Complex Numbers

Complex Numbers Definition: Complex numbers, a fundamental concept in Mathematics, extend the realm of real numbers by introducing the imaginary unit “i”(iota), defined as the square root of –1. They comprise of real and imaginary parts, typically represented as a + ib, here a and b are real numbers. Complex numbers find wide-ranging applications in fields such as electrical engineering, Physics, and signal processing, elucidating phenomena beyond the scope of real numbers alone.

1.0What is a Complex Number?

A Complex Number is a Mathematical entity that combines a real number and an imaginary number. It is expressed in the form a + ib, where a and b are real numbers and i (iota) represents the imaginary unit, defined as the square root of –1. Complex numbers extend the number system beyond real numbers, allowing for the representation of quantities that involve both real and imaginary components. They are essential in various branches of Mathematics and have widespread applications in fields like engineering, physics, and signal processing.

Example: Z = 3 + 4i; Z = –7 + 3i

2.0Conjugate of a Complex Number

The conjugate operation applied to a complex number , Z = a + ib is represented as  and it is derived by negating the imaginary part. (ib becomes –ib).

The conjugate of a complex number is another complex number obtained by changing the sign of the imaginary part. For a complex number z = a + ib, its conjugate is a – ib. In other words, if the original complex number lies in the complex plane, its conjugate is reflected across the real axis. The conjugate of a complex number plays a vital role in various mathematical operations, such as simplifying expressions and finding roots, as well as in applications like signal processing and control theory.

Examples: $\mathrm{Z}=4+3\mathrm{i}\text{ so, }\bar{Z}=4-3\mathrm{i}$

${\mathrm{Z}}_1=-3-\mathrm{i}\text{ so, }\overline{Z_2}=-3+\mathrm{i}$

3.0Modulus of a Complex Number

The modulus of a complex number denoted as |z|, represents its distance from the origin in the complex plane. It is a measure of the magnitude or length of the complex number.

Given a complex number z = a + ib, where a is the real part and b is the imaginary part, the modulus |z| is calculated as:

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

It involves finding the square root of the sum of squares of both the real and imaginary components of the complex number.

Example: $Z=3+4i\text{ so, }|Z|=\sqrt{9+16}=5$

$Z_1=-7+2\mathrm{i}\text{ so, }\left|Z_1\right|=\sqrt{49+4}=\sqrt{53}$

The modulus provides valuable geometric information about the complex number's position in the complex plane. It is always a non-negative real number. If the modulus is zero, the complex number lies at the origin, while larger values indicate greater distance from the origin.

The modulus of a complex number is also used in various mathematical operations involving complex numbers, such as finding the distance between two complex numbers, and determining the magnitude of vectors.

4.0Polar Form of a Complex Number

The polar form of a complex number is represented in terms of its magnitude (modulus) and argument (angle). By utilizing trigonometric functions, we can express a complex number z as r (cos θ + i sin θ), where r is the modulus and θ is the argument.

It provides an alternative way to express complex numbers, particularly useful for simplifying certain operations like multiplication and exponentiation.

The magnitude r of the complex number represents its distance from the origin within the complex plane, computed by the square root of the sum of squares of both the real and imaginary components of the complex number, where a and b are the real and imaginary parts of the complex number, respectively.

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

5.0Argument of a Complex Number

The argument of a complex number indicates the angle it forms with the +ve real axis within the complex plane. This angle offers essential geometric details regarding the positioning and alignment of the number.

Given a complex number z = a + ib, where a is the real part and b is the imaginary part, the argument θ is calculated using trigonometric functions:

$\tan \theta =\frac{b}{a}$

However, this formula alone doesn't account for all possible cases. It's important to consider the quadrant in which the complex number lies, as the angle may need adjustment based on the signs of a and b.

For instance:

• If a > 0 and b > 0, the argument lies in the first quadrant $\left(0<\theta <\frac{\pi }{2}\right)$ 
• If a < 0 and b > 0, the argument lies in the second quadrant $\left(\frac{\pi }{2}<\theta <\pi \right)$ .
• If a < 0 and b < 0, the argument lies in the third quadrant. $\left(\pi <\theta <\frac{3\pi }{2}\right)$ .
• If a > 0 and b < 0, the argument lies in the fourth quadrant $\left(\frac{3\pi }{2}<\theta <2\pi \right)$ .

