Algebraic Operations on Complex Numbers

Introduction

Algebraic operations on complex numbers are given by arithmetic operations are addition, subtraction, multiplication, and division. Complex numbers make it simpler to find the square root of negative values. The concept of complex numbers was first presented when Hero of Alexandria, a Greek mathematician, attempted to compute the square root of a negative number in the first century.

Numerous scientific research, including those involving signal processing, electromagnetism, fluid physics, quantum mechanics, and vibration analysis, have made use of complex numbers. In this tutorial, we will discuss algebraic operations on complex numbers.

Complex Numbers

Real and imaginary numbers are combined to form complex numbers. The format for complex numbers is a + ib, where ib is the imaginary part and a is the real part.

Therefore, a complex number is a simple expression of the addition of two numbers, real and imaginary numbers. a is purely real and b is purely imaginary.

Real numbers − All numbers that exist in the notation of the number system, such as positive, negative, zero, whole, rational, irrational, and fractional, are real numbers.

Imaginary numbers: The numbers that are not real are imaginary. Square the imaginary number to get a negative result it is represented as i=√(-1).

Operations on Complex Numbers

Algebraic operations on complex numbers are represented in mathematics by four basic arithmetic operations: addition, subtraction, division, and multiplication. A complex number is the product of a real and an imaginary number.

Addition and Subtraction

• Complex number addition

  In addition to complex numbers, the real and imaginary parts are added separately.

  If, Z₁=m+in and Z₂=p+iq , then

  $$\mathrm{Z_1+Z_2=(m+p)+i(n+q)}$$

• Subtraction of two complex numbers

  In the subtraction of two complex numbers real and imaginary parts are subtracted separately.

  If, Z₁=m+in and Z₂=p+iq,then

  $$\mathrm{Z_1-Z_2=(m-p)+i(n-q)}$$

Multiplication

The distributive property is used to multiply two or more complex numbers. If we have two complex numbers, z = a + ib and w = c + id, then the multiplication of complex numbers z and w is expressed as z.w = (a + ib) (c + id). To find the product of complex numbers, we employ the distributive property of multiplication.

$$\mathrm{z.w = (a + ib) (c + id). }$$

$$\mathrm{z.w = (ac-bd)+i(ad+bc) }$$

Division

To divide complex numbers, we must discover a term that can multiply the numerator and denominator and eliminate the imaginary part of the denominator, resulting in a real number in the denominator.

If z = a + ib and w = c + id

Then $\mathrm{\frac{z}{w}=\frac{a + ib}{c + id}}$

Now multiply and divide by c - id

$$\mathrm{\frac{z}{w}=\frac{a + ib}{c + id}×\frac{c - id}{c - id}}$$

$$\mathrm{\frac{z}{w}=\frac{(ac+bd)}{c^2+d^2}+i \frac{(bc-ad)}{c^2+d^2}}$$

Polar Form

The polar form of complex numbers represents complex numbers in a different way than the rectangular form. Complex numbers are a combination of real and imaginary numbers. The format for complex numbers is a + ib.

However, complex numbers are represented as a mix of modules and arguments in polar form. The absolute value (modulus) is another name for a complex number's modulus. The coordinate system's real and imaginary (hypothetical) polar coordinates are used to express this polar form.

Here, the vertical axis represents the hypothetical (imaginary) axis, while the horizontal axis represents the real axis. Calculations are made for r to determine the real and complex components of the coordinates. r is the vector's length, and θ is the angle that is made with the real axis.

$$\mathrm{z = x + iy.}$$

$$\mathrm{z= r (cosθ + i sinθ)}$$

For complex numbers, r means absolute or modulus, the angle θ is known as a complex argument $\mathrm{r=|z|=\sqrt{x^2+y^2}.}$

Operations in Polar Form

Follow these methods to add/subtract complex numbers in polar form −

• Convert all complex numbers from polar to rectangular notation.

• Do addition or subtraction on complex numbers in rectangle form

• Convert the result back into polar form for your solution after evaluating it.

Solved Examples

1) Simplify, (1+i)×(4+3i)

Answer: Given equation is (1+i)×(4+3i)

$$\mathrm{\Rightarrow (4-3)+i(3+4)}$$

$$\mathrm{\Rightarrow 1+7i}$$

2) Simplify, (2+i)×(2+2i)

Answer: Given equation is (2+i)×(2+2i)

$$\mathrm{\Rightarrow (4-2)+i(4+2)}$$

$$\mathrm{\Rightarrow 2+6i}$$

3) Simplify, (11+2i)-(14+3i)

Answer: In the subtraction of two complex numbers real and the imaginary parts are subtracted separately.

