Rationalizing 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 the square root of -1.

Front

What does it mean to rationalize a complex number?

Back

Rationalizing a complex number involves eliminating the imaginary part from the denominator of a fraction by multiplying the numerator and denominator by the conjugate of the denominator.

Front

What is the conjugate of a complex number?

Back

The conjugate of a complex number a + bi is a - bi. It is obtained by changing the sign of the imaginary part.

Front

How do you multiply complex numbers?

Back

To multiply complex numbers, use the distributive property (FOIL method) and apply the fact that i^2 = -1.

Front

What is the formula for the modulus of a complex number?

Back

The modulus of a complex number a + bi is given by |a + bi| = √(a² + b²).

Front

How do you add complex numbers?

Back

To add complex numbers, combine the real parts and the imaginary parts separately: (a + bi) + (c + di) = (a + c) + (b + d)i.

Front

What is the difference between real and imaginary parts of a complex number?

Back

In a complex number a + bi, 'a' is the real part and 'b' is the imaginary part.

Front

What is the process to rationalize the denominator of 1/(2 + 3i)?

Back

Multiply the numerator and denominator by the conjugate of the denominator: (1/(2 + 3i)) * ((2 - 3i)/(2 - 3i)) = (2 - 3i)/(4 + 9) = (2 - 3i)/13.

Front

What is the result of (3 + 4i)(2 - i)?

Back

Using the distributive property: (3 + 4i)(2 - i) = 6 - 3i + 8i - 4i² = 6 + 5i + 4 = 10 + 5i.

Front

How do you subtract complex numbers?

Back

To subtract complex numbers, subtract the real parts and the imaginary parts separately: (a + bi) - (c + di) = (a - c) + (b - d)i.