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### Course: Precalculus>Unit 3

Lesson 3: Complex conjugates and dividing complex numbers

# Dividing complex numbers

Sal divides (6+3i) by (7-5i). Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• How do you know what the conjugate is? Do you just flip the middle sign?
• By definition, the conjugate is the form "a-bi" of an imaginary number "a+bi". You just have to change the sign of the IMAGINARY part.
• Why always the emphasis on the denominator? Sal always seems quite concerned about simplifying his denominators (more than his numerators), but I do not really see the exact need of getting rid of the imaginary number in the denominator, other than maybe it is more practical if one is trying to do a very complex, mutli-step problem. And in any case, there is still an imaginary number in the numerator. Am I missing something or is he just showing us a possible tool to use here??
• The reason for getting rid of the complex parts of the equation in the denominator is because its not easy to divide by complex numbers, so to make it a real number, which is a whole lot easier to divide by, we have to multiply it by a number that will get rid of all the imaginary numbers, and a good number to use is the conjugate.
• Why is 1/i = -i ? Intuition or proof please. (I've never seen this definition on any of the Khan Academy videos. But first came across it under 'hints' of the "Dividing Complex Numbers" exercise.)
• You can multiply and divide the term 1/i by 'i'. This gives you:
(1 x i)/(i x i) , and as i^2 is -1, denominator becomes -1 and numerator is i. Hence it simplifies to -i.
• Why is it not good to have a complex number in the denominator?
• managing to get rid of the complex number in the denominator is useful because then you can write it with the form a + bi with a and b rationnals
• Help! I don't get it. I want to be smart like Sal. Explain please! I need different ways to do this.
• You are smart. You have to work for it :)
• Yes. It works both ways. If the original denominator was 7+5i, then it's cognate is 7-5i. Just remember that you have to switch the sign of the imaginary number. The real number always stays the same.
• At sal says 0/0 is undefined why?
• Division by zero is an operation for which you cannot find an answer, so it is disallowed. You can understand why if you think about how division and multiplication are related.
12 divided by 6 is 2 because: 6 times 2 is 12
Now image 12/0:
12 divided by 0 is x would mean that: 0 times x = 12

But no value would work for x because 0 times any number is 0. So division by zero doesn't work.
• What's the difference between a reciprocal and a conjugate?
• A reciprocal for a number n is 1/n or n^-1. It is also called the multiplicative inverse. The reciprocal function y=1/x forms a hyperbola graph, except for x=0.

Now, as Sal says, conjugate is an algebraic tool and is used to rationalize the denominator of a fraction. We applied this in Grade 8 Math. Take this example:
(1+2^1/2) / (2+2*2^1/2)
= (1+2^1/2) / (2+2*2^1/2) * (2-2*2^1/2) / (2-2*2^1/2)
= (1+2^1/2)(2-2*2^1/2) / (2+2*2^1/2)(2-2*2^1/2)
= {(2 - 2*(2^1/2) + 2*(2^1/2) - 2*(2^1/2)*(2^1/2)} / -4
=-2 / -4
=1/2

Now here (2-2*2^1/2) is the conjugate of (2+2*2^1/2). Conjugate is a binomial formed by negating the second term of the originial binomial. For complex numbers, a-bi is the conjugate of a+bi.

I hope I have been able to make the difference clear.
• Why is it in the excercise you can divide 20i by 5 and get 4i? When Sal has been saying all along that i is in it's own category and you cannot just put it together with real numbers.
• Because 20i is 20 times i. The one we are dividing by 5 is not i but 20.
This might help.
20i=
20*i
4*5*i
That's why.....
20i/5=4i
because.....
4*5*i/5---------------->we can see that the 5s will cancel out
4i----------------------->and that's how we got our answer.