MAC1105 Section 1.3 - Complex Numbers
August 27, 2023
1
Multiplication of Square Roots
The square root of a product can be converted to a product of square roots. For example:
p
√ √
(5)(3) = ( 5)( 3)
Also remember, a square root and a square will cancel.
√
52 = 5
In this way, square roots can be simplified.
√
p
p
p
√
√
150 = (5)(5)(3)(2) = (5)2 (3)(2) = ( 52 )( (3)(2)) = 5 6
1.1
Practice
Simplify the following:
√
1. 12 =
2.
3.
4.
5.
√
√
√
√
36 =
60 =
24 =
72 =
1
2
Division of Square Roots
The square root of a fraction can be converted to a fraction of square roots.
r
√
2
2
=√
3
3
It is standard to avoid having a square root in a denominator, so we would simplify in this way:
r
√
2
2
=√
3
3
r
√ !
2
2
= √
(1)
3
3
r
√ ! √ !
2
2
3
√
= √
3
3
3
r
√ √ 2
3
2
=
√ 2
3
3
r
√
2
6
=
3
3
2.1
Practice
Simplify the following:
q
5
1.
2 =
2.
3.
q
8
3
q
75
6
=
=
2
3
Imaginary Numbers
If we consider the square root of a negative number, it would need to be a number which can be multiplied
with itself to produce a negative. However, when we look for where this square root would be on our
number line, we run into a problem. Since a negative multiplied with a negative is a positive, and no two
positives multiplied together would give a negative, we appear to be out of options. The numbers found
in our number line are called real numbers, and there is no such real number that equals, for example,
the square root of -1. In order to express these missing values, a kind of number called an imaginary
number is used. Imaginary numbers are written using a symbol i, where
√
i = −1.
To convert a square root of any negative into this notation, we take the following steps:
√
−5 =?
Start with our square root of a negative
p
√
−5 = (−1)(5)
Pull out a factor of -1 from the negative
√
√
√
−5 = ( −1)( 5)
Separate a square root of -1
√
√
√
−5 = i 5
Simplify the −1 into an i
The rules for simplifying from the previous page still apply. So for example,
√
−150 =?
p
√
−150 = (−1)(2)(3)(5)(5)
√
p
p
√
−150 = ( (−1))( (2)(3))( 52 )
√
√
−150 = i5 6
It is common to write the whole number to the left of the i, and the square root to the right of the i. So
we would ideally write this:
√
√
−150 = 5i 6
Furthermore, fractions are typically written to the left of the i, like the following:
r
√
25
5 2
−
=
i
2
2
3.1
Practice
Simplify the following:
√
1. −12 =
2.
3.
√
√
−60 =
−18 =
3
4.
q
− 53 =
5.
q
− 72 =
6.
q
− 12
5 =
7.
q
− 21
10 =
4
4
Powers of imaginary numbers
Given the definition
√
−1 = i, we can figure out powers of imaginary numbers. For example,
√
−1 = i
√ 2
−1 = −1
√ 3 √ 2√
−1 = −1 −1 = (−1)(i) = −i
√ 4
√ 2
−1 = ( −1 )2 = (−1)2 = 1
Further, if we continue to
√
5
−1 , we find that
√
√
√ 4
5
−1 = ( −1)( −1 )
√
= ( −1)(1)
√
= −1
=i
√
√ 4
So for any powers of −1 that are greater than 4, we can pull out factors of −1 until we have one of
√ 2 √ 3
√ 4
√
−1,
√ −1 , −1 , or −1 . So we have found that there are only four possible values for any power
of −1.
4.1
Practice
Simplify the following:
1. i =
2. i2 =
3. i3 =
4. i4 =
5. i5 =
6. i6 =
7. i7 =
8. i8 =
9. i9 =
10. i10 =
5
5
Multiplying Imaginary Numbers
We can multiply imaginary numbers using the powers of i that we just found. For example,
(5i)(2i) = (5)(i)(2)(i)
= (5)(2)(i)(i)
2
5.1
Separate into factors
Rearrange so that i factors are grouped together
= 10(i )
Combine the factors again
= 10(−1)
Remember that i2 = −1
= −10
Done
Practice
Simplify the following:
1. (5i)(3) =
2. (3i)(5i) =
3. (20i)(10i) =
4. (2i)(3i)(4i) =
5. (i)(3i)(5i)(2i) =
6
6
Complex Numbers
We create complex numbers when we combine real numbers with imaginary numbers via addition. For
example,
2 + 3i
is a complex number, because it is the sum of a real and an imaginary number. This is also the standard
form for writing a complex number. If you are given a complex number in some other form and asked to
convert it to standard, you will want to separate it into a sum of real and imaginary parts. For example,
2 1
2+i
= + i
3
3 3
All simplification rules from before still apply here, so you will always want to completely simplify both
the real and imaginary parts in your answers.
7
7
Multiplying Complex Numbers
Complex numbers can be multiplied by distributing and using what we have learned about multiplying
imaginary numbers. For example,
(1 + 2i)(2 + i) = (1)(2 + i) + (2i)(2 + i)
Distribute
= (1)(2) + (1)(i) + (2i)(2) + (2i)(i)
2
7.1
Distribute again
= 2 + i + 4i + 2i
Simplify
= 2 + i + 4i + 2(−1)
Remember i2 = −1
= 2 + i + 4i − 2
Simplify
= 2 − 2 + i + 4i
= 0 + 5i
Gather terms
Simplify
= 5i
Done
Practice
Simplify the following:
1. (2 + 2i)(3 + 5i) =
2. (2 + 3i)(1 + i) =
8
3. (3 − i)(1 + i) =
4. (1 − i)(1 + i) =
5. (5 + 2i)(5 − 2i) =
9