For the imaginary unit i, where i^2 = -1, the complex number w is defined as w = \frac{6 +… — ACT

For the imaginary unit $i$ , where $i^2 = -1$ , the complex number $w$ is defined as $w = \frac{6 + 4i}{2i}$ . What is the imaginary part of $w$ ?

Answer: -3

The imaginary part of

w

-3

Dividing each term in the numerator of

\frac{6 + 4i}{2i}

by the denominator

2i

yields

\frac{6}{2i} + \frac{4i}{2i}

, which simplifies to

\frac{3}{i} + 2

. Since

i^2 = -1

, the term

\frac{3}{i}

can be rationalized to

-3i

. Thus, the complex number in standard form is

2 - 3i

. The imaginary part is the real coefficient of

i

, which is

-3

Divide each term in the numerator by the denominator.

w = \frac{6}{2i} + \frac{4i}{2i}

This separates the quotient into two simpler terms that can be simplified individually.

Simplify both terms.

w = \frac{3}{i} + 2

Reduce the fractions by dividing out common factors in both the numerators and the denominators.

Rationalize the denominator of the imaginary term.

\frac{3}{i} \cdot \frac{i}{i} = \frac{3i}{i^2} = \frac{3i}{-1} = -3i

Multiply the numerator and denominator by

i

to eliminate the imaginary unit from the denominator, using the property

i^2 = -1

Combine the real and imaginary parts into standard form

a + bi

w = 2 - 3i

Group the real constant and the simplified imaginary term together.

Identify the imaginary part of the complex number.

-3

The imaginary part of a complex number

a + bi

is the real coefficient

b

of the imaginary unit

i

Simplifying a quotient of complex numbers by dividing by a pure imaginary number.

Question