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 = √(-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 to simplify the division of complex numbers.

Front

How do you multiply complex numbers?

Back

To multiply complex numbers, use the distributive property (FOIL method) and combine like terms. For example, (a + bi)(c + di) = ac + adi + bci + bdi^2.

Front

What is the formula for dividing complex numbers?

Back

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

Front

Simplify \( \frac{4-9i}{4+14i} \)

Back

\( -\frac{55}{106}-\frac{23i}{53} \)

Front

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

Back

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

Front

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

Back

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

Front

What is the process to find the conjugate of a complex number?

Back

To find the conjugate of a complex number a + bi, simply change the sign of the imaginary part, resulting in a - bi.

Front

What is the significance of the conjugate in division of complex numbers?

Back

The conjugate is used to eliminate the imaginary part in the denominator, making it possible to express the result as a + bi.

Front

Simplify \( \frac{1}{-10+12i} \)

Back

\( -\frac{5}{122}-\frac{3i}{61} \)