Operations with 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 i = √-1.

Front

What is the imaginary unit 'i'?

Back

The imaginary unit 'i' is defined as the square root of -1, i.e., i = √-1.

Front

How do you simplify the square root of a negative number, such as √-4?

Back

To simplify √-4, express it as √(4) * √(-1) = 2i.

Front

What is the conjugate of a complex number?

Back

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

Front

What is the conjugate of -2 + 5i?

Back

The conjugate of -2 + 5i is -2 - 5i.

Front

How do you multiply two complex numbers, such as 4i and (-2 - 8i)?

Back

To multiply, use the distributive property: 4i * (-2) + 4i * (-8i) = -8i - 32 = 32 - 8i.

Front

What is the result of (10 + 15i) - (48 - 30i)?

Back

The result is (10 - 48) + (15i + 30i) = -38 + 45i.

Front

How do you simplify √-28?

Back

To simplify √-28, express it as √(28) * √(-1) = 2i√7.

Front

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

Back

The modulus is given by |a + bi| = √(a^2 + b^2).

Front

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

Back

To add, combine the real parts and the imaginary parts: (3 + 1) + (4i + 2i) = 4 + 6i.