Imaginary Numbers

Front

What is 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 √-18?

Back

To simplify √-18, factor out the negative: √-18 = √(18) * √(-1) = 3√2 * i.

Front

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

Back

The result is 1 - 5i.

Front

What is the value of i^4?

Back

i^4 = 1.

Front

How do you add two complex numbers, such as (3 + 2i) and (1 - 4i)?

Back

To add complex numbers, combine the real parts and the imaginary parts separately: (3 + 1) + (2i - 4i) = 4 - 2i.

Front

What is the conjugate of a complex number, such as 2 + 3i?

Back

The conjugate of a complex number a + bi is a - bi. For 2 + 3i, the conjugate is 2 - 3i.

Front

How do you multiply two complex numbers, such as (1 + 2i) and (3 + 4i)?

Back

To multiply complex numbers, use the distributive property: (1 + 2i)(3 + 4i) = 3 + 4i + 6i + 8i^2 = 3 + 10i - 8 = -5 + 10i.

Front

What is the modulus of a complex number, such as 3 + 4i?

Back

The modulus of a complex number a + bi is given by √(a^2 + b^2). For 3 + 4i, the modulus is √(3^2 + 4^2) = √(9 + 16) = √25 = 5.

Front

What is the argument of a complex number, such as 1 + i?

Back

The argument of a complex number a + bi is the angle θ in the polar form, calculated as θ = arctan(b/a). For 1 + i, θ = arctan(1/1) = π/4 or 45 degrees.