Real Numbers

Introduction

The collection of all Rational Numbers and Irrational Numbers together forms the set of Real Numbers.

Definition and Notation

The Real Number Line

Every real number is represented by a unique point on the number line. Also, every point on the number line represents a unique real number.

Laws of Exponents for Real Numbers

For a > 0 being a real number and p, q being rational numbers:

• a^p × a^q = a^p+q (Product Law)
• (a^p)^q = a^pq (Power of a Power)
• a^p / a^q = a^p-q (Quotient Law)
• a^p × b^p = (ab)^p (Power of a Product)
• a⁰ = 1
• a^-n = 1 / aⁿ

Rationalization

If the denominator of an expression contains a square root (making it irrational), we convert it into an equivalent expression with a rational denominator.
Example: 1/√2 = √2/2.
To rationalize 1/(a+√b), multiply numerator and denominator by (a-√b).