Would it be correct to say that the number 0 is a real number, pure imaginary number and a complex number all at the same time?

Logically, one could make an argument that 0 is neither real nor imaginary, since is has neither an imaginary nor a real part. But excluding it from either the real or the imaginary axis would be extremely awkward; so we define "purely imaginary" in a negative way, not as a number that HAS only an imaginary part, but as one that DOES NOT have any (non-zero) real part: So, yes, 0 is a real number, pure imaginary number (0i) and a complex number (0i) all at the same time.

Where can you apply the complex plane in real life?

Not much. But it comes out a lot in future mathematics.

Are complex numbers included in real numbers? If so, are they irrational?

It is actually the other way around. The Real numbers are a subset of the set that contains all of Complex numbers, so are the Imaginary numbers. Imagine a big circle with 2 small circles inside it that don't intersect with each other, that would be the set of the Complex number (big circle) and the Real and Imaginary sets (small circles).

for something complex it isn’t as complex as i thought

It's just a name. It has nothing to do with the level of difficulty.

Why do imaginary numbers exist on the y axis?

There is no particular rule that says you must put imaginary numbers on the y-axis. You can also plot them on the x-axis. But imaginary numbers are mostly plotted on the y-axis and hence this has become somewhat of a convention.

What is the complex plane for?

The complex plane is used to visualize complex numbers. You’ll learn later on that multiplying complex numbers can be thought of as rotations around the plane, similar to how negative numbers are like reflections on the number line. So, the complex plane helps visualize complex numbers and certain operations you can do with them. Hope this helps!

are numbers like -1 and -3 complex?

Yes. All real numbers are in the set of complex numbers, but their imaginary component is 0. -1 written in standard form for a complex number is: -1+0i. Similarly, -3 written in standard form for a complex number is -3+0i.

Can we use the why axis instead of the x axis as the real number plane?

The real axis is by convention the horizontal axis of the complex plane. There's nothing really stopping you from doing it the other way around, but that just complicates things unnecessarily and makes it harder to communicate your ideas with others.

Main content

Algebra 2

Course: Algebra 2 > Unit 2

Lesson 3: The complex plane

The complex plane

Google Classroom

Learn what the complex plane is and how it is used to represent complex numbers.

The Imaginary unit, or i, is the number with the following equivalent properties:

i, squared, equals, minus, 1
square root of, minus, 1, end square root, equals, i

A complex number is any number that can be written as start color #1fab54, a, end color #1fab54, plus, start color #11accd, b, end color #11accd, i, where i is the imaginary unit and start color #1fab54, a, end color #1fab54 and start color #11accd, b, end color #11accd are real numbers.

start color #1fab54, a, end color #1fab54 is called the start color #1fab54, start text, r, e, a, l, end text, end color #1fab54 part of the number, and start color #11accd, b, end color #11accd is called the start color #11accd, start text, i, m, a, g, i, n, a, r, y, end text, end color #11accd part of the number.

The complex plane

Just like we can use the number line to visualize the set of real numbers, we can use the complex plane to visualize the set of complex numbers.

The complex plane consists of two number lines that intersect in a right angle at the point left parenthesis, 0, comma, 0, right parenthesis.

The horizontal number line (what we know as the x-axis on a Cartesian plane) is the real axis.

The vertical number line (the y-axis on a Cartesian plane) is the imaginary axis.

Plotting a complex number

Every complex number can be represented by a point in the complex plane.

For example, consider the number 3, minus, 5, i. This number, also expressed as start color #1fab54, 3, end color #1fab54, plus, left parenthesis, start color #11accd, minus, 5, end color #11accd, right parenthesis, i, has a real part of start color #1fab54, 3, end color #1fab54 and an imaginary part of start color #11accd, minus, 5, end color #11accd.

The location of this number on the complex plane is the point that corresponds to start color #1fab54, 3, end color #1fab54 on the real axis and start color #11accd, minus, 5, end color #11accd on the imaginary axis.

