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

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How do you add two complex numbers?

Back

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

Front

How do you subtract two complex numbers?

Back

To subtract two 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 i^2 = -1.

Front

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

Back

The conjugate of a complex number a + bi is a - bi. It is obtained by changing the sign of the imaginary part.

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How do you multiply a complex number by its conjugate?

Back

Multiplying a complex number by its conjugate results in a real number: (a + bi)(a - bi) = a^2 + b^2.

Front

What is the simplified form of (-3 + 2i)(1 - 4i)?

Back

The simplified form is 5 + 14i.

Front

What is the simplified form of (11 - 7i) - (-3 + 12i)?

Back

The simplified form is 14 - 19i.

Front

What is the simplified form of (-9 + 5i) + (3 - 2i)?

Back

The simplified form is -6 + 3i.