$$\mathrm{(11+2i)-(14+3i)}$$

$$\mathrm{\Rightarrow (11-14)+(2i-3i)}$$

$$\mathrm{\Rightarrow -3-i}$$

4) Simplify (8+5i)+(6+9i)

Answer: In addition to complex numbers the real and imaginary parts are added separately.

$$\mathrm{(8+5i)+(6+9i)}$$

$$\mathrm{ \Rightarrow (8+6)+(5i+9i)}$$

$$\mathrm{ \Rightarrow 14+14i}$$

5) Simplify: (1+2i)×(2+3i)+(3+2i)

Answer: Given equation is (1+2i)×(2+3i)+(3+2i)

$$\mathrm{\Rightarrow (2-6)+i(4+3)+(3+2i)}$$

$$\mathrm{\Rightarrow (-4+7i)+(3+2i)}$$

$$\mathrm{\Rightarrow -1+9i}$$

6) Find the modulus of complex number z=11+12i

Answer: Given complex number is z=1+2i we have to find a modulus that is

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

$$\mathrm{|z|=\sqrt{5}}$$

7) Simplify the complex number $\mathrm{z=\frac{3+4i}{\sqrt{2}+i}}$

Answer: Given complex number is $\mathrm{z=\frac{3+4i}{\sqrt{2}+i}}$

Now multiply by its conjugate in the numerator as well as denominator

$$\mathrm{z=\frac{3+4i}{√2+i}×\frac{√2-i}{√2-i}}$$

$$\mathrm{z=\frac{3√2+4√2 i-3i+4}{2-1}}$$

$$\mathrm{z=(3√2+4)+i(4√2-3)}$$

8) Find the square of the complex number z=4+3i.

Answer: Given complex number is z=4+3i.

$$\mathrm{|z|=\sqrt{(4)^2+(3)^2}}$$

$$\mathrm{|z|=\sqrt{25}}$$

$$\mathrm{|z|=5}$$

and a=4, b=3.

Now use the formula for the square root of the complex number

$$\mathrm{\sqrt{a+ ib}=±(\sqrt{\frac{|z|+a}{2}}±\frac{ib}{|b|} \sqrt{\frac{|z|-a}{2}})}$$

Now put the value

$$\mathrm{\sqrt{a+ ib}=±(\sqrt{\frac{5+4}{2}}±\frac{i3}{|3|} \sqrt{\frac{5-3}{2}})}$$

$$\mathrm{\sqrt{a+ ib}=±(\frac{9}{2}+i)}$$

9) Find the root of the quadratic equation x²-x+1=0

Answer: We know that the quadratic formula is $\mathrm{x=\frac{-b±\sqrt{b^2-4ac}}{2a}}$

Now apply this formula for the given equation,

$$\mathrm{x=\frac{1±\sqrt{1^2-4.1.1}}{2.1}}$$

$$\mathrm{x=\frac{1±\sqrt{-3}}{2}}$$

$$\mathrm{x=\frac{1±i\sqrt{3}}{2}}$$

10) Find the root of the quadratic equation x²-3x+3=0

Answer: We know that the quadratic formula is $\mathrm{x=\frac{-b±\sqrt{b^2-4ac}}{2a}}$

Now apply this formula for the given equation,

$$\mathrm{x=\frac{3±\sqrt{(-3)^2-4.1.3}}{2.1}}$$

$$\mathrm{x=\frac{3±\sqrt{9-12}}{2}}$$

$$\mathrm{x=\frac{3±i\sqrt{3}}{2}}$$

Conclusion

Complex numbers are created by adding imaginary and real numbers together. A complex number normally has the form a + ib and is represented by the symbol z.

Re(z) is used to represent the value "a" as the real component in this instance because both a and b are real integers, Im(z) is used to represent the value "b" as the imaginary part, and ib are also referred to as an imaginary number.

FAQs

1.What do you mean by complex numbers?

Real and imaginary numbers are combined to form complex numbers. The format for complex numbers is a + ib. where ib is the imaginary part and a is the real part.

2.What is iota in complex numbers?

Iota is an imaginary unit represented by i, and the value of iota is √(-1) or i=√(-1).

3.What do you mean by the polar form of complex numbers?

In contrast to the rectangular form, complex numbers are represented differently in the polar form. Complex numbers are usually expressed in the form z = x + iy.

4.What is the modulus in complex numbers?

The separation (distance) of the complex number from the origin in the Argand plane is known as the modulus of the complex number z = a + ib. It is denoted by the symbol | z | and equals $\mathrm{| z | =\sqrt{(a)^2+(b)^2}}$

5.What is Euler's formula?

Euler's formula describes the relationship between complex exponential functions and trigonometric functions.

$$\mathrm{z=r( cos x + i sin x)}$$

$$\mathrm{z=r e^{ix}}$$