Dividing Complex Numbers with conjugates

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 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

Why do we multiply by the conjugate when dividing complex numbers?

Back

Multiplying by the conjugate helps eliminate the imaginary part from the denominator, making it a real number.

Front

What is the formula for multiplying two complex numbers?

Back

If z1 = a + bi and z2 = c + di, then z1 * z2 = (ac - bd) + (ad + bc)i.

Front

How do you simplify a complex fraction?

Back

To simplify a complex fraction, multiply the numerator and denominator by the conjugate of the denominator.

Front

What is the result of multiplying (2 + 3i) by its conjugate?

Back

The result is 2^2 + 3^2 = 4 + 9 = 13.

Front

What is the denominator after multiplying (1 + 2i) by its conjugate (1 - 2i)?

Back

The denominator is 1^2 + (2)^2 = 1 + 4 = 5.

Front

What is the numerator after multiplying (2 - 3i) by (1 + 2i)?

Back

The numerator is (2)(1) + (2)(-3i) + (-3i)(1) + (-3i)(2i) = 2 - 6i + 3 + 6 = 5 - 3i.

Front

What is the standard form of a complex number?

Back

The standard form of a complex number is a + bi, where a is the real part and b is the imaginary part.

Front

How do you find the modulus of a complex number?

Back

The modulus of a complex number a + bi is given by √(a^2 + b^2).