Number Systems

• Introduction to Number Systems

  • Natural Numbers ($\mathbb{N}$)
    • Counting numbers: $1, 2, 3, \dots$
  • Whole Numbers ($\mathbb{W}$)
    • Natural numbers including zero: $0, 1, 2, 3, \dots$
  • Integers ($\mathbb{Z}$)
    • All whole numbers and negative numbers: $\dots, -2, -1, 0, 1, 2, \dots$

• Rational Numbers ($\mathbb{Q}$)

  • Definition
    • Numbers of the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$
  • Properties
    • Equivalent (or rational) forms
    • Infinitely many rational numbers between any two rational numbers
  • Decimal Representation
    • Terminating decimals (e.g., $0.25$)
    • Non-terminating recurring (repeating) decimals (e.g., $0.333\dots$)

• Irrational Numbers

  • Definition
    • Numbers that cannot be expressed in the form $\frac{p}{q}$
    • Decimal representation is non-terminating and non-recurring
  • Examples
    • $\sqrt{2}, \sqrt{3}, \pi$
    • Locating irrational numbers like $\sqrt{2}$ on the number line

• Real Numbers ($\mathbb{R}$)

  • Definition
    • Collection of all rational and irrational numbers
  • Real Number Line
    • Every real number corresponds to a unique point, and vice-versa (Completeness Property)
  • Operations on Real Numbers
    • Algebraic laws of addition and multiplication
    • Identities involving square roots:
      • $\sqrt{ab} = \sqrt{a}\sqrt{b}$
      • $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$
      • $(a+\sqrt{b})(a-\sqrt{b}) = a^2 - b$

• Rationalisation

  • Concept
    • Converting the denominator of an expression from an irrational number to a rational number
  • Rationalising Factor
    • Multiplying numerator and denominator by a suitable conjugate (e.g., factor for $a + \sqrt{b}$ is $a - \sqrt{b}$)

• Laws of Exponents for Real Numbers

  • Basic Rules (where $a > 0$ is a real number and $p, q$ are rational exponents)
    • Product Rule: $a^p \cdot a^q = a^{p+q}$
    • Quotient Rule: $\frac{a^p}{a^q} = a^{p-q}$
    • Power of a Power: $(a^p)^q = a^{pq}$
    • Power of a Product: $a^p \cdot b^p = (ab)^p$
    • Zero Exponent: $a^0 = 1$
    • Negative Exponent: $a^{-p} = \frac{1}{a^p}$