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

To multiply two complex numbers (a + bi) and (c + di), use the distributive property: (a + bi)(c + di) = ac + adi + bci + bdi^2, which simplifies to (ac - bd) + (ad + bc)i.

Front

What is the result of multiplying (3 + 4i) by its conjugate (3 - 4i)?

Back

The result is 3^2 + 4^2 = 9 + 16 = 25, which is a real number.

Front

How do you simplify the expression (8 + 6i) / (29)?

Back

To simplify, divide both the real and imaginary parts by the denominator: (8/29) + (6/29)i.

Front

What is the process of rationalizing the denominator?

Back

Rationalizing the denominator involves multiplying the numerator and denominator by the conjugate of the denominator to eliminate any imaginary numbers in the denominator.

Front

What is the denominator after multiplying (5 + 2i) by its conjugate (5 - 2i)?

Back

The denominator is 5^2 + 2^2 = 25 + 4 = 29.

Front

What is the numerator after multiplying (3 + 4i) by (5 - 2i)?

Back

The numerator is (3*5) + (3*-2i) + (4i*5) + (4i*-2i) = 15 - 6i + 20i + 8 = 23 + 14i.

Front

What is the imaginary unit 'i'?

Back

The imaginary unit 'i' is defined as the square root of -1, and it is used to extend the real number system to include complex numbers.