# 1. The Square Root of Minus One!

If we want to calculate the square root of a negative number, it rapidly becomes clear that neither a positive or a negative number can do it.

For example, −1 ≠ ±1, since 12 = (−1)2 = +1.

To find −1 we must introduce a new quantity, i, defined to be such that i2 = −1. (Note that engineers often use the notation j.)

Example 1:

(a)

√−25 Since (5i)2 = 5i = 52 × i2 = 25 × (−1) = −25

(b)

√− 16/9 Since (4/3i)2 = 4/3i = 16/9 × (i)2 = −16/9

# 2. Real, Imaginary and Complex Numbers

Real numbers are the usual positive and negative numbers.

If we multiply a real number by i, we call the result an imaginary number. Examples of imaginary numbers are: i, 3i and −i⁄2. If we add or subtract a real number and an imaginary number, the result is a complex number. We write a complex number as:

 z = a + ib

- where a and b are real numbers.

# 3. Adding and Subtracting Complex Numbers

If we want to add or subtract two complex numbers, z1 = a + ib and z2 = c + id, the rule is to add the real and imaginary parts separately:

 z1 + z2 = a + ib + c + id = a + c + i(b + d) z1 − z2 = a + ib − c − id = a − c + i(b − d)

Example 2:

(a)
 (1 + i) + (3 + i) = 1 + 3 + i(1 + 1) = 4 + 2i
(b)
 (2 + 5i) − (1 − 4i) = 2 + 5i − 1 + 4i = 1 + 9i

Exercise 1: Add or subtract the following complex numbers:

(a)(3 + 2i) + (3 + i)

Solution:

 (3 + 2i) + (3 + i) = 3 + 2i + 3 + i = 3 + 3 + 2i + 2i = 6 + 3i
(b)(4 − 2i) − (3 − 2i)

Solution: Here we need to be careful with the signs!

 (4 − 2i) − (3 − 2i) = 4 − 2i − 3 + 2i = 4 − 3 − 2i + 2i = 1

- a purely real result.

(c)(−1 + 3i) + ½(2 + 2i)

Solution: The factor of 1 ⁄ 2 multiplies both terms in the complex number:

 (−1 + 3i) + ½(2 + 2i) = −1 + 3i + 1 + i = 4i
- a purely imaginary result.
(d) 1/3 (2 − 5i) − 1/6 (8 − 2i)

Solution:

 1/3 (2 − 5i) − 1/6 (8 − 2i) = 2/3 − 5/3 i − 8/6 + 2/6 i = 2/3 − 5/3 i − 4/3 + 1/3 i = 2/3 − 4/3 − 5/3 i + 1/3 i = − 2/3 − 4/3 i

- which we may also write as 2/3(1 + 2i).

Quiz 1: To which of the following does the expression (4 − 3i) + (2 + 5i) simplify?

(a)6 − 8i Incorrect - please try again!
(b)6 + 2iCorrect - well done!
(c)6 + 8i Incorrect - please try again!
(d)9 − i Incorrect - please try again!

Quiz 2: To which of the following does the expression (3 − i) − (2 − 6i) simplify?

(a)3 − 9i Incorrect - please try again!
(b)1 − 7i Incorrect - please try again!
(c)1 + 5iCorrect - well done!
(d)1 − 5i Incorrect - please try again!

# 4. Multiplying Complex Numbers

We multiply two complex numbers just as we would multiply expressions of the form (x + y) together (see the module on Brackets):

 (a + ib) (c + id) = ac + a(id) + (ib)c + (ib)(id) = ac + iad + ibc − bd = ac − bd + i(ad + bc)

Example 3:

 (2 + 3i) (3 + 2i) = 2 × 3 + 2 × 2i + 3i × 3 + 3i × 2i = 6 + 4i + 9i − 6 = 13i

Exercise 2: Multiply the following complex numbers:

(a)(3 + 2i) (3 + i)

Solution:

 (3 + 2i) (3 + i) = 3 × 3 + 3 × i + 2i × 3 + 2i × i = 9 + 3i + 6i − 2 = 9 − 2 + 3i + 6i = 7 + 9i
(b)(4 − 2i) (3 − 2i)

Solution:

 (4 − 2i) (3 − 2i) = 4 × 3 + 4 × (−2i) − 2i × 3 − 2i × (−2i) = 12 − 8i − 6i − 4 = 12 − 4 −8i − 6i = 8 −14i
(c)(−1 + 3i) (2 + 2i)

Solution:

 (−1 + 3i) (2 + 2i) = −1 × 2 − 1 × 2i + 3i × 2 + 3i × 2i = −2 − 2i + 6i − 6 = −2 − 6 − 2i + 6i = −8 + 4i
(d)(4 − 5i) (8 − 2i)

Solution:

 (2 - 5i) (8 - 3i) = 2 × 8 + 2 × (−3i) − 5i × 8 − 5i × (−3i) = 16 − 6i − 40i − 15 = 16 − 15 −6i − 40i = 1 − 46i

Quiz 3: To which of the following does the expression (2 − i) (3 + 4i) simplify?

(a)5 + 4i Incorrect - please try again!
(b)6 + 11i Incorrect - please try again!
(c)10 + 5iCorrect - well done!
(d)6 + i Incorrect - please try again!