So the number start color #1fab54, 3, end color #1fab54, plus, left parenthesis, start color #11accd, minus, 5, end color #11accd, right parenthesis, i is associated with the point left parenthesis, start color #1fab54, 3, end color #1fab54, comma, start color #11accd, minus, 5, end color #11accd, right parenthesis. In general, the complex number start color #1fab54, a, end color #1fab54, plus, start color #11accd, b, end color #11accd, i corresponds to the point left parenthesis, start color #1fab54, a, end color #1fab54, comma, start color #11accd, b, end color #11accd, right parenthesis in the complex plane.

Check your understanding

Problem 1

Plot the complex number minus, 4, plus, 7, i.

Problem 2

Plot the complex number 6, i, plus, 1.

Problem 3

Plot the complex number minus, i, minus, 3.

Problem 4

Plot the complex number 4, i.

Problem 5

Plot the complex number minus, 7.

Connections to the real number line

In Pythagoras's days, the existence of irrational numbers was a surprising discovery! They wondered how something like square root of, 2, end square root could exist without an accurate complete decimal expansion.

The real number line, however, helps rectifying this dilemma. Why? Because square root of, 2, end square root has a specific location on the real number line, showing that it is indeed a real number. (If you take the diagonal of a unit square and place one end on 0, the other end corresponds to the number square root of, 2, end square root.)

Likewise, every complex number does indeed exist because it corresponds to an exact location on the complex plane! Perhaps by being able to visualize these numbers, we can understand that calling these numbers "imaginary" was an unfortunate misnomer.

Complex numbers exist and are very much a part of mathematics. The real number line is simply the real axis on the complex plane, but there is so much beyond that single line!

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g1y
Posted 7 years ago. Direct link to g1y's post “Are complex numbers inclu...”
Are complex numbers included in real numbers? If so, are they irrational?
Button navigates to signup pageComment on g1y's post “Are complex numbers inclu...”
(9 votes)
Answer
- Hugo Barbosa
  Posted 3 years ago. Direct link to Hugo Barbosa's post “It is actually the other ...”
  It is actually the other way around. The Real numbers are a subset of the set that contains all of Complex numbers, so are the Imaginary numbers. Imagine a big circle with 2 small circles inside it that don't intersect with each other, that would be the set of the Complex number (big circle) and the Real and Imaginary sets (small circles).
  Comment on Hugo Barbosa's post “It is actually the other ...”
  (60 votes)
Muramalla Lasya Gayathri
Posted 6 years ago. Direct link to Muramalla Lasya Gayathri's post “"If you take the diagonal...”
"If you take the diagonal of a unit square and place one end on 0, the other end corresponds to the number √2." explain please !!
Button navigates to signup pageButton navigates to signup page
(23 votes)
Answer
- kubleeka
  Posted 6 years ago. Direct link to kubleeka's post “A unit square is a square...”
  A unit square is a square of side length 1. The diagonal of this square has length √2, which you can prove with Pythagorean theorem.
  
  Now look at the number line. If we take that diagonal of length √2 and place one end at 0, then the other end will fall a distance of √2 from 0. So it will fall on the point √2.
  Comment on kubleeka's post “A unit square is a square...”
  (35 votes)
kyalopatrick24
Posted 6 years ago. Direct link to kyalopatrick24's post “Would it be correct to sa...”
Would it be correct to say that the number 0 is a real number, pure imaginary number and a complex number all at the same time?
Button navigates to signup pageButton navigates to signup page
(10 votes)
Answer
- Ron Joniak
  Posted 6 years ago. Direct link to Ron Joniak's post “Logically, one could make...”
  Logically, one could make an argument that 0 is neither real nor
  imaginary, since is has neither an imaginary nor a real part. But
  excluding it from either the real or the imaginary axis would be
  extremely awkward; so we define "purely imaginary" in a negative way,
  not as a number that HAS only an imaginary part, but as one that DOES
  NOT have any (non-zero) real part:
  
