Number system

Introduction

• Natural numbers
• Whole numbers
• Integers
  • Z - German word - "Zahlen" - to count
• Rational numbers - EX - 1.1
  • Q - Quotient
  • Rational - Ratio
  • Equivalent rational numbers or fractions
  • p/q , are co prime
  • infinitely many rational numbers between 2 rational numbers
  • exercise:
    • state whether the number is rational or not - 0
    • find the rational numbers in between
• Irrational numbers
• Real numbers
  • Collection of rational and irrational numbers
  • R - Represented by

Every point in the number line is represented by unique number 

Every real number is represented by a unique point on the number line

Irrational numbers - EX - 1.2

• First to discover the irrational numbers
• cannot be written in p/q form

Locating irrational number on the number line - EX - 1.2

For √2:

• mark 0 and 1 in the number line
• make a vertical line of 1 unit from point 1 and mark it as A
• then connect A and O
• Complete the arc drawn from the A on the number line beside 1
• this marks the point for √2 For √3:
• √ [ (√2)^2 + 1^2 ]= √3 = pythagoreas theorem
• mark 0 and 1 in the number line
• make a vertical line of 1 unit from point 1 and mark it as A
• make a perpendicular line from A with unit distance, and mark it as B
• join OB
• now, complete the arc drawn from B on the number line beside 1.
• this point marks the √3

Real number and decimal expansion - EX - 1.3

• terminating vs non terminating
• recurring vs non recurring

• remainders in recurring will always be less than the divisor
• if the remainder repeat then we get a recurring set of quotients

Operations on real numbers - EX - 1.4

• add
• subtract
• multiply
• divide Facts:
• sum or difference of rational and irrational = irrational
• product or quotient of rational and irrational = irrational
• add or sub or multiply or divide 2 irrationals = may be rational or irrational
• rationalise the denominator
• laws for irrational numbers
  

Laws of exponents for real numbers - EX - 1.5

RSA or RDS