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## Algebra (all content)

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

Lesson 7: Absolute value & angle of complex numbers

# Absolute value of complex numbers

Sal finds the absolute value of (3-4i). Created by Sal Khan.

## Want to join the conversation?

• so you have 5 = |3 - 4i|, that means 3 - 4i = 5 or -5. I evalulated i and got either 2 or 1/2. i'm soooo confused! whats happening!
• You do not solve for i - it is one of those symbols that has a set meaning like pi. It is not a variable. i=โ(-1) always.

The absolute value of a complex number is found using
โ(a^2 + b^2)
• When he draws out that right triangle, why is that one side "4" and not "4i"?
• He's treating the triangle as a regular triangle in geometry. Pretend that the complex number 3-4i is instead the point (3, -4) and we're trying to find the distance between (3, -4) and (0, 0). When you draw the side that goes down from (0, 0) to (0, -4), it's not going to be 4y or something special: The length is just going to be 4.
• I am confused, Shouldn't the pair of numbers be expressed 3,-4i instead of 3-4i
• No, because 3-4i is a single number (that is equal to 25). The fact here is that you can plot 3-4i on the complex plane separating the imaginary part from the real part.
• Can anyone please give a good real life application of the concept of getting the absolute value of a complex number?
• Yes, it is used to model capacitors and inductors in AC circuits. Trying to explain how it works is beyond the scope of an answer here, but I found this page which seems to cover it, at least the basics: http://hyperphysics.phy-astr.gsu.edu/hbase/electric/impcom.html#c3 That link is bookmarked to the spot where they actually use the absolute value operation.
• Wait... I'm confused. How can the absolute value of an imaginary number be a real number?
• Because of the way the absolute value is defined.
As always, the absolute value is defined as a distance, in this case, the distance on a complex plane. If you plot a number on the complex plane and calculate the distance from the origin, the is no complex component, so you end up with a real number.
• can somebody define what exactly the absolute value of a complex number is?
• Well first let's forget about the absolute value of complex numbers. What is the absolute value of a real number?
If we plot the real numbers on the real number line, the absolute value of any real number is simply its distance from 0 on the real number line.
Similarly, we plot the complex numbers on the complex plane. In the complex plane, the origin represents the number 0. Thus, the absolute value of a complex number is the distance between that number and the origin (0) on the complex plane.
Comment if you have questions!
• how to get real and imaginary parts of (a+ib/a-ib)..... pls help me
• Hmmm, I would have thought you had to multiply by (a+ib)/(a+ib) in order to distinctly obtain the real and imaginary segments. I.e. (I'm assuming the initial expression is (a+ib)/(a-ib))
(a+ib)/(a-ib) * (a+ib) = (a^2+2aib-b^2)/(a^2+b^2)
Through splitting the numerator, the real and imaginary parts can be obtained:
(a^2-b^2)/(a^2+b^2) + i(2ab/(a^2+b^2))
• Is there a video that explains how to simplify the roots?
• Simplifying roots is just a form of factoring. Factor what is inside the square root, and if there are any pairs, you can simplify:
For example,
โ(28) = โ(2โข2โข7) =2โ7
Another example:
โ(180) = โ(18โข10) =โ([2โข3โข3]โข[2โข5]) = โ(2โข2โข3โข3โข5) = 2โข3โ5 = 6โ5
• Does it make sense that the absolute value of a complex number is a real number?
How can i just be ignored?
This is stretching my imagination.
• The absolute value of a complex number, say x + iy is the distance from the origin. The same as the absolute value of a normal number on the number line. x + iy Is simply graphed as the point x on the x axis and y on the y axis. So abs(x + iy) = sqrt(x^2+y^2) and the I isn't put in the square because it would through off the distance from origin calculation.
• When carrying out the absolute value of the complex number in the example "sqrt((3-4i)^2)" I noticed the exponent was distributed to the real and imaginary parts of the complex #, which looks a lot like a polynomial, making this step counter intuitive to me. Just to see what would happen I treated the complex number like a polynomial, distributed twice (foil'ed), and after reducing the expression ended up with -5-24i which doesn't satisfy the equal sign, even so I don't understand why its suddenly okay to distribute exponents for any reason other than it doing so satisfied the equal sign in the example (5=5.) Thanks!

## Video transcript

I have the complex number 3 minus 4i. I've plotted it on the complex plane. We see that the real part is 3, so we've gone 3 along the horizontal axis, or the real axis. And the imaginary part is negative 4, so we've gone down 4 along the vertical axis. So this right here is the point 3 minus 4i. Now what I want to think about is what the absolute value of 3 minus 4i is. And just as a reminder, absolute value literally means-- whether we're talking about a complex number or a real number, it literally just means distance away from 0. So the absolute value of 3 minus 4i is going to be the distance between 0, between the origin on the complex plane, and that point, and the point 3 minus 4i. So this distance right over here is going to be the absolute value of 3 minus 4i. So how can we think about that? Well, we could literally just set up a right triangle and then use the Pythagorean theorem. So let's think about it. If we wanted to set up a right triangle, the height here, the distance between 0 and negative 4, well, that distance is going to be 4. And then the base of this triangle, the distance between 0 and 3, is just going to be 3. And this is definitely a right angle. This is a horizontal line. This is a vertical line. We can now use the Pythagorean theorem to figure out the absolute value of 3 minus 4i. The distance between this point and 0-- it's the hypotenuse of this right triangle. So we just use the Pythagorean theorem. This side squared, 3 squared, plus this side squared, plus 4 squared, is going to be equal to the absolute value of 3 minus 4i squared, the absolute value squared. So 3 squared plus 4 squared, that's 9 plus 16, which is 25. So you get 25 is equal to the absolute value of 3 minus 4i squared. And we know if you take the absolute value of something, this is just a distance. It's going to be positive. So we want to take the positive square root, the principal square root, of both sides of this. And so we're going to be left with-- well, the principal square root, the positive square root of 25, is 5, is equal to the absolute value of 3 minus 4i. So another way of saying it, this thing right over here is going to be equal to 5. This distance right over here is equal to 5. Now, without having to draw it, one way you could just think about this is I'll have my real part. I have my imaginary part. I could literally take each of those parts, square them, take the sum, and take the square root. So another way of taking it, if you didn't want to visualize all this-- but this is really what we're doing. You could say, well, this is just going to be equal to take the real part squared. Take the imaginary part squared-- so let me write this. Add them together, and then take the square root, or the principal root. The principal root's just the positive square root. So that's going to be the square root of 9 plus 16, which, once again, is equal to 5.