  So, yes, 0 is a real number, pure imaginary number (0i) and a complex number (0i) all at the same time.
  Comment on Ron Joniak's post “Logically, one could make...”
  (28 votes)
rylan.wetsell
Posted 4 months ago. Direct link to rylan.wetsell's post “What does it mean by "√2,...”
What does it mean by "√2, has a specific location on the real number line, showing that it is indeed a real number. (If you take the diagonal of a unit square and place one end on 0, the other end corresponds to the number √2.)"?
Button navigates to signup pageComment on rylan.wetsell's post “What does it mean by "√2,...”
(10 votes)
Answer
- William Ravanesi
  Posted 4 months ago. Direct link to William Ravanesi's post “Imagine the diagonal of a...”
  Imagine the diagonal of a square as its own line, then rotate it 45 degrees and place it parallel to a number line with one end at zero. You can see that since the other end is at √2, the number actually exists, albeit in a form that gives us an infinite number of decimals.
  Button navigates to signup page
  (13 votes)
nikhilsivajiss
Posted 3 months ago. Direct link to nikhilsivajiss's post “Where can you apply the c...”
Where can you apply the complex plane in real life?
Button navigates to signup pageComment on nikhilsivajiss's post “Where can you apply the c...”
(10 votes)
Answer
- Hummingbird.
  Posted 3 months ago. Direct link to Hummingbird.'s post “Not much. But it comes ou...”
  Not much. But it comes out a lot in future mathematics.
  Comment on Hummingbird.'s post “Not much. But it comes ou...”
  (10 votes)
kitty-chan
Posted 2 months ago. Direct link to kitty-chan's post “What is the complex plane...”
What is the complex plane for?
Button navigates to signup pageButton navigates to signup page
(6 votes)
Answer
- Tanner P
  Posted 2 months ago. Direct link to Tanner P's post “The complex plane is used...”
  The complex plane is used to visualize complex numbers. You’ll learn later on that multiplying complex numbers can be thought of as rotations around the plane, similar to how negative numbers are like reflections on the number line. So, the complex plane helps visualize complex numbers and certain operations you can do with them.
  
  Hope this helps!
  Button navigates to signup page
  (13 votes)
2027nayakneal
Posted 5 months ago. Direct link to 2027nayakneal's post “for something complex it ...”
for something complex it isn’t as complex as i thought
Button navigates to signup pageButton navigates to signup page
(9 votes)
Answer
- Venkata
  Posted 5 months ago. Direct link to Venkata's post “It's just a name. It has ...”
  It's just a name. It has nothing to do with the level of difficulty.
  Button navigates to signup page
  (4 votes)
Naveen Giri
Posted 6 years ago. Direct link to Naveen Giri's post “Can we use the why axis i...”
Can we use the why axis instead of the x axis as the real number plane?
Button navigates to signup pageButton navigates to signup page
(0 votes)
Answer
- andrewp18
  Posted 6 years ago. Direct link to andrewp18's post “The real axis is by conve...”
  The real axis is by convention the horizontal axis of the complex plane. There's nothing really stopping you from doing it the other way around, but that just complicates things unnecessarily and makes it harder to communicate your ideas with others.
  Comment on andrewp18's post “The real axis is by conve...”
  (23 votes)
Arpan Agarwala
Posted 3 months ago. Direct link to Arpan Agarwala 's post “Why do imaginary numbers ...”
Why do imaginary numbers exist on the y axis?
Button navigates to signup pageComment on Arpan Agarwala 's post “Why do imaginary numbers ...”
(7 votes)
Answer
- Quantum
  Posted 3 months ago. Direct link to Quantum's post “There is no particular ru...”
  There is no particular rule that says you must put imaginary numbers on the y-axis. You can also plot them on the x-axis. But imaginary numbers are mostly plotted on the y-axis and hence this has become somewhat of a convention.
  Button navigates to signup page
  (6 votes)
rehadam31
Posted 5 months ago. Direct link to rehadam31's post “are numbers like -1 and -...”
are numbers like -1 and -3 complex?
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(4 votes)
Answer
- Kim Seidel
  Posted 5 months ago. Direct link to Kim Seidel's post “Yes. All real numbers ar...”
  Yes. All real numbers are in the set of complex numbers, but their imaginary component is 0.
  -1 written in standard form for a complex number is: -1+0i. Similarly, -3 written in standard form for a complex number is -3+0i.
  Button navigates to signup page
  (8 votes)