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### Course: Precalculus>Unit 3

Lesson 5: Modulus (absolute value) and argument (angle) of complex numbers

# Absolute value & angle of complex numbers

Sal finds the modulus (which is the absolute value) and the argument (which is the angle) of √3/2+1/2*i. Created by Sal Khan.

## Want to join the conversation?

• I don't understand how e just popped into the equation. Is there another video series I should be referencing to understand where this came from? I've already watched the compound interest series that introduces e but that didn't prep me for this.
• This video should be in Calculus playlist, not in Precalculus. Because you're not supposed to have even heard of Euler's formula if you are learning precalculus
• we know real infinity(positive.negative) but is there a complex infinity?
• Excellent question!
Yes there is, in the stereographic mapping of the Riemann Sphere to the complex plane. The "Point at Infinity" corresponds to the "North Pole" of the Riemann Sphere. Link: http://en.wikipedia.org/wiki/Complex_plane
• At Sal writes +1, even though i^2=-1

Am I wrong or right?
• He calculated the absolute value of z, |z|, where you square the real parts of z, and then add them and take the square root.

So,
if z = a + bi
then the real parts are a and b

In this case z = √(3)/2 + i
Then a = √(3)/2
and b = 1, because the real part of i is 1, just as the real part of 2i is 2

The absolute value of z is:
|z| = √(a^2 + b^2)

Which gives:
|z| = √(3/4 + 1)

Hope that helped!
• At , Sal states that imaginary numbers in exponential form should be measured in radians. Why?
• Degrees are a contrived unit, radians are not. From this point on, you will be usually dealing with radians not degrees. When you move on in math, you won't be having degrees at all, just radians.

The reason for considering an exponent with an imaginary unit as an angle is because of this relationship:
e^(i*x) = cos x + i sin x
(this only works if x is in radians)
• What if the complex number is not on the z=a+bi form? For example simply -2i. If I then wanted to find the argument, wouldn't I then end up with phi=atan(-2/0), which is undefined?
• Actually, -2i IS in the z=a+bi form. It just means a=0. Just had that one on a trig test about a month ago.
• In the practice questions that follow, when the angle measure must be given between -180 degrees and 180 degrees, how do you know when to add or subtract 180 degrees from the result of taking the inverse tangent to get the final answer?
• First, figure out which quadrant the point a + bi lies in. You can do this by thinking about the signs of a (the real part) and b (the imaginary part):
- Quad I: a is + , b is +
- Quad II: a is - , b is +
- Quad III: a is - , b is -
- Quad IV: a is + , b is -

Next, work the problem and get the result.

Finally, think about which quadrant your final answer should be in - this will be the same quadrant that the original point lies in. Remember that instead of a circle that goes from 0° to 360°, we are starting at -180° (the negative x-axis... or Re-axis) and traveling counter-clockwise to +180°.

Now, check to see if your result is in the same quadrant as the original point a + bi. If not, just add or subtract 180° to get it into the correct quadrant.

You can double check to make sure that adding or subtracting 180° (or doing nothing) gives you an answer in the same quadrant as the original point.
- Quad I will be 0° to 90°
- Quad II will be 90° to 180°
- Quad III will be -180° to -90°
- Quad IV will be -90° to 0°

Hope this helps!
• wait so for the example you gave at the 9 minute mark where z=sqrt(3)/2 + i, when calculating r = |z|= sqrt((3/4) + (i^2)) wouldnt i^2 equal -1 or (-4/4) as oposed to it equaling 1 or (4/4). since i^2 is equal to -1 according to the imaginary number system ?