Modulus and Conjugate of a Complex Number: Definitions, Properties, Applications

Modulus and Conjugate of a Complex Number: When you consider an integer, its modulus or absolute value is the distance of that number from the number zero on a number line. When it comes to two dimensions, the modulus can be considered the magnitude of a position vector of a point with respect to the origin. When it comes to complex numbers, which have varied scientific applications in fluid dynamics and quantum mechanics, the modulus of a given complex number plays an important role in various calculations. When you deal with quadratic equations with complex roots, they appear in conjugate pairs. Since quadratic equations have many applications in Physics, it is of utmost importance to learn about the modulus and conjugate of complex numbers.

Representing a Complex Number on a Complex Plane

A complex plane is a two-dimensional plane with a number line of real numbers in the horizontal axis and a number line of complex numbers in the vertical axis intersecting at the origin. Like a Cartesian plane, a complex number can be plotted on a complex plane by plotting the real part along the \(x\)-axis and the imaginary part along the \(y\)-axis.

The complex number \(4+6 i\) can be plotted as:

Definition of Modulus of a Complex Number

The modulus of a complex number \(z=a+b i\) is the shortest distance between the origin and the point \((a, b)\) on a complex plane. It is denoted by the symbol \(|z|\).

Consider the complex number \(4+6 i\). The modulus would be the length of the hypotenuse \(\overline{O P}\) of the right triangle \(\Delta O P R\).

By the Pythagorean Theorem, the length of the hypotenuse is equal to the sum of squares of the perpendicular sides.

That is \(|4+6 i|=\sqrt{4^{2}+6^{2}}\) or \(\sqrt{52}\).

Formula to Find the Modulus of a Complex Number

The modulus of a complex number is defined as the non-negative square root of the sum of squares of the real and imaginary parts of the complex number.

That is, the modulus of the complex number \(z=a+b i\) is:

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

The modulus of the complex number \(-5+8 i\) is:

\(|-5+8 i|=\sqrt{(-5)^{2}+8^{2}}\) or \(\sqrt{89}\).

The modulus of a complex number is also known as its absolute value.

Definition of Conjuate of Complex Number

The complex conjugate is also known as the conjugate of a complex number, is also a complex number. It has the same real part and an imaginary part that is the additive inverse of the given complex number. It is denoted by the symbol \(\bar{z}\).

That is, the conjugate of the complex number \(z=a+b i\) is \(\bar{z}=a-b i\)

Representing Conjugates on a Complex Plane

If the position vector connecting the complex number \(z=a+b i\) to the origin makes an angle \(\theta\) with the positive \(x\)-axis, then its conjugate \(\bar{z}=a-b i\) makes an angle of \(-\theta\) with the positive \(x\)-axis.

Therefore, the conjugate of a complex number is its reflection about the \(x\)-axis.

Alternate Definition of Conjugate of a Complex Number

For a complex number \(z\), it’s conjugate \(\bar{z}\) is defined as the number such that

1. \(z+\bar{z} \in R\)
2. \(z \cdot \bar{z} \in R\)

However, one can prove that if \(z=a+b i\), its conjugate is a unique number with the above properties and is equal to \(\bar{z}=a-b i\), making the two definitions equivalent to each other.

Properties of Conjugates

Consider the complex number \(z=a+b i\) then its conjugate is \(\bar{z}=a-b i\).
Then:

1. \(z+\bar{z}=a+b i+a-b i\)

\(=2 a\)

2. \(z-\bar{z}=a+b i-(a-b i)\)

\(=a+b i-a+b i\)

\(=2 b i\)

3. \(z \cdot \overline z = \left( {a + bi} \right)\left( {a – bi} \right)\)

Using the identity \((a+b)(a-b)=a^{2}-b^{2}:\)
\(=a^{2}-(b i)^{2}\)

Using the fact that \(i^{2}=-1\)

\(=a^{2}+b^{2}\)

\(=|z|^{2}\)

So: \(z \cdot \overline z = {\left| z \right|^2}\)

Dividing both sides by \(|z|^{2}\) when the complex number has a non-zero modulus,

\(\frac{z \cdot \bar{z}}{|z|^{2}}=\frac{|z|^{2}}{|z|^{2}}\)

\(z \cdot\left(\frac{\bar{z}}{|z|^{2}}\right)=1\)

That is, the product of \(z\) and \(\frac{\bar{z}}{|z|^{2}}\) is \(1\).

Thus: \(\frac{\bar{z}}{|z|^{2}}\) is the multiplicative inverse of \(z\)

\(z^{-1}=\frac{\bar{z}}{|z|^{2}}\)

Note that the inverse of complex numbers exists only for the ones that have non-zero modulus.

4. The conjugate of a conjugate is the number itself. If \(z=a+b i\) is a complex number, then its conjugate is \(\bar{z}=a-b i\), and the conjugate of \(\bar{z}=a-b i\) is \(\overline{(\bar{z})}=a+b i\) which is the same as the number we started with.