Here θ is called the argument or amplitude of Z and we write it as arg(z) = θ. The Unique Value Of θ such that –π ≤ θ ≤ π is called the Principal argument.

In practice, the argument is often expressed in radians or degrees, depending on the context. It's a fundamental concept in complex analysis, providing insight into the behavior of complex functions and their properties.

6.0Complex Number Formula

Various formulas govern arithmetic operations involving complex numbers, including addition, subtraction, multiplication, and division. Understanding these formulas is essential for manipulating complex quantities effectively.

7.0Multiplicative Inverse of a Complex Number

The multiplicative inverse (reciprocal) of a complex number z is another complex number w such that their product equals 1. It is given by.

$\omega =\frac{1}{z}=\frac{\overline{\mathrm{z}}}{|z{|}^2}$

 Properties of Complex Numbers:

Complex numbers exhibit various properties, including commutativity, associativity, distributivity, and the existence of additive and multiplicative identities. These properties form the foundation for performing operations on complex numbers.

Complex numbers possess several properties that govern their arithmetic and algebraic operations. Some of the key properties include:

1. Closure: The sum, difference, and product of two complex numbers are also complex numbers. In other words, complex numbers are closed under addition, subtraction, and multiplication.
2. Commutativity and Associativity: Addition and multiplication of complex numbers are commutative and associative. That is, for any complex numbers z₁ and z₂:

• z₁ + z₂ = z₂ + z₁
• z₁ × z₂ = z₂ × z₁
• (z₁ + z₂) + z₃ = z₁ + (z₂ + z₃)
• (z₁ × z₂) × z₃= z₁ × (z₂ × z₃) 

3. Distributive Property: Multiplication distributes over addition for complex numbers. That is, for any complex numbers z₁, z₂, and z₃:

• z₁ × (z₂ + z₃) = z₁ × z₂ + z₁ × z₃

4. Additive Identity: The complex number 0 acts as the additive identity, meaning that for any complex number z, z + 0 = z.
5. Multiplicative Identity: The complex number 1 acts as the multiplicative identity, meaning that for any complex number z, z × 1 = z.
6. Additive Inverse: For every complex number z, there exists an additive inverse –z, such that z + (–z) = 0.
7. Multiplicative Inverse (Non-zero Complex Numbers): For every non-zero complex number z, there exists a multiplicative inverse z^–1, such that z × z^–1 = 1.
8. Conjugate Property: The multiplication of a complex number by its conjugate yields the square of its modulus.

The product of a complex number and its conjugate is equal to the square of its modulus. For any complex number z = a + ib, its conjugate $\overline{\mathrm{Z}}=\mathrm{a}-\mathrm{ib}$, and its 

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

$\mathrm{z}\times \overline{\mathrm{z}}=|\mathrm{z}{|}^2$

These properties form the basis for performing operations and simplifying expressions involving complex numbers in various mathematical contexts.

8.0Euler's Formula

Euler's formula is a profound result in complex analysis that establishes a deep connection between the exponential function, trigonometric functions, and complex numbers. It states that for any real number x:

e^ix = cos(x) + i sin(x) 

where e denotes the base of the natural logarithm, i is iota, cos(x) represents the cosine function, and sin(x) represents the sine function.

This formula beautifully explains the relationship between exponential growth and circular motion, illustrating how complex numbers can be expressed using trigonometric functions. It also provides a compact and elegant way to express complex numbers in their polar form.

By utilizing Euler's formula, complex numbers can be expressed in the form:

$\mathrm{z}=\mathrm{r}.{\mathrm{e}}^{\mathrm{i}\theta }$

where r is the magnitude (modulus) of the complex number and θ is its argument (angle). This representation, known as the exponential form of a complex number, simplifies various operations such as multiplication, exponentiation, and finding roots.

Euler's formula has widespread applications in mathematics, physics, engineering, and signal processing. It is instrumental in solving differential equations, analyzing periodic phenomena, and understanding the behavior of waves and oscillations.

