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Operation with Complex Numbers

Operation with Complex Numbers

So far, we have dealt with real numbers such as:

2, 13, 3, 7.09,...

In this section, we shall look at a new concept called an imaginary number. Consider the square root of 2. We know that this yields the non-repeating decimal

2=1.414213562...

Now, what is the square root of -2? You might think that there is no solution to the square root of a negative number. However, this is not true! In fact, this is where the imaginary number comes into play. The concept of an imaginary number stems from the imaginary unit, denoted by the letter i, and is represented by the following derivation:

i2=-1 -1=i

Thus, the square root of -2 is simply

-2=-1×2-2=-12-2=i2-2=1.414 i (correct to 3 decimal places)

As a matter of fact, we can add real and imaginary numbers together. This structure of numbers leads us to the idea of a complex number.

A complex number is an algebraic expression that includes the factor i = √-1 and is written in the form z = a + bi.

Standard Form of Complex Numbers

The standard form of complex numbers is

z = a+ib

where

  • Re (z) = a is the real part of the complex number z

  • Im (z) = b is the imaginary part of the complex number z

This is also denoted by

z=Re (z)+Im (z) i=a+bi

Real and Imaginary Numbers

There are two important subclasses of complex numbers: for a complex number z = a + bi

  • If Im (z) = 0, then z = a is a real number

  • If Re (z) = 0, then z = bi is said to be purely imaginary

Why are Complex Numbers Important?

Complex numbers have a range of applications. For instance, they are widely used in the field of electrical engineering and quantum mechanics. Complex numbers also help us solve polynomial equations that do not have any real solutions: have a look at Graph and Solve Quadratic Equations which explains how to do this.

We can conduct basic arithmetic operations with complex numbers such as addition, subtraction, multiplication, and division.

Operations with Complex Numbers; Addition and Subtraction

In this section we will explain the most important operations you should be able to perform with complex numbers:

  • Addition and subtraction of complex numbers
  • Scalar multiplication
  • Multiplication and division of complex numbers

Addition and Subtraction of Complex Numbers

To add complex numbers, simply add the corresponding real and imaginary parts. The same rule applies when subtracting complex numbers.

Let z1 and z2 be two complex numbers with z1 = a + bi and z2 = c + di, where a, b, c, and d are real numbers.

Addition of Complex Numbers Formula

z1+z2=(a+bi)+(c+di)

Distributing the positive sign in the second term (to both the real and imaginary parts) and collecting like terms, we obtain

z1+z2=(a+c)+(b+d)i

Subtraction of Complex Numbers Formula

z1-z2=(a+bi)-(c+di)

Distributing the negative sign in the second term (to both the real and imaginary parts) and collecting like terms, we obtain

z1-z2=(a-c)+(b-d) i

Let α = 3 - 2i and β = 5 + 7i be two complex numbers

Calculate α + β

α+β=(3-2i)+(5+7i)α+β=(3+5)+(-2i+7i)α+β=8+5i

Determine α - β

α-β=(3-2i)-(5+7i)α-β=(3-5)+(-2i-7i)α-β=-2-9iα-β=-(2+9i)

Scalar Multiplication of Complex Numbers

The Scalar Multiplication of Complex Numbers is the multiplication of a real number and a complex number. In this case, the real number is also called the scalar.

To multiply a complex number by a scalar, simply multiply both the real and imaginary parts by the scalar separately.

Let z = a + bi be a complex number and c be a scalar, where a, b and c are real numbers.

Scalar Multiplication of Complex Numbers Formula

c×z=c (a+bi)=ca+cbi

Let α = 3 - 2i and β = 5 + 7i be two complex numbers

Find 7α

In this case, we are multiplying the complex number α by the real number 7 (also called scalar).

7α=7(3-2i)7α=21-14i

Evaluate 2β

In this case, we are multiplying the complex number β by the real number 2 (also called scalar).

2β=2(5+7i)2β=10+14i

Multiplication of Complex Numbers

Multiplying complex numbers is exactly the same as the binomial expansion technique: apply the FOIL method and combine like terms.

Multiplication of Complex Numbers Formula

z1×z2=(a+bi) (c+di)=ac-bd+(cb+ad) i

This is how the FOIL method works, step-by-step.

Let z1 and z2 be two complex numbers with z1 = a + bi and z2 = c + di, where a, b, c, and d are real numbers. To multiply them

  1. Write both in the standard form.
  2. Perform the binomial expansion.
  3. Combine like terms.
z1×z2=(a+bi)(c+di)z1×z2=ac+bci+adi+bdi2

Noting that i2 = -1, we obtain

z1×z2=ac+bci+adi+bd(-1)z1×z2=ac+bci+adi-bd

Simplifying this, we get

ac-bd+(cb+ad) i

Let α = 3 - 2i and β = 5 + 7i be two complex numbers.

