Evaluate the expressions, writing the result as a simplified complex number. (3+2 i)/(1+2 i)-(2-

Evaluate the expressions, writing the result as a simplified...

Key Concepts

Simplification of Complex Expressions

After performing operations such as multiplication using the complex conjugate, it is necessary to combine like terms by grouping real and imaginary parts separately. This final step ensures the expression is presented in its simplest standard form, which is generally written as a + bi.

Rationalization of Denominators

Rationalization is the process of eliminating the imaginary unit from the denominator of a complex fraction. This is achieved by multiplying the numerator and the denominator by the conjugate of the denominator, transforming the denominator into a real number and thereby simplifying the expression.

Complex Numbers Arithmetic

Complex numbers consist of a real part and an imaginary part and are used to represent numbers beyond the one-dimensional number line. Operations such as addition, subtraction, multiplication, and division extend the rules of arithmetic to account for the imaginary unit i, where i² = -1. This fundamental understanding is essential for problems involving the manipulation and simplification of complex expressions.

Complex Conjugate

The complex conjugate of a number is obtained by changing the sign of its imaginary part. It is a crucial concept when working with divisions involving complex numbers because multiplying by the conjugate can help eliminate the imaginary part from the denominator, leading to a simplified result in standard a + bi form.

Transcript

00:03 Here we're asked to evaluate an expression and to write it in simplest form as a complex number.

00:09 Well first i'm going to evaluate the first part, which is a 3 plus 2 i divided by 1 plus 2i.

00:17 And we should simplify this first before going on because we want to divide these complex numbers in order to rewrite the result in standard form.

00:27 What i'm going to do here is multiply both the numerator and denominator by the complex conjugate of the denominator, that would be 1 minus 2i.

00:38 Then we're going to foil out the numerator and denominator.

00:43 In the numerator, we'll get 3 times 1, that's 3.

00:46 Then 3 multiplied by negative 2i, that's minus 6i, 2i times 1, that's 2i, and then finally 2i times negative 2i, that's minus 4i squared.

01:04 In the denominator, we'll get 1 times 1, that's 1, followed by 1 minus negative 2i.

01:11 That's minus 2i, then 2i times 1, which is 2i.

01:17 And finally, 2i times negative 2i, that's negative 4i squared.

01:23 Now we'll simplify this further, remembering that i squared is negative 1.

01:27 So in the numerator we'll have 3 plus 4, because i squared is negative 1.

01:33 That's the real part, and the imaginary part will be negative 6i plus 2i, that's negative 4i.

01:41 In the denominator, we have 1 minus 4i squared, that's 3.

01:45 Just 1 plus 4 because i squared is negative 1.

01:50 And then we have negative 2i plus 2i.

01:53 Those 2 will cancel out.

01:55 So we just have 1 plus 4.

01:57 And this will become 7 minus 4i all over 5.

02:02 Now that's the first part.

02:06 Make sure to write this down somewhere because i'll be going to another screen.

02:12 So for the next part, we're going to calculate the second part of the expression.

02:17 It is again a quotient of two complex numbers.

02:20 It's 2 minus 3i divided by 3 plus i.

02:24 Like before, we're going to multiply both the numerator and denominator by the complex conjugate of the denominator.

02:31 And in this case, it's going to be 3 minus i.

02:46 So in the numerator, we have, we're going to foil it out again.

02:49 We'll have 2 times 3, that's 6.

02:52 Then you have 2 times negative i, that's minus 2i.

02:57 Then negative 3i times 3, that's negative 9i...

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