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 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 in the denominator, allowing us to express the result as a standard complex number.

Front

What is the formula for multiplying two complex numbers (a + bi)(c + di)?

Back

(a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i.

Front

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

Back

(2 + 3i)(2 - 3i) = 4 + 9 = 13.

Front

How do you simplify the expression (2 + 3i)/(1 - 2i)?

Back

Multiply the numerator and denominator by the conjugate of the denominator: (2 + 3i)(1 + 2i) / (1 - 2i)(1 + 2i) = (2 + 4i + 3i + 6i^2) / (1 + 4) = (-4 + 7i) / 5.

Front

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

Back

The denominator is 1 + 4 = 5.

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 complex numbers.

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 a + bi?

Back

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