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### Course: Algebra 2>Unit 2

Lesson 2: Complex numbers introduction

# 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?
• 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.
• What happens if you put i to the power of i?
• 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?
• 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.
• at I 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?
• 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.
• uhh, i don't mean to be dumb, but what on earth is E?
did I miss something?
• e (not E) is called Euler's number. Euler's number is an irrational number that is approximately equal to 2.718.
• 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 ?
• 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.
• 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)?
• 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!
• What if I could create a new type of number that can describe a complex number * a real number?
• 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).
• why is it called imaginary if it comes out to be -1 or 1?
• 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.
• I didn't understand imaginary. What is imaginary?
• 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