Operations with complex numbers as algebraic

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.

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?

Back

The product of two complex numbers (a + bi) and (c + di) is given by (ac - bd) + (ad + bc)i.

Front

What is the conjugate of a complex number?

Back

The conjugate of a complex number a + bi is a - bi. It is used to simplify the division of complex numbers.

Front

How do you multiply complex numbers?

Back

To multiply complex numbers, use the distributive property (FOIL method): (a + bi)(c + di) = ac + adi + bci + bdi^2, and remember that i^2 = -1.

Front

What is the simplified form of (3 + 4i) + (5 - 2i)?

Back

The simplified form is (3 + 5) + (4 - 2)i = 8 + 2i.

Front

What is the simplified form of (2 - 3i) - (4 + 5i)?

Back

The simplified form is (2 - 4) + (-3 - 5)i = -2 - 8i.

Front

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

Back

The result is (1*3 - 1*4i + 2i*3 + 2i*-4i) = 3 + 6i + 8 = 11 + 6i.