Radicals:

If is a positive integer and and are real numbers such that then we write  where  is the radical sign, is the radicand and is an index.

Properties of radicals:

1) For any positive integer  and for a real number a

2) For any positive integer and positive real numbers and we have

3) For any positive integer  and positive real numbers and we have  .

Complex Numbers:

For any two real numbers an ordered pair  is defined as a complex number and is denoted by . Here and are just symbols.

Complementary angles:

Two angles are called complementary if their sum equals to 90 degrees.

Algebra of complex numbers:

1)

2)

3)

 Conjugate of a complex number:

A pair of complex numbers are said to be conjugates of each other if they have identical real parts and imaginary parts that are identical except for having opposite signs.

 Conjugate Complex Numbers:

If are any two complex numbers, then we have the following properties

1)

2)

3) The modulus of a complex number is defined as

Flash Movie	Flash Game	Flash Card
Core Concept Tutorial	Problem Solving Drill	Review Cheat Sheet

This tutorial describes roots, radicals and complex numbers.  First, the properties of radicals are discussed. The constraints and special cases of radicals are presented in this tutorial. Applications of radicals are mentioned in the examples.

Next, complex numbers are presented in some of the examples. Last, complex numbers and the ways that radicals are used are mentioned in this tutorial.

Specific Tutorial Features:
• Radical Properties are mentioned in the examples.
• Special types of radicals are shown in the example problems.
• Complex conjugate numbers are used in examples.

Series Features:
• Concept map showing inter-connections of new concepts in this tutorial and those previously introduced.
• Definition slides introduce terms as they are needed.
• Visual representation of concepts
• Animated examples—worked out step by step
• A concise summary is given at the conclusion of the tutorial.

Radicals and Roots
Definition of radicals
Special cases of radicals
Constraints of radicals
Properties of radicals
Complex Numbers
Definition of complex numbers
Algebra of Complex Numbers
Conjugate Complex Numbers