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

How do you add two complex numbers?

Back

To add two complex numbers, you add their real parts and their imaginary parts separately. For example, (a + bi) + (c + di) = (a + c) + (b + d)i.

Front

What is the result of (5 - 2i) + (-7 + 8i)?

Back

The result is -2 + 6i.

Front

How do you subtract two complex numbers?

Back

To subtract two complex numbers, you subtract their real parts and their imaginary parts separately. For example, (a + bi) - (c + di) = (a - c) + (b - d)i.

Front

What is the result of (4 - 2i) - (3 + 6i)?

Back

The result is 1 - 8i.

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 express complex numbers.

Front

What is i²?

Back

i² = -1.

Front

What is i³?

Back

i³ = i * i² = i * -1 = -i.

Front

What is i⁴?

Back

i⁴ = (i²)² = (-1)² = 1.

Front

How do you simplify powers of i?

Back

Powers of i can be simplified using the cycle: i, -1, -i, 1. For example, to simplify i^31, find the remainder of 31 divided by 4, which is 3, so i^31 = i³ = -i.