Simplifying 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 i² = -1.

Front

What is the imaginary unit 'i'?

Back

The imaginary unit 'i' is defined as the square root of -1. It is used to extend the real number system to include solutions to equations that do not have real solutions.

Front

How do you add complex numbers?

Back

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

Front

How do you subtract complex numbers?

Back

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

Front

What is the product of two complex numbers (a + bi)(c + di)?

Back

The product is given by (ac - bd) + (ad + bc)i, using the distributive property and the fact that i² = -1.

Front

How do you simplify the expression (7 + 2i)(9 - 6i)?

Back

The simplified form is 75 - 24i.

Front

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

Back

The result is -38 + 45i.

Front

How do you simplify the square root of a negative number, such as √-108?

Back

To simplify √-108, factor out the negative: √-108 = √(108) * √(-1) = √(36 * 3) * i = 6i√3.

Front

What is the result of simplifying 2i - 7i + 10?

Back

The result is -5i + 10.

Front

How do you find the sum of two complex numbers, such as (5 - 2i) + (-7 + 8i)?

Back

The sum is found by adding the real parts and the imaginary parts: (5 - 7) + (-2 + 8)i = -2 + 6i.