Equations and Inequalities

Complex Numbers and Imaginary Numbers

Complex numbers extend the real number system by including the imaginary unit i, where i is defined as the square root of -1. Every complex number can be written in the form a + bi, where a and b are real numbers. The set of all such numbers is called the set of complex numbers, and a + bi is called the standard form of a complex number.

• Imaginary unit:

• Standard form:

• Real part:

• Imaginary part:

Operations on Complex Numbers

Complex numbers can be added, subtracted, and multiplied using rules similar to those for binomials. The imaginary unit obeys the property .

• Addition/Subtraction: Combine like terms (real with real, imaginary with imaginary).

• Multiplication: Use distributive property and substitute when necessary.

Example: Adding and Subtracting Complex Numbers

Example: Multiplying Complex Numbers

Conjugate of a Complex Number

The conjugate of a complex number is . Multiplying a complex number by its conjugate results in a real number:

• Formula:

Complex Number Division

To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part from the denominator. The result is then written in standard form.

• Step 1: Multiply numerator and denominator by the conjugate of the denominator.

• Step 2: Simplify using .

• Step 3: Write the result in standard form .

Principal Square Root of a Negative Number

The principal square root of a negative number is defined using the imaginary unit. For any positive real number :

Operations Involving Square Roots of Negative Numbers

When performing operations with square roots of negative numbers, express each square root in terms of and simplify as with other complex numbers.

Example: Adding Square Roots of Negative Numbers

Example: Squaring a Complex Number with Square Roots