Perform the following operations and express your answer in the form a+b i. (2-i)/(3+2 i)

Name: Problem 14, Chapter 10: Higher Level Mathematics
Uploaded: 2022-01-19T18:09:29-08:00
Duration: 6 min 56 s
Channel: Asif Khan
Description: Perform the following operations and express your answer in the form a+b i. (2-i)/(3+2 i)

Perform the following operations and express your answer in...

Key Concepts

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part, usually expressed in the form a + bi where a and b are real numbers and i is the imaginary unit satisfying i² = -1. This concept is fundamental in many areas of mathematics and engineering as it extends the idea of one-dimensional number lines to two dimensions.

Division of Complex Numbers

Dividing complex numbers involves rewriting the expression so that there is no complex number in the denominator. This process often requires multiplying the numerator and the denominator by the conjugate of the denominator, thereby making use of the properties of complex numbers to simplify the expression.

Complex Conjugate

The complex conjugate of a complex number is obtained by changing the sign of its imaginary part. Multiplying by the conjugate is a key technique in complex number division because it eliminates the imaginary part in the denominator, resulting in a real number. This technique leverages the fact that the product of a complex number and its conjugate has no imaginary component.

Rectangular Form

Expressing a complex number in rectangular form means writing it as a + bi, where 'a' corresponds to the real part and 'b' to the coefficient of the imaginary part. This form is particularly useful for addition, subtraction, and for clearly observing the real and imaginary components of the number.

Transcript

00:01 Hi, today in this video we will learn the concept of dividing two complex numbers and we are given to complex number and division form like this one two minus aorta divided by three plus two aorta dividing the complex numbers is a little bit difficult as compared to dividing a complex number by our real number for example i will tell you an example i will show you an example like if we have two plus six aorta let's take this is just an example and we have you have we have to divide by real number two then it varies it's very easy just divided two by two and then six aorta by two we will get one plus simply two three times three a two so by real numbers very easy so we will try to get rid of this aorta that is the imaginary part from the denominator of the given question rather i would say simply we will try to get a real number in the denominator of your given complex number and for doing so we know that when we have we have a complex number a plus b a uta if we multiply this complex number by its complex conjugate that is a minus b a we get a real number that is a square plus b square this is the one of the fact and the complex conjugate is nothing but just you can you have to change the sign of the imaginary part you can see here is plus then it you have to make it minus and if it is minus you have to make it plus just these are the complex conjugates of each other so we are getting complex and yours when we will multiply together with each other we get a real number so i will use this concept here and i will try to multiply this denominator by its complex conjugate that is simply three minus two eight but to keep the value of fraction unchanged i have to multiply this with numerator as well that is 3 minus 2 aorta now a numerator will be multiplied with numerator it will be like 2 minus aorta will be multiplied with 3 minus 2 aota and in the denominator you can see here this is just i will show you here if we take 3 as a a a a a a a a a b then you can see a plus b is here and a minus b is here and we know that we have a formula that is a plus b into a minus b that's equal to a square minus b square i will use this formula here simply so this…

Please give Ace some feedback