Real Numbers Cheat Sheet

Key Definitions

• Real Numbers: All numbers on the number line, including rational and irrational numbers.

• Rational Numbers: Numbers that can be expressed as p/q where p and q are integers and q ≠ 0.

• Irrational Numbers: Numbers that cannot be expressed as a fraction (e.g., √2, π).

• Natural Numbers: Positive integers starting from 1 (1, 2, 3, ...).

• Whole Numbers: Natural numbers including zero (0, 1, 2, ...).

• Integers: Whole numbers and their negatives (..., -2, -1, 0, 1, 2, ...).

Important Concepts

Properties of Real Numbers

• Closure Property: Sum or product of two real numbers is a real number.

• Commutative Property: a + b = b + a and a × b = b × a.

• Associative Property: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c).

• Distributive Property: a × (b + c) = a × b + a × c.

Key Formulas

• Square of a Binomial:

 
  (a + b)² = a² + 2ab + b² 
  

• Difference of Squares:

 
  a² - b² = (a + b)(a - b) 
  

Common Mistakes to Avoid

Common Mistake: Forgetting that irrational numbers cannot be expressed as fractions.

Memory Tricks

Tip: "NICE" —

• Natural Numbers

• Integers

• Complex Numbers

• Everything else (Rational & Irrational)

Key Diagrams

Number Line: 
<---|---|---|---|---|---|---|---|---|--->
   -3  -2  -1   0   1   2   3   4   5

• Natural Numbers: Right of 0

• Whole Numbers: Includes 0

• Integers: Both sides of 0

• Rational Numbers: All fractions on the line

• Irrational Numbers: Points that cannot be pinpointed on the line (e.g., √2)