved Examples Based on Modulus of Complex Number
Modulus of Complex Number - Properties, Formula, Graph and Examples
Modulus of Complex Number - Properties, Formula, Graph and Examples

What is Modulus of Complex Number?

If z = x + iy is a complex number. Then, the modulus of z, denoted by | z |, is the distance of z from the origin in the Argand plane, and it is a non-negative real number equal to $\sqrt{x^2+y^2}$.

i.e. |z| =$\sqrt{\mathrm{x}^2+\mathrm{y}^2}$.

Every complex number can be represented as a point in the argand plane with the x-axis as the real axis and the y-axis as the imaginary axis.

$|z|=\sqrt{x^2+y^2}=r$ (length r from origin to point (x,y))

Modulus of Complex Number Formula

The modulus of a complex number $z=x+i y$, denoted by |z|, is given by the formula $|z|=\sqrt{ }\left(x^2+y^2\right)$ where x is the real part and y is the imaginary part of the complex number z. The modulus of the complex number z can also be calculated using the conjugate of z. Since $z . \bar{z}=(x+i y)(x-i y)=x^2+y^2 \space{\text {we have }} z \cdot \bar{z}=|z|^2 \Rightarrow \sqrt{z \cdot \bar{z}}$

Modulus of a complex number using graph

The distance between the complex number's coordinates and the origin on a complex plane, as shown in a graph of a complex number $\mathbf{z}$, is known as the complex number's modulus. The modulus of a complex number is the distance of the number expressed as a point on the argand plane (x, y). This distance, expressed as $r=\sqrt{(x^2+y^2) }$, is a linear distance between the origin $(0,0)$ and the point $(x, y)$.

Moreover, this can be interpreted in terms of the Pythagorean theorem, which states that the hypotenuse, base, and height of a right-angled triangle reflect the modulus, real portion, and imaginary part, respectively, of a complex number. The magnitude (or length) of the vector that represents $x + yi$ is equal to the modulus of a complex number, $\mathrm{x}+\mathrm{yi}$.

Properties of Modulus

i) $|z| \geq 0$

ii)$|z|=0$, iff $z=0$ and $|z|>0$, iff $z \neq 0$

iii)$-|z| \leq \operatorname{Re}(z) \leq|z|$ and $-|z| \leq \operatorname{Im}(z) \leq|z|$

iv) $z|=| \bar{z}|=|-z|=|-\bar{z} \mid$

v) $z \bar{z}=|z|^2$

vi) $z_1 z_2|=| z_1|| z_2 \mid$. Thus,$\left|z^n\right|=|z|^n$

vii) $\left|\frac{z_1}{z_2}\right|=\frac{\left|z_1\right|}{\left|z_2\right|}$

viii) $\left|z_1 \pm z_2\right| \leq\left|z_1\right|+\left|z_2\right|$ (Triangle inequality) This can be generalized for n complex numbers.

iх) $\left|z_1 \pm z_2\right| \geq|| z_1|-| z_2||$

Summary

We concluded that the modulus is an important term in complex numbers that are used to find the distance from the origin. Like the distance formula in coordinate geometry, it seems to be similar to this for calculating distance. It is also applied in real-life scenarios to find the distance between two points in a complex plane or argand plane.

Recommended Video :

Solved Examples Based on Modulus of Complex Number

Example 1: The complex number $z$ satisfying $|z|=2$ will be:

1) $z=1+i$

2) $z=\sqrt{3}-i$

3) $\sqrt{2}+i$

4) $z=1-i$

Solution:

As we learned in

Definition of Modulus of z(Complex Number) -

$|z|=\sqrt{a^2+b^2}$ is the distance of z from the origin in the Argand plane

- wherein

Real part of $z=\operatorname{Re}(z)=$ a \& Imaginary part of $z=\operatorname{Im}(z)=b$

Now,

if $z=1+i$, then $|z|=\sqrt{1^2+1^2}=\sqrt{2}$

if $z=\sqrt{3}-i$, then $|z|=\sqrt{(\sqrt{3})^2+(-1)^2}=\sqrt{3+1}=2$

if $z=\sqrt{2}+i$, then $|z|=\sqrt{(\sqrt{2})^2+1^2}=\sqrt{2+1}=\sqrt{3}$

if $z=1+i$ then $|z|=\sqrt{1^2+(-1)^2}=\sqrt{1+1}=\sqrt{2}$

Hence, the answer is the option (2).