Therefore, \(\overline{(\bar{z})}=z\).

5. The modulus of a complex number and its conjugate is the same.

That is, if \(z=a+b i\) is a complex number, then \(\bar{z}=a-b i\) and

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

\(|\bar{z}|=\sqrt{a^{2}+b^{2}}\)

\(|\bar{z}|=|z|\)

Why Are Conjugates Important?

Consider the quadratic equation \(x^{2}+2 x+5=0\).

Using the quadratic formula, the roots of this equation can be calculated as:

\(x=\frac{-2 \pm \sqrt{2^{2}-4(1)(5)}}{2(1)}\)

\(=\frac{-2 \pm \sqrt{-16}}{2}\)

\(=-1+4 i,-1-4 i\)

The numbers \(-1+4 i\) and \(-1-4 i\) are complex conjugates of each other. That is, the roots of the quadratic equation \(x^{2}+2 x+5=0\) are complex conjugates of each other.

In general, complex roots of quadratic equations occur in conjugate pairs. Since quadratic equations have vast applications in practical life situations, complex conjugates play an important role.

Algebra of Modulus and Conjugate of Complex Numbers

If \(z_{1}=a_{1}+b_{1} i\) and \(z_{2}=a_{2}+b_{2} i\), then, with the basic algebraic calculations, it can be shown that:

1. \(\left|z_{1} \pm z_{2}\right|=\left|z_{1}\right| \pm\left|z_{2}\right|\)

2. \(\left|z_{1} \cdot z_{2}\right|=\left|z_{1}\right| \cdot\left|z_{2}\right|\)

3. \(\left|\frac{z_{1}}{z_{2}}\right|=\frac{\left|z_{1}\right|}{\left|z_{2}\right|}\) where \(\left|z_{2}\right| \neq 0\)

4. \(\overline{z_{1} \pm z_{2}}=\overline{z_{1}} \pm \overline{z_{2}}\)

5. \(\overline{z_{1} \cdot z_{2}}=\overline{z_{1}} \cdot \overline{z_{2}}\)

6. \(\overline{\left(\frac{z_{1}}{z_{2}}\right)}=\frac{\overline{z_{1}}}{\overline{z_{2}}}\) where \(\overline{z_{2}} \neq 0\)

Solved Examples on Modulus and Conjugate of a Complex Number

Q.1. Find the modulus of the complex number \(6+8 i\).
Ans: The modulus of a complex number \(a+b i\) is given by \(\sqrt{a^{2}+b^{2}}\).
\(|6+8 i|=\sqrt{6^{2}+8^{2}}\)
\(=\sqrt{36+64}\)
\(=\sqrt{100}\)
\(=10\)
Therefore, \(|6+8 i|=10\)

Q.2. What possible complex numbers can have a real part \(-7\) and modulus is \(\sqrt{65}\) ? What is the relation between them?
Ans: Given that the real part of the complex number is \(-7\) and the modulus is \(\sqrt{65}\). Let the imaginary part of the complex number be \(k\).
Then, \(\sqrt{(-7)^{2}+k^{2}}=\sqrt{65}\).
Squaring both sides: \(49+k^{2}=65\)
Solving for \(k\):
\(k^{2}=65-49\)
\(=16\)
\(k=\sqrt{16}\)
\(=\pm 4\)
Therefore the possible complex numbers are \(-7+4 i\) and \(-7-4 i\). Since the two complex numbers have equal real parts and the imaginary parts that are additive inverses, the two numbers are complex conjugates of each other.

Q.3. Does the multiplicative inverse of the complex number \(-5+3 i\) exist? If it does, find it.
Ans: The inverse of complex numbers exists only for the ones that have non-zero modulus. Here, \(z=-5+3 i\). The modulus of a complex number \(z=a+b i\) is given by \(|z|=\sqrt{a^{2}+b^{2}}\).
So, for \(z=-5+3 i\):
\(|z|=\sqrt{(-5)^{2}+(3)^{2}} \neq 0\)
Thus, the multiplicative index does exist for the complex number \(-5+3 i\). The multiplicative inverse of a complex number \(z\) is given by the formula:
\(z^{-1}=\frac{\bar{z}}{|z|^{2}}\)
Here, \(z=-5+3 i\).
The conjugate, \(\bar{z}=-5-3 i\) and the square of the modulus
\(|z|^{2}=(-5)^{2}+(3)^{2}\)
\(=25+9\)
\(=34\)
So: \(z^{-1}=\frac{\bar{z}}{|z|^{2}}\)
\(=\frac{-5-3 i}{34}\)
\(=-\frac{5}{34}-\frac{3}{34} i\)

