Imaginary and Complex Numbers

Front

What is the definition of 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 standard form of a complex number?

Back

The standard form of a complex number is written as a + bi, where 'a' is the real part and 'bi' is the imaginary part.

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, where 'i' is the imaginary unit.

Front

What is the result of squaring a complex number, such as (-6 + i)²?

Back

The result of squaring the complex number (-6 + i) is 35 - 12i.

Front

What is the imaginary part of the complex number a + bi?

Back

The imaginary part of the complex number a + bi is 'b', which is the coefficient of 'i'.

Front

How do you multiply two complex numbers, such as (-5 - 5i)(1 + 5i)?

Back

To multiply two complex numbers, use the distributive property (FOIL) and combine like terms. The result of (-5 - 5i)(1 + 5i) is 20 - 30i.

Front

What is the product of two imaginary numbers, such as (3i)(-2 + 4i)?

Back

The product of (3i)(-2 + 4i) is -12 - 6i.

Front

What is the conjugate of a complex number a + bi?

Back

The conjugate of a complex number a + bi is a - bi.

Front

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

Back

To add two complex numbers, combine their real parts and their imaginary parts separately. The result of (3 + 2i) + (1 + 4i) is 4 + 6i.

Front

What is the modulus of a complex number a + bi?

Back

The modulus of a complex number a + bi is given by the formula √(a² + b²).