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, and it is used to extend the real number system to include solutions to equations that do not have real solutions.

Front

What is the value of i^2?

Back

i^2 = -1.

Front

How do you multiply two complex numbers?

Back

To multiply two complex numbers, use the distributive property (FOIL method) and combine like terms. For example, (a + bi)(c + di) = ac + adi + bci + bdi^2.

Front

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

Back

The result is -12 - 6i.

Front

How do you simplify the square root of a negative number?

Back

To simplify the square root of a negative number, factor out -1 and use the imaginary unit 'i'. For example, √-25 = √(25)√(-1) = 5i.

Front

What is the simplified form of (8 - 3i)^2?

Back

The simplified form is 55 - 48i.

Front

What is the result of multiplying two imaginary numbers, (2i)(3i)?

Back

The result is -6.

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?

Back

To add two complex numbers, simply add their real parts and their imaginary parts separately. For example, (a + bi) + (c + di) = (a + c) + (b + d)i.