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 .
For any two real numbers an ordered pair is defined as a complex number and is denoted by . Here and are just symbols.
Two angles are called complementary if their sum equals to 90 degrees.
Algebra of complex numbers:
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
3) The modulus of a complex number is defined as
Rapid Study Kit for "Title":
Core Concept Tutorial
Problem Solving Drill
Review Cheat Sheet
"Title" Tutorial Summary :
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.
• 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.
"Title" Topic List:
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