Find α x β

α×β=(3-2i)(5+7i)α×β=15+21i-10i-14i2α×β=15+11i-14(-1)α×β=15+11i+14α×β=29+11i

Division of Complex Numbers

If you have a fraction of complex numbers, multiply the numerator and denominator by the complex conjugate of the denominator.

For a complex number z = a + bi, the complex conjugate of z is denoted by z* = a - bi.

After that, expand and simplify the expression to the standard form of complex numbers. The result is given by the following formula:

Division of Complex Numbers Formula

z1z2=a+bic+di=ac+bd+(bc-ad) ic2+d2=ac+bdc2+d2+bc-adc2+d2i

When dividing complex numbers be sure to write the final answer in its standard form.

Let's see in practice and step-by-step how to perform complex numbers division. Let z1 and z2 be two complex numbers with z1 = a + bi and z2 = c + di, where a, b, c, and d are real numbers. Dividing z1 by z2, we obtain

z1z2=a+bic+di

The complex conjugate of the denominator, z2 is z2* = c - di.

Now multiplying both the numerator and denominator by z2*, we get

z1z2=a+bic+di×c-dic-diz1z2=(a+bi)(c-di)(c+di)(c-di)

Expanding this expression, we obtain

z1z2=ac+bci-adi-bdi2c2+cdi-cdi-d2i2z1z2=ac+bci-adi-bd(-1)c2+cdi-cdi-d2(-1)z1z2=ac+bci-adi+bdc2+d2

Finally, combining like terms, we have

z1z2=ac+bd+(bc-ad) ic2+d2

Let α = 3 - 2i and β = 5 + 7i be two complex numbers. Here, β is the denominator. The complex conjugate of β is β* = 5 - 7i.

Calculate α ÷ β

αβ=3-2i5+7i

Here, β is the denominator. The complex conjugate of β is β* = 5 - 7i. Thus, multiplying the numerator and denominator by β* yields:

αβ=3-2i5+7i×5-7i5-7iαβ=(3-2i)(5-7i)(5+7i)(5-7i)αβ=15-21i-10i+14i225-35i+35i-49i2αβ=15-31i+14(-1)25-49(-1)αβ=15-31i-1425+49αβ=1-31i74αβ=174-3174i

Operation with Complex Numbers - Key takeaways

OperationFormula
Additionz1+z2=(a+bi)+(c+di)=(a+c)+(b+d) i
Subtractionz1-z2=(a+bi)-(c+di)=(a-c)+(b-d) i
Scalar Multiplicationc×z=c (a+bi)=ca+cbi
Multiplicationz1×z2=(a+bi) (c+di)=ac-bd+(cb+ad) i
Divisionz1z2=a+bic+di=ac+bdc2+d2+bc-adc2+d2i

Frequently Asked Questions about Operation with Complex Numbers

How to do operations with complex numbers: To conduct operations with complex numbers, we must first identify the real and imaginary parts of the complex number.

Operations with complex numbers include addition, subtraction, multiplication and division.

How to solve operations with complex numbers: To solve operations with complex numbers, we must first identify the real part and imaginary part of the complex number and then perform the given arithmetic procedure

How to divide complex numbers:

  1. Multiply the numerator and denominator by the complex conjugate
  2. Expand and simplify the expression
  3. Write the final answer in standard form as a + bi

The rules of a complex number refer to the relationship between a given complex number say z = a + bi and its complex conjugate z* = a - bi.

Final Operation with Complex Numbers Quiz

Question

Solve (2 + 7i) + (3 − 4i) 

Show answer

Answer

5 + 3i

Show question

Question

Solve (6 + 9i) + (12 − 17i) 

Show answer

Answer

18 - 8i

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Question

Calculate (9 + 5i) − (4 + 7i) 

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Answer

5 − 2i

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Question

Calculate (2 + 7i) − (13 + 5i) 

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Answer

-11 + 2i

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Question

Given a complex number z = 5 - 2i, find 6z

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Answer

30 - 12i

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Question

Given a complex number z = 9 + 3i, find -2z

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Answer

-18 - 6i

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Question

Evaluate (3 + 2i) (5 + 6i) 

Show answer

Answer

3 + 28i

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Question

Evaluate (5 - 9i) (4 + 2i) 

Show answer

Answer

38 -26 i

Show question

Question

What is the complex conjugate of -2 + i?

Show answer

Answer

-2 - i

Show question

Question

What is the complex conjugate of 15 - 4i?

Show answer

Answer

15 + 4i

Show question

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