Download The imaginary unit

1/31/2017
(67) Intro to the imaginary numbers | What are the imaginary numbers? | Introduction to complex numbers | Algebra II | Khan Academy
Intro to the imaginary numbers
Learn about the imaginary unit i, about the imaginary numbers, and about square roots of
negative numbers.
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In your study of mathematics, you may have noticed that some quadratic
equations do not have any real number solutions.
For example, try as you may, you will never be able to find a real
2
number solution to the equation x
= −1. This is because it is impossible to
square a real number and get a negative number!
2
However, a solution to the equation x
= −1 does exist in a new number
system called the complex number system.
The imaginary unit
The backbone of this new number system is the imaginary unit, or the
number i.
The following is true of the number i:
i = √−1
2
i = −1
The second property shows us that the number i is indeed a solution to the
2
equation x = −1. The previously unsolvable equation is now solvable with
the addition of the imaginary unit!
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Pure imaginary numbers
The number i is by no means alone! By taking multiples of this imaginary unit,
we can create infinitely many more pure imaginary numbers.
For example, 3i, i√5, and −12i are all examples of pure imaginary numbers,
or numbers of the form bi, where b is a nonzero real number.
Taking the squares of these numbers sheds some light on how they relate to
the real numbers. Let's investigate this by squaring the number 3i. The
properties of integer exponents remain the same, so we can square 3i just as
we'd imagine.
2
2 2
(3i) = 3 i
2
= 9i
2
Using the fact that i
= −1, we can simplify this further as shown.
2
= 9i
= 9(−1)
= −9
2
The fact that (3i)
= −9 means that 3i is a square root of −9.
Check your understanding
2
What is (4i) ?
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(67) Intro to the imaginary numbers | What are the imaginary numbers? | Introduction to complex numbers | Algebra II | Khan Academy
Check
[I need help!]
Which of the following is a square root of −16?
−4
4i
Check
[I need help!]
In this way, we can see that pure imaginary numbers are the square roots of
negative numbers!
Simplifying pure imaginary numbers
The table below shows examples of pure imaginary numbers in both
unsimplified and in simplified form.
Unsimplified form
Simplified form
√ −9
3i
√ −5
i√5
−√ −144
−12i
But just how do we simplify these pure imaginary numbers?
Let's take a closer look at the first example and see if we can think through
the simplification.
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Original
Thought process
equivalence
√ −9 = 3i
The square root of −9 is an imaginary number. The square
root of 9 is 3, so the square root
of negative 9 is 3 imaginary units, or 3i.
The following property explains the above "thought process" in mathematical
terms.
For a
> 0, √−a
= i√a
If we put this together with what we already know about simplifying radicals,
we can simplify all pure imaginary numbers. Let's look at an example.
Example
Simplify √−18 .
Solution
First, let's notice that √−18 is an imaginary number, since it is the square
root of a negative number. So, we can start by rewriting √−18 as i√18.
Next we can simplify √18 using what we already know about simplifying
radicals.
The work is shown below.
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(67) Intro to the imaginary numbers | What are the imaginary numbers? | Introduction to complex numbers | Algebra II | Khan Academy
−−−−
−−
√ −18 = i√ 18
For a > 0, √ −−
−a = i√ a
−−−−
=i⋅√ 9⋅2
9 is a perfect square factor of 18
= i√ 9 ⋅ √ 2
−−
√ ab = √ a ⋅ √ b when a, b ≥ 0
=i⋅3⋅√ 2
√9=3
= 3i√ 2
Multiplication is commutative
So it follows that √−18
= 3i√2.
[I'm still confused. I'd like to see another example.]
Let's practice some problems
Problem 1
Simplify √−25 .
Check
[I need help!]
Problem 2
Simplify √−10 .
Check
[I need help!]
Problem 3
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Problem 3
Simplify √−24 .
Check
[I need help!]
Why do we have imaginary numbers anyway?
The answer is simple. The imaginary unit i allows us to find solutions to many
equations that do not have real number solutions.
This may seem weird, but it is actually very common for equations to be
unsolvable in one number system but solvable in another, more general
number system.
Here are some examples with which you might be more familiar.
With only the counting numbers, we can't solve x + 8
integers for this!
With only the integers, we can't solve 3x − 1
numbers for this!
= 1; we need the
= 0; we need the rational
2
With only the rational numbers, we can't solve x
numbers and the real number system!
2
And so, with only the real numbers, we can't solve x
imaginary numbers for this!
= 2. Enter the irrational
= −1. We need the
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As you continue to study mathematics, you will begin to see the importance
of these numbers.
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