Imaginary Numbers

Front

What is the definition of an imaginary number?

Back

An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit 'i', where i is defined as the square root of -1.

Front

What is the value of i^2?

Back

i^2 = -1.

Front

How do you simplify the square root of a negative number, such as √-x?

Back

To simplify √-x, you can express it as i√x.

Front

What is the value of √-1?

Back

√-1 = i.

Front

What is the simplified form of √-4?

Back

√-4 = 2i.

Front

How do you express √-24 in terms of i?

Back

√-24 = 2i√6.

Front

What is the simplified form of √-12?

Back

√-12 = 2i√3.

Front

What is the pattern of powers of i?

Back

The powers of i cycle every four: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1, and then it repeats.

Front

What is the value of i^5?

Back

i^5 = i.

Front

How do you convert a complex number in the form a + bi to polar form?

Back

To convert a complex number a + bi to polar form, calculate the modulus r = √(a^2 + b^2) and the argument θ = arctan(b/a). Then express it as r(cos θ + i sin θ).