Complex Numbers

Definition and Standard Form

A complex number is a number of the form a + bi, where a and b are real numbers. The value a is called the real part and bi is called the imaginary part of the complex number.

• Imaginary number: A complex number of the form a + bi where b is nonzero.

• Standard form: Every complex number can be written as a + bi (or a + ib).

Example: is a complex number with real part 3 and imaginary part 4i.

Definition of i

• i is defined as the imaginary unit:

• It follows that

Example:

Square Roots of Negative Numbers

• If , then

Example:

Operations with Complex Numbers

Addition and Subtraction of Complex Numbers

To add or subtract complex numbers, combine the real parts and the imaginary parts separately. The result should be written in the form a + bi.

1. Combine the real parts.

2. Combine the imaginary parts.

3. Write the result in the form a + bi.

Example 1:

• a)

• b)

Multiplication of Complex Numbers

To multiply complex numbers, use the distributive property (FOIL method), substitute , and combine like terms.

1. Multiply as if they are two binomials (FOIL method).

2. Substitute for .

3. Combine like terms and write the result in the form a + bi.

Example 2:

• a)

• b)

Multiplying a complex number by its conjugate:

Division of Complex Numbers

To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator to obtain a real number in the denominator.

1. Write the division as a fraction.

2. Multiply numerator and denominator by the conjugate of the denominator.

3. Multiply and simplify the numerator (using FOIL). Multiply and simplify the denominator to a real number (using FOIL).

4. Write the result in the form a + bi.

Conjugate: The conjugate of is .

General formula:

Example 3:

• a)

• b)

For each, multiply numerator and denominator by the conjugate of the denominator, then simplify.

Summary Table: Operations with Complex Numbers

Additional info: The above notes provide a comprehensive overview of basic operations with complex numbers, which are foundational in Precalculus and higher mathematics. Mastery of these operations is essential for solving equations involving complex solutions and for understanding further topics such as polar form and complex plane representations.