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 i = √(-1). It is used to represent the square root of negative numbers.

Front

What is the value of i^2?

Back

i^2 = -1.

Front

What is the value of i^3?

Back

i^3 = i^2 * i = -1 * i = -i.

Front

What is the value of i^4?

Back

i^4 = (i^2)^2 = (-1)^2 = 1.

Front

How do you multiply two complex numbers?

Back

To multiply two complex numbers (a + bi)(c + di), use the distributive property: ac + adi + bci + bdi^2, and simplify using i^2 = -1.

Front

What is the result of (4 - 3i)(7 + 2i)?

Back

(4 - 3i)(7 + 2i) = 28 + 8i - 21i - 6i^2 = 34 - 13i.

Front

How do you simplify a complex expression like 3i - (4 + 7i)?

Back

To simplify, distribute the negative sign: 3i - 4 - 7i = -4 - 4i.

Front

What is the value of i^7?

Back

i^7 = i^(4+3) = i^4 * i^3 = 1 * -i = -i.

Front

What is the value of i^66?

Back

i^66 = i^(64+2) = (i^4)^16 * i^2 = 1 * -1 = -1.