Imaginary Numbers

Front

What is an imaginary number?

Back

An imaginary number is a 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 squared (i^2)?

Back

i^2 = -1.

Front

How do you solve the equation x^2 + 36 = 0?

Back

To solve x^2 + 36 = 0, subtract 36 from both sides to get x^2 = -36. Taking the square root gives x = ±6i.

Front

What is the solution to the equation 33 = 18 - 3x^2?

Back

To solve 33 = 18 - 3x^2, rearrange to get 3x^2 = 18 - 33, which simplifies to 3x^2 = -15. Dividing by 3 gives x^2 = -5, so x = ±i√5.

Front

What is the product of two imaginary numbers, such as (3i)(4i)?

Back

The product (3i)(4i) = 12i^2 = 12(-1) = -12.

Front

What is the conjugate of an imaginary number, such as 5i?

Back

The conjugate of 5i is -5i.

Front

How do you express the square root of a negative number, such as √-49?

Back

√-49 can be expressed as 7i, since √-49 = √(49) * √(-1) = 7i.

Front

What is the significance of the imaginary unit 'i' in complex numbers?

Back

The imaginary unit 'i' allows for the extension of the real number system to include solutions to equations that do not have real solutions, such as x^2 + 1 = 0.

Front

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

Back

To add (2 + 3i) and (4 - 2i), combine the real parts and the imaginary parts: (2 + 4) + (3i - 2i) = 6 + i.

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.