Example 2: Let $z=-12-5 i$ and $w=2 \sqrt{11}+i y$ such that z and w both are equidistant from origin then y equals Where $y \in R$

1) $2 \sqrt{5}$
2) $3 \sqrt{5}$
3) $4 \sqrt{5}$
4) $5 \sqrt{5}$

Solution

As we learned in

Definition of Modulus of z(Complex Number) -

$|z|=\sqrt{a^2+b^2}$ is the distance of z from the origin in the Argand plane

- wherein

Real part of $z=\operatorname{Re}(z)=a$ \& Imaginary part of $z=\operatorname{Im}(z)=b$

$\because$ z & w are equidistant from origin so $|z|=|w|$

$\begin{aligned} & \Rightarrow \sqrt{144+25}=\sqrt{44+y^2} \\ & \Rightarrow \sqrt{169}=\sqrt{44+y^2} \\ & \Rightarrow 169=44+y^2 \\ & \Rightarrow y^2=125 \\ & \Rightarrow y= \pm 5 \sqrt{5}\end{aligned}$

Hence, the answer is $5 \sqrt{5}$

Example 3: Let $w(\operatorname{Im} w \neq 0)$ be a complex number. Then the set of all complex numbers z satisfying the equation $w-\bar{w} z=k(1-z)$ for some real number k is :

1) $\{z:|z|=1\}$

2) $\{z: z=\bar{z}\}$

3) $\{z: z \neq 1\}$

4) $\{z:|z|=1, z \neq 1\}$

Solution

As we have learned

Property of Modulus of z(Complex Number) -

$\frac{z_1}{z_2} \left\lvert\,=\frac{\left|z_1\right|}{\left|z_2\right|}\right.$

- where |.| denotes the modulus of a complex number

Now,

$\omega-\bar{\omega} z=k(1-z)$

$\Rightarrow \omega-k=\bar{\omega} z-k z$

$\Rightarrow z=\frac{\omega-k}{\bar{\omega}-k} \neq$

And

$\Rightarrow|z|=\frac{|\omega-k|}{|\bar{\omega}-k|}$

$=\frac{|x+i y-k|}{|x-i y-k|}$

$=\frac{\sqrt{(x-k)^2+y^2}}{\sqrt{(x-k)^2+y^2}}=1$

Hence, the answer is the option 4.

Example 4: If a>0 and $z=\frac{(1+i)^2}{a-i}$ , has magnitude $\sqrt{\frac{2}{5}}$ , then $[\bar{z} \mathrm{i}$ is equal to :

Solution:

Multiplication of Complex Numbers -

(a+ib)(c+id)=(ac-bd)+i(bc+ad)

Definition of Modulus of z(Complex Number) -

$|z|=\sqrt{a^2+b^2}$ is the distance of z from the origin in the Argand plane

- wherein

Real part of z = Re (z) = a & Imaginary part of z = Im (z) = b

$z=\frac{(1+i)^2}{a-i}$

$=\frac{1+2 i-1}{a-i}=\frac{2 i}{a-i}$

$z=\frac{2 i}{a-i} \times \frac{a+i}{a+i}=\frac{2 i(a+i)}{a^2+1}$

given that

$|z|=\sqrt{\frac{2}{5}}$

$\left|\frac{2 i(a+i)}{a^2+1}\right|=\sqrt{\frac{2}{5}}$

$\frac{2}{\sqrt{a^2+1}}=\sqrt{\frac{2}{5}}$

$a=3$

$z=-\frac{1}{5}-\frac{3 i}{5}$

Hence, the answer is $-\frac{1}{5}-\frac{3}{5} i$

Example 5:The equation $|z-i|=|z-1|, i=\sqrt{-1}$ represents:

1) a circle of radius $\frac{1}{2}$.

