Multiplying 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 imaginary unit 'i'?

Back

The imaginary unit 'i' is defined as the square root of -1. It is used to extend the real number system to include solutions to equations that do not have real solutions.

Front

How do you multiply two complex numbers (a + bi) and (c + di)?

Back

To multiply two complex numbers, use the distributive property: (a + bi)(c + di) = ac + adi + bci + bdi^2. Since i^2 = -1, this simplifies to (ac - bd) + (ad + bc)i.

Front

What is the product of (1 + 2i) and (3 + 4i)?

Back

(1 + 2i)(3 + 4i) = 3 + 4i + 6i + 8i^2 = 3 + 10i - 8 = -5 + 10i.

Front

What is the product of (2 - 3i) and (1 + i)?

Back

(2 - 3i)(1 + i) = 2 + 2i - 3i - 3i^2 = 2 - i + 3 = 5 - i.

Front

What is the conjugate of a complex number a + bi?

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 does multiplying by the conjugate help in division of complex numbers?

Back

Multiplying by the conjugate helps eliminate the imaginary part in the denominator, making it a real number, which simplifies the division of complex numbers.

Front

What is the product of (3 + 2i) and (4 - 5i)?

Back

(3 + 2i)(4 - 5i) = 12 - 15i + 8i - 10i^2 = 12 - 7i + 10 = 22.

Front

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

Back

(2 + i)(-3 - 4i) = -6 - 8i - 3i - 4i^2 = -6 - 11i + 4 = -2 - 11i.

Front

What is the product of (-2 + 7i) and (1 - 3i)?

Back

(-2 + 7i)(1 - 3i) = -2 + 6i + 7i - 21i^2 = -2 + 13i + 21 = 19 + 13i.