Perform the following operations and express your answer in the form a+b i. (12 i)/(3+4 i)

Name: Problem 22, Chapter 10: Higher Level Mathematics
Uploaded: 2022-01-26T17:57:04-08:00
Duration: 8 min 45 s
Channel: Asif Khan
Description: Perform the following operations and express your answer in the form a+b i. (12 i)/(3+4 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 represents the real component and b represents the imaginary component. They are used to solve equations that have no solutions in the set of real numbers.

Standard Form (a + bi)

The standard form of a complex number is a + bi, which clearly distinguishes the real part (a) from the imaginary part (b). Expressing complex numbers in this form helps simplify and standardize calculations and comparisons among them.

Complex Conjugate

The complex conjugate of a complex number a + bi is defined as a - bi. It is particularly useful in division of complex numbers because multiplying the numerator and denominator by the conjugate of the denominator eliminates the imaginary part from the denominator, thereby converting it to a real number.

Division of Complex Numbers

Division of complex numbers involves multiplying the numerator and denominator by the complex conjugate of the denominator to rationalize it. This process simplifies the expression into the standard form a + bi, making the result easier to interpret and use in further calculations.

Transcript

00:01 Hi, today in this video we will learn the concept of division of two complex numbers and we are given a question that is 12 iota divided by 3 plus 4 aorta is both are the complex numbers and they are in division form and we have to express this given question in a plus b a form that is in complex number form before going to solve this question i would like to tell you you a little bit about the complex number.

00:38 Complex number is simply denoted by a small z and that is equals to a plus b aota.

00:49 And here a and b are the real numbers and this a part, the first part that is said where a is mentioned, this is called real part as the real number is there.

01:02 So that's why.

01:03 And we have a i shaped letter is there.

01:06 And we call this a iota and overall b aorta is this part is said to be imaginary part a imaginary part of a complex number where a and b itself themselves are real numbers and this whole part this whole z is equals to a plus b aoid is said to be complex number so here we have a two complex numbers in the numerator simply it has a real number you can say that it's zero simply so division of a complex number is a little difficult as compared to dividing any complex number by real number so in order to make it easy i would like to get just get rid of this complex number from the denominator and for do so i will do one thing i will use a fact one of the fact that is we know that if we have a complex number, that is a plus b aorta, let's say.

02:17 And this is in a denominator, then i know that if i multiply this complex number by its complex conjugate, that is a minus b aota, then after multiplication, i will get a real number that is a square plus b squared...

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