9.0Solved Example of a Complex Number

Example 1: The conjugate of 6i – 5 is

(A) (6i + 5) (B) (–6i –5) (C) (–6i + 5) (D) None of these

Ans. (B)

Solution:

6i – 5 = (–5) + (6i)

⇒ conjugate of 6i – 5 = (–5 – 6i)

Example 2: Find square root of 5 + 12i 

Solution:

Let $a+ib=\sqrt{5+12i}$ ⇒ 5 + 12i = (a + ib)²

5 + 12i = (a² – b²) + 2iab ⇒ a² – b² = 5 ...(i)

and 2ab = 12 ...(ii)

Now (a² + b² )² = (a² – b² )² + 4a² b²

(a² + b² )² = 5² + 12² = 169

a² + b² = 13 (∵ a² + b² > 0) ...(iii)

Solving (i) & (iii), we get

a² = 9 and b² = 4 ⇒ a = ± 3 and b = ± 2

from (ii), 2ab is positive, so a and b are of the same sign.

a = 3, b = 2 or a = –3 and b = –2

Hence, $\sqrt{5+12\mathrm{i}}=\pm (3+2\mathrm{i})$ 

Example 3: Write  $\frac{(1+7i)}{(2-i{)}^2}$  in cartesian form-

(A) 1 + i (B) –1 – i (C) –1 + i (D) 1 – i

Ans. (C)

Solution:

Let $z=\frac{(1+7i)}{(2-i{)}^2}$ . 

Then, $\mathrm{z}=\frac{1+7\mathrm{i}}{4-4\mathrm{i}+{\mathrm{i}}^2}=\frac{1+7\mathrm{i}}{3-4\mathrm{i}}=\left(\frac{1+7\mathrm{i}}{3-4\mathrm{i}}\right)\left(\frac{3+4\mathrm{i}}{3+4\mathrm{i}}\right)=\frac{-25+25\mathrm{i}}{25}=-1+\mathrm{i}$ 

Example 4: Evaluate amplitude and |z| for $z=-1+\sqrt{2}i$. 

Solution: 

$\mathrm{z}=-1+\mathrm{i}\sqrt{2}$

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

$\mathrm{Arg}z=\pi -{\tan }^{-1}\left(\frac{\sqrt{2}}{1}\right)=\pi -{\tan }^{-1}(\sqrt{2})$

Example 5: Exponential form of $\frac{1+i}{1-i}$  is

(A) $e^{-i\frac{\pi }{2}}$ (B) $e^{-i\frac{\pi }{4}}$ (C)  $e^{-\mathrm{i}\frac{3\pi }{4}}$ (D) $e^{i\frac{\pi }{2}}$

Ans. (D)

Solution:

$\mathrm{z}=\frac{1+\mathrm{i}}{1-\mathrm{i}}$

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

Example 6: Real part of $\frac{3}{\cos \theta +i\sin \theta +2}$ is equal to

(A) $\frac{3(2+\cos \theta )}{5+4\cos \theta }$

(B) $\frac{3(2-\cos \theta )}{5+4\cos \theta }$

(C) $\frac{2(3+\cos \theta )}{5+4\cos \theta }$

(D) $\frac{3(2+\cos \theta )}{5-4\cos \theta }$

Ans. (A)

Solution:

$x+iy=\frac{3}{\cos \theta +i\sin \theta +2}$  ...(i)   

$x-iy=\frac{3}{2+\cos \theta -i\sin \theta }$ ...(ii)  

Adding (i) & (ii)

$x=\frac{3(2+\cos \theta )}{5+4\cos \theta }$

Example 7: Find Polar Complex Number Z = 1 + i

Solution: $|Z|=\sqrt{1^2+1^2}=\sqrt{2}$

$\tan \theta =1\Rightarrow \theta =\frac{\pi }{4}$

$\Rightarrow Z=\sqrt{2}\left(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4}\right)$

10.0Sample Questions for Complex Numbers

1. What is the modulus of a complex number?

Ans: The modulus of a complex number z = a + ib is the distance of the complex number from the origin in the complex plane and is denoted as |z|. It is calculated as $|z|=|z|=\sqrt{a^2+b^2}.$

2. How do you find the argument of a complex number?

Ans: The argument of a complex number z = a + ib is the angle it makes with the positive real axis in the complex plane. It is typically calculated as $tan\theta =\frac{b}{a}$, taking into account the signs of a and b.

3. What is Euler's formula?

Ans: Euler's formula states that e^ix = cos(x) + i sin(x), where e is the base of the natural logarithm, i is the iota, and cos(x) and sin(x) are the cosine and sine functions, respectively. It is a fundamental result in complex analysis.