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## Precalculus

### 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? • 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)
• 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!
• 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? • 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 ? • Ok I think I just figured it out,correct me if Im wrong but because we're dealing with |z| = | sqrt(3)/4 +i |, we're only concerned with finding the magnitude of r = |z| in other words the distance or "hypotenuse" therefore if Re(z) or Im(z) is negative in other words if a = sqrt(3)/2 or b = i are negative we simply ignore the minus signs since we're trying to find the distance r =|z|..... another way of looking at it is if we're trying to use the Pythagoras formula r^2 = x^2 + y^2, we ignore the minus signs or in other words we ignore the quadrant that the right angled triangle is formed and calculate the magnitude of the hypotenuse whether x= negative number or y = negative number giving us a reuslt thats alway positive i.e. |r^2| =|x^2| + |y^2| ... i hope this helps anyone who had the same confusion
• how do i write: -1+√3 i in polar form • precalculus video quoting taylor series? something is wrong folks () • I'm trying to skip algebraiii/trigonometry in my school since I already know most of the material. In my textbook it uses r cis theta.
No where it says re^i(phi). What is the correlation between these two? When are they used?
Thanks! • cis theta is just shorthand for cos theta + i sin theta
`c`os theta + `i s`in theta
so, r cis theta just means r times cis theta and is therefore the same as rcos theta+risin theta or r(cos theta + i sin theta)

I'm not so fond of the cis notation because it obscures the really cool things about how sin and cos and theta can be related as complex numbers, but that is just my preference--I love to see how things tick.

Usually re^i(phi) form is taught in calculus or perhaps introduced in precalc, with more about logarithms and transcendentals. Once you get into logarithms, the concept of `re^i(phi) ` becomes a lot easier and it becomes second nature to convert back and forth. It is the exponential form of the same relationship. It is used to find roots of functions, and is a shorthand way of expressing a complex number, for example in calculus. It offers a format that is a lot more promising for getting to a solution because you can use a lot more tools to manipulate it. Natural logs are to the base e for example.

One other thing that is different in your question is the phi and the theta. I am guessing you are not asking about that--phi is often used for the arguments in physics and in Argand diagrams, but theta is also used. If two angles are given, theta is the first by convention and phi is second fiddle.

If you want to skip algebra and trigonometry you may run into several examples of this-- where your textbook has simplified things that you will have to get used to in their more conventional form. It would be good to watch as many of the advanced math videos on Khan Academy as you can to be able to encounter some of these alternate forms.