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### Course: Algebra (all content) > Unit 16

Lesson 2: What are the complex numbers?# Intro to complex numbers

Sal explains how we obtain complex numbers by adding real numbers and imaginary numbers. Created by Sal Khan.

## Want to join the conversation?

- is i multiplied by 0 a real number?(47 votes)
- 0i = 0, so yes it is a real number.

In fact, real numbers are a special type of complex numbers that can be written as:

a + 0i.(116 votes)

- What happens if you put i to the power of i?(27 votes)
- i guess you can take it as (i)root(-1) which becomes 4th root of i which does not make sense(1 vote)

- Could π be a complex number? Also, can you use variables other than i to represent a complex number?(20 votes)
- Yes, π is a complex number. It has a real part of π and an imaginary part of 0.

The letter i used to represent the imaginary unit is not a variable because its value is not prone to change. It is fixed in the complex plane at coordinates (0,1). However, there are other symbols that can be used to represent the imaginary unit. One of the common contexts in which complex numbers are useful is electrical circuits, for which it is customary to use j for the imaginary unit, since i commonly stands for the induced current in the circuit.(34 votes)

- at1:37I got curious. can we create a new type of number (like we did with imaginary numbers) to describe an imaginary number multiplied by a real number?(17 votes)
- An imaginary number multiplied by a real number is still an imaginary number. For example, 5 times i is 5i and is still considered an imaginary number and is not a different type of number.(8 votes)

- uhh, i don't mean to be dumb, but what on earth is E?

did I miss something?(11 votes)**e**(not E) is called Euler's number. Euler's number is an irrational number that is approximately equal to 2.718.(14 votes)

- I still don't get it..................if the imaginary no. is i and multiples of i like -i and 2i , then why in the complex no. the only imaginary part is 3. I got the logic of the complex plane but still it doesn't suits me, because the imaginary no. is i not 3 ?(8 votes)
- Complex numbers are of the form:

a + bi

Where i is the imaginary unit, and a and b are real numbers.

a is the real part

b is imaginary part

So if you have a complex number that is a multiple of i, it will be of the complex form bi (because a will be zero).

Therefore the imaginary part is the coefficient of the imaginary unit.(13 votes)

- So when you do add them together you don't get a number? Or is it like you get a number that doesn't exist such as (z)=5+3i would you just get (8i)?(6 votes)
- When you add two complex numbers, you usually get another complex number.

Example:

(3 + 5i) + (-1 + 7i)

= (3 - 1) + (5i + 7i)

= 2 + 12i

(6 + 2i) + (1 + i)

= (6 + 1) + (2i + i)

= 7 + 3i

Sometimes you might get a pure imaginary number or a real number:

(3 + 8i) + (-3 + 2i)

= (3 - 3) + (8i + 2i)

= 0 + 10i

= 10i

(*See how the real parts add up to zero? The result is the complex number 0 + 10i, which is equivalent to the***pure imaginary number***10i.*)

(8 + 4i) + (4 - 4i)

= (8 + 4) + (4i - 4i)

= 12 + 0i

= 12

(*See how the imaginary parts add up to zero? The result is the complex number 12 + 0i, which is equivalent to the***real number***12.*)

Hope this helps!(13 votes)

- What if I could create a new type of number that can describe a complex number * a real number?(5 votes)
- We already have that. It is called a complex number:

c (a+bi) = ac + bci

Keep in mind that a real number is a special type of complex number (one in which the imaginary component is 0).(12 votes)

- why is it called imaginary if it comes out to be -1 or 1?(3 votes)
- Really just a name. Rene Descartes came up with it in his book where he coined the term "imaginary" and meant it to be derogatory, as he thought it was useless. Though eventually, more mathematicians found a use for the number, but the name stayed.(14 votes)

- I didn't understand imaginary. What is imaginary?(5 votes)
- An imaginary number is the square root of -1 (or i). Taking a square root of a negative isn't possible, so i was used to make it possible.

For example, sqrt(-16) = 4i

Also, since i = square root of -1:

i^1 = sqrt(-1) = i

i^2 = -1

i^3 = -sqrt(-1) = -i

i^4 = 1

Hope this helps(8 votes)

## Video transcript

Voiceover:Most of your mathematical lives you've been studying real numbers. Real numbers include
things like zero, and one, and zero point three
repeating, and pi, and e, and I could keep listing real numbers. These are the numbers that you're kind of familiar with. Then we explored something interesting. We explored the notion of what if there was a number that if I squared it I would get negative one. We defined that thing
that if we squared it we got negative one, we
defined that thing as i. So we defined a whole new class of numbers which you could really view as multiples of the imaginary unit. So imaginary numbers
would be i and negative i, and pi times i, and e times i. This might raise another
interesting question. What if I combined
imaginary and real numbers? What if I had numbers
that were essentially sums or differences of
real or imaginary numbers? For example, let's say
that I had the number. Let's say I call it z, and z tends to be the most used variable when we're talking about what I'm about to talk
about, complex numbers. Let's say that z is equal to, is equal to the real number five plus the imaginary number three times i. So this thing right over here we have a real number
plus an imaginary number. You might be tempted to
add these two things, but you can't. They won't make any sense. These are kind of going in different, we'll think about it visually in a second, but you can't simplify this anymore. You can't add this real number to this imaginary number. A number like this, let me make it clear, that's real and this is
imaginary, imaginary. A number like this we
call a complex number, a complex number. It has a real part and an imaginary part. Sometimes you'll see notation like this, or someone will say what's the real part? What's the real part of
our complex number, z? Well, that would be the
five right over there. Then they might say, "Well, what's the imaginary part? "What's the imaginary part
of our complex number, z? And then typically the
way that this function is defined they really want to know what multiple of i is this imaginary part right over here. In this case it is going to
be, it is going to be three. We can visualize this. We can visualize this in two dimensions. Instead of having the traditional two-dimensional Cartesian plane with real numbers on the horizontal and the vertical axis, what we do to plot complex numbers is we on the vertical axis we plot the imaginary part, so
that's the imaginary part. On the horizontal axis
we plot the real part. We plot the real part just like that. We plot the real part. For example, z right over here which is five plus three i, the real part is five so we would go one, two, three, four, five. That's five right over there. The imaginary part is three. One, two, three, and so
on the complex plane, on the complex plane we would visualize that number right over here. This right over here is how we would visualize z on the complex plane. It's five, positive five
in the real direction, positive three in the imaginary direction. We could plot other complex numbers. Let's say we have the complex number a which is equal to let's
say it's negative two plus i. Where would I plot that? Well, the real part is negative two, negative two, and the imaginary part is going to be you could imagine this as plus one i so we go one up. It's going to be right over there. That right over there
is our complex number. Our complex number a
would be at that point of the complex, complex, let me write that, that point of the complex plane. Let me just do one more. Let's say you had a complex number b which is going to be, let's say it is, let's say
it's four minus three i. Where would we plot that? Well, one, two, three, four, and then let's see minus one, two, three. Our negative three gets
us right over there. That right over there would be the complex number b.