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## Algebra 2

### Course: Algebra 2>Unit 2

Lesson 3: The complex plane

# The complex plane

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.
A coordinate plane where the x-axis is labeled the real axis and the y-axis is labeled the imaginary-axis. Both axes are scaled by one.
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.
A coordinate plane where the x-axis is labeled the real axis and the y-axis is labeled the imaginary-axis. Both axes are scaled by one. A point is at (three, negative five i). A vertical dashed line extends from the three on the real axis to the point and a horizontal line extends from negative five on the imaginary-axis to the point.
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.

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!

## Want to join the conversation?

• 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).
• "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 !!
• 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.
• 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.
• 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.
• Would zero be a complex number or would it just be a real number?
• adding on to Kim Sindel, Imaginary numbers are complex numbers too. Complex just requires a real part and an imaginary part. Both of those parts can be zero though.
• So numbers are always complex plus whether or not it has the imaginary or real part?
• Yes, you are right!

All complex numbers have a real part and an imaginary part. Because either part can be 0, all real numbers and all imaginary numbers are also complex numbers.

Examples of real numbers and their complex forms:
4 = 4 + 0i
87 = 87 + 0i
-5 = -5 + 0i
√2 = √2 + 0i

Examples of imaginary numbers and their complex forms:
3i = 0 + 3i
99i = 0 + 99i
-12i = 0 - 12i
√6i = 0 + √6i

Hope this helps!
• What is a Cartesian plane, is it the traditional coordinate plane we use in Algebra and Geometry?
• Yes, a Cartesian plan is the traditional coordinate plane used in Algebra & Geometry. DeCarte came up with the concept of the coordinate plane which is why it is also called the Cartesian plane.