Operations with Complex Numbers

Operations with Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are usually written in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit, defined as the square root of -1. This topic covers the basic operations you can perform with complex numbers: addition, subtraction, multiplication, and division.

Addition of Complex Numbers

To add two complex numbers, you add their real parts and their imaginary parts separately. For example, consider the complex numbers z1 = 3 + 4i and z2 = 2 + 5i.

The addition can be calculated as follows:

z1 + z2 = (3 + 4i) + (2 + 5i)

Now, add the real parts and the imaginary parts:

Real part: 3 + 2 = 5

Imaginary part: 4 + 5 = 9

Thus, the sum is:

z1 + z2 = 5 + 9i

Subtraction of Complex Numbers

Subtraction works similarly to addition. You subtract the real parts and the imaginary parts separately. Using the same complex numbers z1 = 3 + 4i and z2 = 2 + 5i, the subtraction is calculated as follows:

z1 - z2 = (3 + 4i) - (2 + 5i)

Now, subtract the real parts and the imaginary parts:

Real part: 3 - 2 = 1

Imaginary part: 4 - 5 = -1

Thus, the difference is:

z1 - z2 = 1 - 1i

Multiplication of Complex Numbers

To multiply two complex numbers, use the distributive property (also known as the FOIL method for binomials). For example, let’s multiply z1 = 3 + 4i and z2 = 2 + 5i:

z1 × z2 = (3 + 4i)(2 + 5i)

Using the distributive property:

z1 × z2 = 3 × 2 + 3 × 5i + 4i × 2 + 4i × 5i

Calculating each term gives:

= 6 + 15i + 8i + 20i^2

Since i^2 = -1, replace 20i^2 with -20:

= 6 + 15i + 8i - 20

Now, combine like terms:

Real part: 6 - 20 = -14

Imaginary part: 15i + 8i = 23i

Thus, the product is:

z1 × z2 = -14 + 23i

Division of Complex Numbers

To divide complex numbers, you multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number a + bi is a - bi. Let’s divide z1 = 3 + 4i by z2 = 2 + 5i:

z1 / z2 = (3 + 4i) / (2 + 5i)

Multiply the numerator and denominator by the conjugate of the denominator:

z1 / z2 = (3 + 4i)(2 - 5i) / (2 + 5i)(2 - 5i)

Calculate the denominator first:

(2 + 5i)(2 - 5i) = 2^2 - (5i)^2 = 4 - 25(-1) = 4 + 25 = 29

Now calculate the numerator:

(3 + 4i)(2 - 5i) = 3 × 2 + 3 × (-5i) + 4i × 2 + 4i × (-5i)

Calculating each term gives:

= 6 - 15i + 8i - 20i^2

Replace -20i^2 with +20:

= 6 - 15i + 8i + 20

Combine like terms:

Real part: 6 + 20 = 26

Imaginary part: -15i + 8i = -7i

The numerator becomes:

26 - 7i

So, we have:

z1 / z2 = (26 - 7i) / 29

This can be separated into real and imaginary parts:

z1 / z2 = 26/29 - (7/29)i

Summary

• Addition: Combine real parts and imaginary parts separately.
• Subtraction: Subtract real parts from real parts and imaginary parts from imaginary parts.
• Multiplication: Use the distributive property and replace i^2 with -1.
• Division: Multiply by the conjugate of the denominator.

Check your understanding

1. What is the sum of the complex numbers 4 + 3i and 2 - 7i?
2. How do you subtract the complex numbers 5 + 6i and 3 + 2i?
3. Calculate the product of the complex numbers 1 + 2i and 3 + 4i.
4. Divide the complex number 7 + 8i by 1 - i.