# 5. Complex Conjugation

For any complex number, z = a + ib, we define the complex conjugate to be: z* = aib. It is very useful since:

 z + z* zz* = a + ib + (a − ib) = 2a = (a + ib)(c − id) = a2 + iab − iab − a2 − (ib)2 = a2 + b2

The modulus of a complex number is defined as: |z| = zz

Exercise 3: Combine the following complex numbers and their conjugates:

(a)If z = (3 + 2i), find z + z

Solution:

 (3 + 2i) + (3 + 2i)∗ = (3 + 2i) + (3 − 2i) = 3 + 2i + 3 − 2i = 3 + 3 + 2i − 2i = 6
(b)If z = (3 − 2i), find zz

Solution:

 (3 - 2i) (3 - 2i)∗ = (3 - 2i) (3 + 2i) = 9 + 6i − 6i − 2i × (2i) = 9 −4i2 = 9 + 4 = 13
(c)If z = (−1 + 3i), find zz

Solution:

 (−1 + 3i) (−1 + 3i)∗ = (−1 + 3i) (−1 − 3i) = (−1) × (−1) + (−1)(−3i) + 3i(−1) + 3i(−3i) = 1 + 3i − 3i − 9i2 = 1 + 9 = 10
(d)If z = (4 − 3i), find |z|

Solution:

 √(4 − 3i) (4 + 3i) = √42 + 4 × 3i − 3i × 4 − 3i × 3i = √16 + 12i − 12i − 9i2 = √16 + 9 = √25 = 5

Quiz 4: Which of the following is the modulus of 4 − 2iANS: NONE OF THEM!

(a)5 + 4i Incorrect - please try again!
(b)6 + 11i Correct - well done!
(c)10 + 5i Incorrect - please try again!
(d)6 + i Incorrect - please try again!

Explanation:

# 6. Dividing Complex Numbers

The 'trick' for dividing two complex numbers is to multiply top and bottom by the complex conjugate of the denominator:

z1/z2 = z1/z2 × z*2/z*2 = z1z*2/z2z*2

The denominator, z2z2, is now a real number.

Example 4:

 1/i = 1/i × −i/−i = −i/i × (−i) = −i/1 = −i

Example 5:

 (2 + 3i)/(1 + 2i) = (2 + 3i) /(1 + 2i) × (1 − 2i)/(1 − 2i) = (2 + 3i) (1 − 2i)/(1 + 4) = 1/5(2 + 3i) (1 − 2i) = 1/5(2 − 4i + 3 i + 6) = 1/5(8 − i)

Exercise 4: Perform the following divisions:

(a)(2 + 4i)/i

Solution:

 (2 + 4i)/i = (2 + 4i)/i × −i/−i = (2 + 4i) × (−i)/+1 = (2 + 4i) (−i) = −2i −i2 = 4 − 2i
(b)(−2 + 6i)/ (1 + 2i)

Solution:

 (−2 + 6i)/ (1 + 2i) = (−2 + 6i)/ (1 + 2i) × (1 − 2i)/(1 − 2i) = (−2 + 6i) (1 − 2i)/ (1 + 4) = 1/5 (−2 + 6i) (1 − 2i) = 1/5 (−2 + 4i + 6i − 12i2) = 1/5 (−2 + 10i + 12) = 1/5 (−2 + 12 + 10i) = 1/5 (10 + 10i) = 2 + 2i
(c)(1 + 3i)/ (2 + i)

Solution:

 (1 + 3i)/(2 + i) = (1 + 3i)/(2 + i) × (2 − i)/(2 − i) = (1 + 3i) (2 − i)/ (4 + 1) = 1/5 (2 − i + 6i − 3i2) = 1/5 (2 + 3 + 5i) = 1/5 (5 + 5i) = 1 + i
(d)(3 + 2i)/ (3 + i)

Solution:

 (3 + 2i)/(3 + i) = (3 + 2i)/(3 + i) × (3 − i)/(3 − i) = (3 + 2i) (3 − i)/ (9 + 1) = 1/10 (3 + 2i) (3 − i) = 1/10 (9 − 3i + 6i - 2i2) = 1/10 (9 + 2 + 3i) = 1/10 (11 + 3i)

Quiz 5: To which of the following does the expression: 8 − i/2 + i simplify?

(a)3 − 2i Correct - well done!
(b)2 + 3i Incorrect - please try again!
(c)4 − ½i Incorrect - please try again!
(d)4 Incorrect - please try again!

Explanation:

 8 − i/2 + i = 8 − i/2 + i   2 − i/2 − i = (8 − i)(2 − i)/ 22 + 11 = (8 × 2 + 8 × (−i) − i × 2 − i × (−i))/ 5 = 1/ 5 (16 − 8i − 2i − 1) = 1/ 5 (15 − 10i) = 3 − 2i

Quiz 6: To which of the following does the expression −2 + i/2 + i simplify?

(a)−1 Incorrect - please try again!
(b)1/5(−5 + 7i) Incorrect - please try again!
(c)−1 + ½i Incorrect - please try again!
(d) 1/5(−3 + 4i) Correct - well done!

Explanation:

 −2 + i/2 + i = −2 + i/2 + i   2 − i/2 − i = (−2 + i)(2 − i)/ 22 + 11 = 1/ 5 (−2 × 2 − 2 × (−i) + i × 2 + i × (−i)) = 1/ 5 (−4 + 2i + 2i + 1 = 1/ 5 (−3 + 4)i

# 7. Complex Numbers Quiz

In each of the following, simplify the expression and choose the solution from the options given.

1.(3 + 4i) − (2 − 3i)
(a)3 −i
(b)5 + 7i
(c)1 + 7i
(d)1 − i
2.(3 + 3i) (2 − 3i)
(a)6 − 8i
(b)6 + 8i
(c)−3 + 3i
(d)15 − 3i
3.|12 − 5i|
(a)13
(b)7
(c)119
(d)−12.5
4. (7 − 17i)/(5 − i)
(a)135 + 17i
(b)3 + i
(c)3 + 2i
(d)2 − 3i