Dividing Complex Numbers

Front

What is a complex number?

Back

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as i^2 = -1.

Front

What is the conjugate of a complex number?

Back

The conjugate of a complex number a + bi is a - bi. It is used in division to eliminate the imaginary part from the denominator.

Front

How do you multiply complex numbers?

Back

To multiply complex numbers (a + bi)(c + di), use the distributive property: ac + adi + bci + bdi^2, and simplify using i^2 = -1.

Front

What is the formula for dividing complex numbers?

Back

To divide complex numbers \( \frac{a + bi}{c + di} \), multiply the numerator and denominator by the conjugate of the denominator: \( \frac{(a + bi)(c - di)}{(c + di)(c - di)} \).

Front

Simplify \( \frac{8 + 5i}{1 - 7i} \).

Back

\( \frac{-27 + 61i}{50} \)

Front

Simplify \( \frac{12}{8i} \).

Back

\( -\frac{3i}{2} \)

Front

Simplify \( \frac{14}{7 - 6i} \).

Back

\( \frac{98 + 84i}{85} \)

Front

Simplify \( \frac{-4 + 12i}{-9 + 5i} \).

Back

\( \frac{48 - 44i}{53} \)

Front

Simplify \( \frac{10 - 10i}{2 + 2i} \).

Back

\( -5i \)

Front

What is the modulus of a complex number?

Back

The modulus of a complex number a + bi is given by \( \sqrt{a^2 + b^2} \). It represents the distance from the origin in the complex plane.