2) a circle of radius $1.$

3) the line through the origin with slope $1.$

4) the line through the origin with slope $-1.$

Solution:

Definition of Complex Number -

$z=x+i y, x, y \in R \& i^2=-1$

- wherein

Real part of $z=\operatorname{Re}(z)=x \&$ Imaginary part of $z=\operatorname{Im}(z)=y$

Definition of Modulus of z(Complex Number) -

$|z|=\sqrt{a^2+b^2}$ is the distance of z from the origin in the Argand plane

- wherein

Real part of $z=\operatorname{Re}(z)=a$ \& Imaginary part of $z=\operatorname{Im}(z)=b$

$\begin{aligned} & |z-i|=\mid z-1 \\ & |x+i y-i|=|x+i y-1| \\ & |x+i(y-1)|=|(x-1)+i y| \\ & \sqrt{x^2+(y-1)^2}=\sqrt{(x-1)^2+y^2} \\ & x^2+(y-1)^2=(x-1)^2+y^2 \\ & x^2+y^2+1-2 y=x^2+1-2 x+y^2 \\ & -2 y=-2 a\end{aligned}$

$y=x$

The line through the origin with slope 1


Frequently Asked Questions (FAQs)

1. How is the modulus of a complex number calculated?
For a complex number z = a + bi, the modulus is calculated using the formula |z| = √(a² + b²), where a is the real part and b is the imaginary part.
2. What is the relationship between the modulus and complex conjugates?
The modulus of a complex number is equal to the modulus of its complex conjugate. If z = a + bi, then |z| = |z*| where z* is the complex conjugate.
3. What is the modulus of the product of two complex numbers?
The modulus of the product of two complex numbers is equal to the product of their individual moduli. |z₁ · z₂| = |z₁| · |z₂|
4. How does squaring a complex number affect its modulus?
When a complex number is squared, its modulus is squared. If |z| is the modulus of z, then |z²| = |z|².
5. How does the modulus behave under complex number division?
For division of complex numbers, the modulus of the quotient is equal to the quotient of their moduli. |z₁ / z₂| = |z₁| / |z₂|
6. What does the modulus represent geometrically?
Geometrically, the modulus represents the length of the vector from the origin to the point representing the complex number on the complex plane.
7. Can the modulus of a complex number be negative?
No, the modulus of a complex number is always non-negative. It represents a distance, which cannot be negative.
8. How does the modulus relate to the polar form of a complex number?
In the polar form r(cos θ + i sin θ), the modulus r is the distance from the origin to the point representing the complex number.
9. How is the modulus used in solving quadratic equations with complex roots?
The modulus helps in understanding the nature of complex roots. If a quadratic equation has complex roots a ± bi, their modulus √(a² + b²) represents the distance of the roots from the origin in the complex plane.
10. How does the modulus help in understanding the properties of complex roots of unity?
All nth roots of unity have a modulus of 1, forming a circle of radius 1 centered at the origin in the complex plane. This property helps in visualizing and understanding their distribution.
11. What is the modulus of a complex number?
The modulus of a complex number is the distance between the point representing the complex number and the origin (0,0) on the complex plane. It represents the absolute value or magnitude of the complex number, regardless of its direction.
12. What is the difference between the modulus and the argument of a complex number?
The modulus represents the magnitude or length of the complex number, while the argument represents the angle the complex number makes with the positive real axis.
13. How does the modulus relate to the absolute value of real numbers?
For real numbers, the modulus is equivalent to the absolute value. If x is a real number, |x| as a complex number is the same as |x| as a real number.
14. What is the modulus of i?
The modulus of i is 1. Since i = 0 + 1i, |i| = √(0² + 1²) = 1.
15. What is the significance of |z| = 1 in the complex plane?
|z| = 1 represents the unit circle in the complex plane. All complex numbers with modulus 1 lie on this circle.
16. What is the triangle inequality in terms of modulus?
The triangle inequality states that for any two complex numbers z₁ and z₂, |z₁ + z₂| ≤ |z₁| + |z₂|. This means the modulus of the sum is less than or equal to the sum of the moduli.
17. How does the modulus relate to the concept of a conformal map?
Conformal maps preserve angles between curves. The modulus helps in understanding how these maps scale distances locally, which is related to the magnitude of the complex derivative.
