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Intro to the imaginary numbers

Learn about the imaginary unit i, about the imaginary numbers, and about square roots of negative numbers.
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 number solution to the equation x2=1. This is because it is impossible to square a real number and get a negative number!
However, a solution to the equation x2=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
  • i2=1
The second property shows us that the number i is indeed a solution to the equation x2=1. The previously unsolvable equation is now solvable with the addition of the imaginary unit!

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, i5, 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.
Using the fact that i2=1, we can simplify this further as shown.
The fact that (3i)2=9 means that 3i is a square root of 9.

Check your understanding

What is (4i)2?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Which of the following is a square root of 16?
Choose 1 answer:

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 formSimplified form
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.
Original equivalenceThought process
9=3iThe 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=ia
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.


Simplify 18.


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 i18.
Next we can simplify 18 using what we already know about simplifying radicals.
The work is shown below.
18=i18For a>0a=ia=i929 is a perfect square factor of 18=i92ab=ab when a,b0=i329=3=3i2Multiplication is commutative
So it follows that 18=3i2.

Let's practice some problems

Problem 1

Simplify 25.

Problem 2

Simplify 10.

Problem 3

Simplify 24.

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=1; we need the integers for this!
  • With only the integers, we can't solve 3x1=0; we need the rational numbers for this!
  • With only the rational numbers, we can't solve x2=2. Enter the irrational numbers and the real number system!
And so, with only the real numbers, we can't solve x2=1. We need the imaginary numbers for this!
As you continue to study mathematics, you will begin to see the importance of these numbers.

Want to join the conversation?

  • blobby green style avatar for user Dlnickelson17
    what would -i^-i be, would it just be 2^2
    (88 votes)
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  • piceratops seed style avatar for user Michael Mendoza
    What is the real world application for this??
    (59 votes)
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    • male robot donald style avatar for user kurt westphal
      design, simulation, analysis of normal and semiconductor circuits, acoustics and speakers, physics., mechanical system vibration, automotive exhaust note tuning, guitar pickups and boutique high power tube/solid state amplifiers, chemical engineering linear/non linear flows, financial modeling, statistics and big data,
      (173 votes)
  • duskpin ultimate style avatar for user Vestige
    my brain blew up
    (45 votes)
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  • blobby green style avatar for user Mojeb Rahman Zameeri
    hard to believe there are people out there that imagine numbers
    (30 votes)
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  • mr pants teal style avatar for user Daisy Dukealoopakiss
    Can you have different answers to simplifying depending on what numbers you take from the original, or would those be wrong? For example: Problem 3, instead of using 4 and 6 I used 8 and 3 and it came out to be 2i x square of 2 x square of 2 x square of 3, but it was counted as wrong. Was it wrong because it wasn't what Kahn had, or because it was just wrong?
    (20 votes)
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  • winston default style avatar for user 27svinay
    How would one use an imaginary number in real life? If it is imaginary, would it have any use cases? If so, how exactly would you need to use it?
    (9 votes)
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    • leaf green style avatar for user Tanner P
      Imaginary numbers are used a lot in electrical engineering. They can also used to prove a lot of formulas that are useful in real life. And they are useful in any field that uses quadratic equations or polynomials.

      When you first learned about negative numbers, they probably seemed weird. How can you have less than nothing? You can’t have -1 apples and you definitely can’t have i apples. But you know now how much math depends on using numbers less than zero, and the same thing goes for imaginary numbers.
      (44 votes)
  • purple pi purple style avatar for user Syeda5
    If imaginary numbers aren't real, how is it possible to use them in real life? You can't count things that don't exist so how do you use them?
    (6 votes)
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    • leaf green style avatar for user kubleeka
      None of the numbers you use in life are real. Can you show me a 3? Not a drawing or a representation of a 3, but the actual number 3? Of course not. It's just an abstraction.

      You mention counting, but most numbers aren't used for counting either. You can't have exactly √2 apples, or any irrational number of apples. That would require splitting atoms and quarks in impossible ways. Yet a vast majority of the real numbers are irrational. They're not about counting either.

      Numbers are just concepts that follow certain rules. The misleadingly-named real numbers are defined as a complete ordered field. The word "field" just means that they follow 9 certain rules, like "for every real number x, x+0=x" Likewise, "ordered" just adds about 3 more rules, and "complete" adds one more. Any relation to real life is just the result of people applying these abstractions to real-world problems.

      To get the complex numbers, we do a similar thing. Take the real numbers and add in
      1. Every real number is complex.
      2. There is a complex number i such that i²= -1.
      3. The sum of two complex numbers is complex.
      4. The product of two complex numbers is complex.
      5. For any two complex numbers a and b, a^b is complex.

      Now we have this concept of "the complex numbers" that we can further explore. Application to reality is not necessary.
      (47 votes)
  • blobby green style avatar for user Sarah Myers
    Does it matter if the i is in front or behind of the solution.
    (11 votes)
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    • leaf blue style avatar for user Stefen
      As long as it is clear what the i is affecting, you can do both.
      EG (2 + 3i) + (4 + 5i) = (2 + 4) + i(3 + 5) or (2 + 4) + (3 + 5)i
      However, there are conventions.
      When we simplify the above we would normally write 6 + 8i, not 6 + i8, but both are fine, but the second one just looks weird. For example, you are used to the notation "1 + 2", but the following notations "+ 1 2" or "1 2 +" are also acceptable in many situations, through they probably looks weird to you now. (The 1st is Polish Notation, the 2nd Reverse Polish Notation)

      Another convention is to place the i before the radical, eg i√8. If you want to place it after, make sure to use parenthesis: (√8)i or √8(i), so as to avoid confusion. If you write √8i, do you mean (√8)i or √(8i)?

      As you keep studying, you will get more and more exposure to the notation conventions we use.
      (32 votes)
  • stelly blue style avatar for user KaBoom
    this make no sense
    (11 votes)
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  • aqualine seedling style avatar for user kitty-chan
    Where is I on the number line?
    (6 votes)
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    • leaf green style avatar for user Tanner P
      Great question! You can't find i on the number line because it only represents real numbers. So, instead we use the complex plane to represent those numbers. On the complex plane, the real-number axis is horizontal and the imaginary axis is vertical.

      And the complex plane opens up a lot of interesting ways to look at complex numbers. For example, the complex number 3+4i would be represented by the point (3,4) on the complex plane. So what would the absolute value of 3+4i be? It would be 5, because the distance from the origin (0,0) to (3,4) is 5.
      (21 votes)