For the imaginary unit i, where i^2 = -1, the complex number z is defined by z = \frac{11 … — ACT

For the imaginary unit $i$ , where $i^2 = -1$ , the complex number $z$ is defined by $z = \frac{11 + 3i}{3 - i}$ . What is the real part of $z$ ?

Answer: 3

Answer

The real part of the complex number is 3.

By multiplying both the numerator and the denominator of

\frac{11 + 3i}{3 - i}

by the conjugate of the denominator,

3 + i

, we obtain

\frac{(11+3i)(3+i)}{(3-i)(3+i)} = \frac{33 + 11i + 9i + 3i^2}{9 - i^2} = \frac{30 + 20i}{10} = 3 + 2i

. The real part of this complex number is the term without

i

, which is 3.

Step-by-Step Solution

Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator, which is

3 + i

z = \frac{(11 + 3i)(3 + i)}{(3 - i)(3 + i)}

To eliminate the imaginary unit from the denominator.

Expand and simplify the numerator using the distributive property and substituting

-1

for

i^2

(11 + 3i)(3 + i) = 33 + 11i + 9i + 3i^2 = 33 + 20i + 3(-1) = 30 + 20i

To combine the real and imaginary terms of the numerator.

Expand and simplify the denominator using the difference of squares property and substituting

-1

for

i^2

(3 - i)(3 + i) = 9 - i^2 = 9 - (-1) = 10

To find the real number denominator.

Divide each term in the simplified numerator by the denominator.

z = \frac{30 + 20i}{10} = 3 + 2i

To express the complex number in the standard form

a + bi

Extract the real part of the resulting complex number

3 + 2i

The real part of a complex number in the form

a + bi

a

Key Concept

Division of complex numbers using the complex conjugate

Alternative Method

Instead of simplifying the fraction directly, assume the resulting complex number is

x + yi

, where

x

represents the real part and

y

represents the imaginary part. We can set up the equation

x + yi = \frac{11 + 3i}{3 - i}

and multiply both sides by

3 - i

to get

(x + yi)(3 - i) = 11 + 3i

. Expanding the left side gives

(3x + y) + (3y - x)i = 11 + 3i

. Equating the real and imaginary parts yields a system of linear equations:

3x + y = 11

and

-x + 3y = 3

. Multiplying the second equation by 3 and adding it to the first equation gives

10y = 20

, which means

y = 2

. Substituting

y = 2

back into the first equation yields

3x + 2 = 11

, which simplifies to

3x = 9

, or

x = 3

. The real part is therefore 3.

Estimated Time:1m 30s

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Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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