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## Integrated math 2

### Course: Integrated math 2>Unit 5

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:
• ${i}^{2}=-1$
• $\sqrt{-1}=i$
A complex number is any number that can be written as $a+bi$, where $i$ is the imaginary unit and $a$ and $b$ are real numbers.
$a$ is called the $\text{real}$ part of the number, and $b$ is called the $\text{imaginary}$ 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 $\left(0,0\right)$.
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 $3-5i$. This number, also expressed as $3+\left(-5\right)i$, has a real part of $3$ and an imaginary part of $-5$.
The location of this number on the complex plane is the point that corresponds to $3$ on the real axis and $-5$ on the imaginary axis.
So the number $3+\left(-5\right)i$ is associated with the point $\left(3,-5\right)$. In general, the complex number $a+bi$ corresponds to the point $\left(a,b\right)$ in the complex plane.

Problem 1
Plot the complex number $-4+7i$.

Problem 2
Plot the complex number $6i+1$.

Problem 3
Plot the complex number $-i-3$.

Problem 4
Plot the complex number $4i$.

Problem 5
Plot the complex number $-7$.

## Connections to the real number line

In Pythagoras's days, the existence of irrational numbers was a surprising discovery! They wondered how something like $\sqrt{2}$ could exist without an accurate complete decimal expansion.
The real number line, however, helps rectifying this dilemma. Why? Because $\sqrt{2}$ 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 $\sqrt{2}$.)
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.
• 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.)"?
• 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.
• Where can you apply the complex plane in real life?
• 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!
• 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.
• 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.
• 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!