18. What is the relationship between the modulus and Liouville's theorem in complex analysis?
Liouville's theorem states that a bounded entire function must be constant. The proof of this theorem relies on properties of the modulus and its behavior under complex differentiation.
19. How is the modulus used in defining the radius of convergence for a power series?
The radius of convergence R for a power series ∑aₙ(z-z₀)ⁿ is often defined as R = 1 / limsup |aₙ|^(1/n). This definition directly involves the modulus of the coefficients.
20. How does the modulus relate to the concept of a harmonic function?
The real and imaginary parts of a holomorphic function are harmonic functions. The modulus of a holomorphic function is related to these harmonic functions through the Cauchy-Riemann equations.
21. What is the role of the modulus in the argument principle?
The argument principle relates the number of zeros and poles of a meromorphic function inside a contour to an integral involving the logarithmic derivative of the function. The modulus is essential in formulating and proving this principle.
22. What is the relationship between the modulus and complex exponentials?
For a complex exponential e^(a+bi), its modulus is e^a. This relates the modulus to the real part of the exponent.
23. How does the modulus function behave on the complex plane?
The modulus function creates concentric circles centered at the origin on the complex plane. Each circle represents points with equal distance from the origin.
24. How can the modulus be used to determine if two complex numbers are equal?
If two complex numbers are equal, their moduli must be equal. However, equal moduli don't necessarily mean the complex numbers are equal, as they could have different arguments.
25. What is the relationship between the modulus and the real and imaginary parts of a complex number?
The square of the modulus is equal to the sum of squares of the real and imaginary parts. For z = a + bi, |z|² = a² + b².
26. How does the modulus relate to the concept of a complex number's norm?
The modulus of a complex number is equivalent to its norm. Both represent the "size" or "magnitude" of the complex number.
27. Can you explain the concept of modulus in terms of energy or power in signal processing?
In signal processing, the square of the modulus of a complex number often represents energy or power. This is particularly useful in analyzing frequency components of signals.
28. How does the modulus help in understanding the growth or decay of complex exponential functions?
For a complex exponential function f(t) = e^(a+bi)t, the modulus |f(t)| = e^at determines whether the function grows (a > 0), decays (a < 0), or maintains constant amplitude (a = 0) over time.
29. What is the relationship between the modulus and complex number multiplication by i?
Multiplying a complex number by i rotates it 90° counterclockwise in the complex plane but does not change its modulus. |iz| = |z| for any complex number z.
30. How does the modulus relate to the concept of a limit in complex analysis?
In complex analysis, limits are often defined in terms of the modulus. For example, lim(z→a) f(z) = L means that |f(z) - L| can be made arbitrarily small as z approaches a.
31. What is the significance of the modulus in defining continuity of complex functions?
A complex function f(z) is continuous at a point z₀ if lim(z→z₀) |f(z) - f(z₀)| = 0. The modulus here measures how close the function values are as z approaches z₀.
32. How is the modulus used in defining complex number inequalities?
Complex number inequalities are often expressed in terms of their moduli. For example, |z - a| < r describes all points inside a circle with center a and radius r.
33. What is the relationship between the modulus and complex number powers?
For a complex number z and a real number n, |z^n| = |z|^n. This property extends the familiar rule for real numbers to the complex plane.
34. How does the modulus relate to the concept of a metric in complex analysis?
The modulus is used to define the standard metric on the complex plane. The distance between two complex numbers z₁ and z₂ is given by |z₁ - z₂|.
35. What is the significance of the modulus in Euler's formula?
