Calculate the (a) complex conjugate of the fraction (3+8 i)^4 /(1+i)^1 0. (b) complex conjugate

Calculate the (a) complex conjugate of the fraction (3+8...

Key Concepts

Exponentiation of Complex Numbers

Exponentiation involves raising a complex number to a power, which can be conveniently performed using its polar form via de Moivre’s theorem. This theorem states that the power of a complex number is obtained by raising its modulus to the given power and multiplying its argument by the exponent, simplifying otherwise tedious algebraic manipulation.

Operations on Complex Numbers

Operations such as multiplication, division, and exponentiation of complex numbers are facilitated by converting them into their polar forms or by using algebraic manipulations like multiplying by the complex conjugate. These operations adhere to specific properties, including distributiveness, associativity, and the use of conjugates to simplify fractions, which are essential for handling complex expressions efficiently.

Complex Conjugate

The complex conjugate of a complex number is obtained by changing the sign of its imaginary part while keeping the real part unchanged. This concept is fundamental because it is used to simplify the division of complex numbers, rationalize denominators, and it has important properties in relation to the modulus and argument of complex numbers.

Complex Modulus

The modulus (or absolute value) of a complex number is defined as the distance from the origin to the point representing the complex number in the complex plane, calculated as the square root of the sum of the squares of its real and imaginary parts. It plays a key role in converting complex numbers between rectangular and polar forms, and it satisfies multiplicative properties over complex number operations.

Transcript

00:01 In this problem, the first thing we're being asked to do is to find the complex conjugate to 1 plus the square of the b.

00:08 Well, first thing, i'm going to simplify the square of the b.

00:11 Well, the perfect square four goes into it two times.

00:14 So we can rewrite this as the square of the four times the square of the two.

00:18 And we know the square of the four is two.

00:20 So this is really just one plus two root two.

00:24 All right.

00:24 Now, in terms of the complex conjugate, so remember, typically our complex numbers in standard form are written as a plus b .i, where a is the real part and b .i is the imaginary part.

00:37 Well, look at our expression, 1 plus 2 root 2.

00:40 That's all the real number.

00:42 Notice, we don't have an i in this case.

00:44 So, that means that in terms of a complex number, a would be 1 plus 2 root 2, which means that b is equal to 0.

00:53 Okay? well, typically, to find the complex conjugate of a plus bi would be a minus bi.

01:00 So in this case, we needed the complex conjugate, we would do our a value, which is 1 plus 2 root 2, minus our b value 0i.

01:10 But we don't really need that 0i there because that value is just equal to 0.

01:15 So our complex conjugate would also happen to be 1 plus 2 root 2...

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