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### Course: Precalculus>Unit 3

Lesson 5: Modulus (absolute value) and argument (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=√(-1) always.

The absolute value of a complex number is found using
√(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,
√(28) = √(2•2•7) =2√7
Another example:
√(180) = √(18•10) =√([2•3•3]•[2•5]) = √(2•2•3•3•5) = 2•3√5 = 6√5