Q.4. Find the conjugate of the complex number \(\frac{5-6 i}{-3+2 i}\).
Ans:
As a first step in finding the conjugate, the number must be written in the standard form \(a+b i\).
Multiplying the numerator and the denominator with the conjugate of the denominator to rationalize the denominator:
\(\frac{5-6 i}{-3+2 i}=\frac{(5-6 i)(-3-2 i)}{(-3+2 i)(-3-2 i)}\)
Using the identity \((a+b)(a-b)=a^{2}-b^{2}\) in the denominator and expanding the numerator using distributive property:
\(=\frac{(5 \times-3)+(5 \times-2 i)+(-6 i \times-3)+(-6 i \times-2 i)}{(-3)^{2}-(2 i)^{2}}\)
\(=\frac{-15-10 i+18 i+12 i^{2}}{9-4 i^{2}}\)
Using the fact that \(i^{2}=-1:\)
\(=\frac{-15+8 i-12}{9+4}\)
\(=\frac{-27+8 i}{13}\)
The conjugate of the complex number \(z=a+b i\) is \(\bar{z}=a-b i.\)
So, the complex conjugate of \(-\frac{27}{13}+\frac{8}{13} i\) is \( – \frac{{27}}{{13}} – \frac{8}{{13}}i\)

Q.5. Simplify: \(\left(\frac{1}{2}-2 i\right)^{3}\)
Ans: Using the identity \((a-b)^{3}=(a-b)\left(a^{2}+a b+b^{2}\right):\)
\(\left(\frac{1}{2}-2 i\right)^{3}=\left(\frac{1}{2}-2 i\right)\left(\left(\frac{1}{2}\right)^{2}+i+(2 i)^{2}\right)\)
\(=\left(\frac{1}{2}-2 i\right)\left(\frac{1}{4}+i+4 i^{2}\right)\)
Using the fact that \(i^{2}=1\) :
\(=\left(\frac{1}{2}-2 i\right)\left(\frac{1}{4}+i-4\right)\)
Expanding using the Distributive Property and simplifying:
\(=\frac{1}{2} \times \frac{1}{4}+\frac{1}{2} \times i+\frac{1}{2} \times(-4)+(-2 i) \times \frac{1}{4}+(-2 i) \times i+(-2 i) \times(-4)\)
\(=\frac{1}{8}+\frac{1}{2} i-2-\frac{1}{2} i-2 i+8 i\)
\(=-\frac{15}{8}+6 i\)

Summary

The article discusses the definition of the modulus as the distance of a complex number from the origin. It also explains, followed by the derivation of the formula to find it. In this article, the conjugate of a complex number is defined with two equivalent definitions. Further, various properties of conjugates are explained in detail. The relationship between the modulus and conjugates of complex numbers helps us find the inverse of complex numbers.

Later the article concludes with details about the algebraic properties of both the modulus and conjugates of a complex number, covering all the basic details of the topic. 

FAQs on Modulus and the Conjugate of a Complex Number

Q.1. How do you find the modulus of a complex number?
Ans: The modulus of the complex number \(z=a+b i\) is:
\(|z|=\sqrt{a^{2}+b^{2}}\)
For example, the modulus of the complex number \(-3+9 i\) is:
\(|-3+9 i|=\sqrt{(-3)^{2}+9^{2}}\) or \(\sqrt{90}\)

Q.2. What is the modulus of a conjugate?
Ans: The modulus of a complex number and its conjugate is the same.
That is if \(z=a+b i\) is a complex number, then \(\bar{z}=a-b i\) and
\(|z|=\sqrt{a^{2}+b^{2}}\)
\(|\bar{z}|=\sqrt{a^{2}+b^{2}}\)
\(|\bar{z}|=|z|\)

Q.3. What is the conjugate of z Bar?
Ans: The conjugate of a conjugate is the number itself. If \(z=a+b i\) is a complex number, then \(\bar{z}=a-b i\) and the conjugate of \(\bar{z}=a-b i\) is \(a+b i\) which is the same as the number we started with. Thus, \(\overline{(\bar{z})}=z\).

Q.4. What is the conjugate of a real number?
Ans: The complex conjugate or simply conjugate of a complex number is another complex number with the same real part and an imaginary part that is the additive inverse of that of the given complex number. A real number is nothing but a complex number with zero as an imaginary part. That is, if \(x\) is a real number, it has the format \(x=a+0 i\). Thus, when the conjugate is calculated, changing the sign of the imaginary part does not affect the number. Therefore, the complex conjugate of a real number is the number itself.

Q.5. What are complex conjugates used for?
Ans: The complex conjugate or simply conjugate of a complex number is another complex number with the same real part and an imaginary part that is the additive inverse of that of the given complex number. A number and its conjugates are reflections of each other about the \(x\)-axis. Complex conjugates are used for rationalizing complex numbers and write them in standard form, separating the real and imaginary parts.

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