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 the square root of -1.

Front

What is the imaginary unit 'i'?

Back

The imaginary unit 'i' is defined as the square root of -1, which means i^2 = -1.

Front

How do you simplify the square root of a negative number, such as √-24?

Back

To simplify √-24, factor out the negative sign: √-24 = √(24) * √(-1) = √(24) * i = 2i√6.

Front

What is the result of multiplying two complex numbers (7 + 2i)(9 - 6i)?

Back

The result is 75 - 24i.

Front

How do you square a complex number, such as (8 - 3i)²?

Back

To square a complex number, use the formula (a + bi)² = a² + 2abi + (bi)². For (8 - 3i)², it equals 64 - 48i - 9 = 55 - 48i.

Front

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

Back

The result is -9 - 12i.

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 + 4i) + (5 + 2i)?

Back

To add complex numbers, combine the real parts and the imaginary parts: (3 + 5) + (4i + 2i) = 8 + 6i.

Front

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

Back

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

Front

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

Back

To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator: (3 + 4i)(1 + 2i) / ((1 - 2i)(1 + 2i)).