In Euler's formula e^(iθ) = cos θ + i sin θ, the modulus is always 1, regardless of θ. This illustrates how complex exponentials with purely imaginary exponents represent rotations on the unit circle.
36. How does the modulus help in understanding the behavior of complex-valued functions?
The modulus helps visualize the magnitude of complex-valued functions. For example, plotting |f(z)| for a complex function f(z) creates a surface over the complex plane, showing how the function's magnitude varies.
37. What is the role of the modulus in defining analytic functions?
The modulus is crucial in the definition of analytic functions. The Cauchy-Riemann equations, which define analyticity, involve partial derivatives of the real and imaginary parts, which are related to the modulus.
38. How does the modulus relate to the concept of a branch cut in complex analysis?
Branch cuts are often defined using the modulus. For example, the principal branch of the complex logarithm is typically defined for complex numbers with arguments between -π and π, which involves restrictions on the modulus and argument.
39. What is the significance of the modulus in understanding Möbius transformations?
Möbius transformations preserve the generalized circle (including straight lines) in the complex plane. The modulus helps in visualizing how these transformations map circles and lines.
40. How is the modulus used in defining complex number series convergence?
The convergence of a complex series ∑aₙ is often determined by examining the series of moduli ∑|aₙ|. If ∑|aₙ| converges, then ∑aₙ converges absolutely.
41. What is the relationship between the modulus and complex number differentiation?
The definition of a complex derivative involves a limit of a quotient of complex numbers. The modulus is crucial in understanding how this limit behaves and in defining the concept of differentiability in the complex plane.
42. What is the role of the modulus in defining holomorphic functions?
Holomorphic functions satisfy the Cauchy-Riemann equations, which involve partial derivatives of the real and imaginary parts. These equations can be expressed in terms of the modulus and argument of the complex derivative.
43. How is the modulus used in the statement of Cauchy's integral formula?
Cauchy's integral formula involves an integral over a closed contour. The modulus is used in estimating this integral and in understanding the behavior of complex functions inside and outside the contour.
44. What is the significance of the modulus in the maximum modulus principle?
The maximum modulus principle states that if f(z) is holomorphic and non-constant in a domain D, then |f(z)| cannot attain a maximum value in D. This principle is fundamental in complex analysis and relies heavily on the properties of the modulus.
45. How does the modulus relate to the concept of a residue in complex analysis?
Residues are calculated using contour integrals, where the modulus plays a crucial role in estimating the integral and understanding the behavior of the function near singularities.
46. What is the role of the modulus in understanding the behavior of meromorphic functions?
The modulus helps in analyzing the behavior of meromorphic functions near their poles and essential singularities. For example, the modulus grows without bound as we approach a pole.
47. What is the significance of the modulus in Rouché's theorem?
Rouché's theorem uses inequalities involving the moduli of complex functions to compare the number of zeros of two functions within a given region.
48. How is the modulus used in defining and understanding Laurent series?
Laurent series expansions involve positive and negative powers of (z-z₀). The modulus is crucial in defining the annulus of convergence and understanding the behavior of the function in different regions.
49. How does the modulus help in understanding the behavior of complex functions near singularities?
The modulus helps classify singularities (removable, poles, essential) by examining how |f(z)| behaves as z approaches the singularity.
50. What is the significance of the modulus in defining and understanding analytic continuation?
Analytic continuation extends the domain of a complex function. The modulus plays a role in determining where such continuation is possible and in understanding the behavior of the extended function.

Articles

Back to top