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

Lesson 10: Multiplying & dividing complex numbers in polar form

# Dividing complex numbers: polar & exponential form

Sal shows how complex division affects the modulus and the argument of the divisor and the dividend. Created by Sal Khan.

## Want to join the conversation?

• At , Sal mentions "Euler's Formula", which is a foreign concept to me. Is there a video or videos that covers it, so I might be able to understand it a little better? Just the link should be fine... thanks!
• Can someone link me to a video were it shows how to use that variable e
• Currently all you need to know about e is that it is a number. It is a really cool number, and it had some properties that are advantageous in calculus that make it the ideal base for an exponential or logarithmic function. But as far as manipulating exponentials with base e, you should use the same principles as you would with 2^x.
• What is e? How would one define the constant?
Also, can someone teach me more about Euler's formula? Isn't that a more advanced calculus topic?

I am working on precalculus now and I found this topic. I'm kind of confused although everything else was crystal clear up until this point.
• Hello Natalie,

As an electrical engineer this is my favorite constant! It makes the math very easy. Wiki “phasor” if you are interested...

Here is a joke for you - How many mathematicians does it take to change a light bulb? Answer : -e^(i*pi)

Wiki has a good explanation of “e” at https://en.wikipedia.org/wiki/E_(mathematical_constant)

Here is a good video especially if you like the Simpsons https://www.youtube.com/watch?v=Yi3bT-82O5s

Know that the constant “e” alive and well. Take a look at your calculator you will see “e^x” and its cousin the natural logarithm “ln”. You will even find e on financial calculators.

BTW the answer to my opening question is one since e^(i*pi) + 1 = 0.

Sorry I couldn’t give a better explanation but I hope the links will get you to a good place.

Regards,

APD
• What is Euler's formula and how does it work? This video is acting like I already know what it is.
• What does this actually mean? (The math is straightforward for converting to polar form and then dividing exponents and subtracting coefficients.) But what 2-dimensional use is this division? He divided a point by another point, and got some seemingly unrelated 3rd point. Is there some geometric relationship that gives meaning to all this?
• Engineers actually use operations on complex numbers to simplify mathematics of complicated problems. Dr Math presents the interesting real world problem of a snowplow. The more snow the plow is pushing, the slower the plow moves because it is pushing a heavier load, therefore the rate of growth of the snow heap slows over time. Instead of trying to track all of the variables, engineers use parametric equations (independent variable time) to calculate A(t), where A = amount of snow at time t.

Dr Math also provided the example of expanding
``(cos (x) + i * sin(x))^3``
. Hairy expansion. Using Euler's formula, you can reduce a multi-step process (where you might make a careless error at any or all steps) to this:
`` since e^(ix) = cos(x) + i *sin(x) and (x^a)^b) = x^(ab),e^(ix)^3 = e^i(3x)= cos(3x) + i * sin(3x)``
• as soon as sal sees the angle 7pi/6 he plots that angle . how?
can any one tell me where i could find a video to teach that stuff
• With radians, you get something easy happening, so that you know how much of a circle the angle sweeps out.
number of radians = (fraction of a circle the angle sweeps out) (2π)
So, just drop the π and divide by 2 and you get the fraction of the circle the angle sweeps out. (Of course, if there is no π in the angle, this isn't easy because you have to divide by π in that case.)

And, once you know what fraction of a circle the angle sweeps out, it is quite easy to graph it.

For example. How much of a circle does the angle 3π/2 sweep out?
Dropping the π and dividing by 2 gives us ¾ of a circle.

Another example: how much of a circle does the angle ⁵⁄₄ π sweep out? Its ⅝ of a circle.

This is but one of many reasons why using radians is far easier than using degrees. With degrees, you have to divide by 360 to find out how much of a circle is swept out by an angle. And obviously finding half of something is far easier than dividing by 360.
• Couldn't You also write this polar form into rectangular form and then conjugate?
• Could you just take the formula at , convert it to rectangular form (a + b * i), and then divide each of the parts?
• Yes, but that requires more work. The point of exponential form is that it makes computation easier in a lot of cases.
• Is there is a specific pattern that emerges when you plot the point of the solution of dividing two complex numbers in relation to the original numbers? For example, at , Sal plots the solution on the imaginary plane, but is there a specific pattern that emerges as to the position of that point?
• Good question! The answer is yes!

The standard trigonometric angle (to the positive x-axis) associated with the quotient of two complex numbers, equals the difference of the standard trigonometric angles associated with them.

The magnitude (or distance from the origin) of the quotient of two complex numbers is the quotient of their magnitudes.
• In the powers of complex numbers question section their is an example question that asks:
1) Find solution of following equation whose argument is between 225 deg - 315 deg ... z^7 = 128i

Now when I click to get a hint it tells me that the modulus is 128 and that the argument is 90 + k * 360 . My question is how do you find that argument?