Number Systems Chapter 1

Number Systems

Introduction to Number Systems

• Students have learned about the number line and representation of various types of numbers on it.
• The number line has infinitely many numbers, including natural numbers, whole numbers, integers, and rational numbers.

Natural Numbers, Whole Numbers, and Integers

• Natural numbers are denoted by the symbol N and include numbers like 1, 2, 3, and so on.
• Whole numbers are denoted by the symbol W and include natural numbers and 0.
• Integers are denoted by the symbol Z and include whole numbers and negative integers.

Rational Numbers

• Rational numbers are denoted by the symbol Q and include numbers that can be written in the form p/q, where p and q are integers and q ≠ 0.
• Rational numbers include natural numbers, whole numbers, and integers.
• Rational numbers do not have a unique representation in the form p/q, where p and q are integers and q ≠ 0.
• Examples of rational numbers include 1/2, 3/4, and 22/7.

Irrational Numbers

• Irrational numbers are numbers that cannot be written in the form p/q, where p and q are integers and q ≠ 0.
• Examples of irrational numbers include √2, √3, and π.
• Irrational numbers were first discovered by the Pythagoreans in ancient Greece.
• Irrational numbers cannot be expressed as a finite decimal or fraction.

Real Numbers

• Real numbers are denoted by the symbol R and include all rational and irrational numbers.
• Real numbers can be represented on the number line.
• Every real number is represented by a unique point on the number line, and every point on the number line represents a unique real number.

Decimal Expansions of Real Numbers

• Rational numbers have a finite or recurring decimal expansion.
• Irrational numbers have a non-recurring and non-terminating decimal expansion.
• Examples of decimal expansions of rational numbers include 0.5, 0.333..., and 0.142857...Here are the study notes for the text:
• Rational and Irrational Numbers*
• The decimal expansion of a rational number is either terminating or non-terminating recurring.
• A number with a non-terminating non-recurring decimal expansion is irrational.
• Examples of rational numbers: 3.142678, 0.3333...
• Examples of irrational numbers: π, √2
• Operations on Rational Numbers*
• Rational numbers satisfy commutative, associative, and distributive laws for addition and multiplication.
• The sum, difference, product, and quotient of two rational numbers are always rational.
• Operations on Irrational Numbers*
• Irrational numbers also satisfy commutative, associative, and distributive laws for addition and multiplication.
• The sum, difference, product, and quotient of a rational and an irrational number are always irrational.
• Example: 2 + √3, 2√3, π - 2 are all irrational numbers.
• Square Roots and nth Roots*
• For any positive real number x, √x is a real number that satisfies (√x)^2 = x.
• Similarly, for any positive real number x and positive integer n, x is a real number that satisfies (^n x)^n = x.
• Geometric construction of square roots and nth roots is possible using circles and triangles.
• Identities for Square Roots*
• Some useful identities relating to square roots:
  • √ab = √a × √b
  • √(a/b) = √a / √b
  • (√a)^2 = a
  • √(a^2) = a
  • (√a + √b)(√a - √b) = a - b### Algebraic Identities
• $(a+b)(a-b) = a^2-b^2$
• $(a+b)^2 = a^2 + 2ab + b^2$
• $(a+b)(c+d) = ac + ad + bc + bd$
• $(a-b)^2 = a^2 - 2ab + b^2$

Rationalising the Denominator

• To rationalise the denominator of $\frac{1}{a+b}$, multiply by $\frac{a-b}{a-b}$
• To rationalise the denominator of $\frac{1}{a-b}$, multiply by $\frac{a+b}{a+b}$
• Example: Rationalise the denominator of $\frac{1}{2+\sqrt{3}}$
  • Multiply by $\frac{2-\sqrt{3}}{2-\sqrt{3}}$
  • Result: $\frac{2-\sqrt{3}}{4-3} = 2 - \sqrt{3}$

Laws of Exponents

• Let $a>0$ be a real number and $n$ be a natural number.
  • $a^m \cdot a^n = a^{m+n}$
  • $(a^m)^n = a^{mn}$
  • $a^m / a^n = a^{m-n}$
• Extend the laws of exponents to negative exponents:
  • $a^{-m} = 1/a^m$
  • $a^0 = 1$
• Let $a>0$ be a real number and $m$ and $n$ be integers.
  • $a^{m/n} = \sqrt[n]{a^m}$
  • $(a^m)^{1/n} = \sqrt[n]{a^m}$

Real Numbers

• A number is rational if it can be expressed in the form $p/q$, where $p$ and $q$ are integers and $q \neq 0$
• A number is irrational if it cannot be expressed in the form $p/q$, where $p$ and $q$ are integers and $q \neq 0$
• The decimal expansion of a rational number is either terminating or non-terminating recurring
• The decimal expansion of an irrational number is non-terminating non-recurring
• All rational and irrational numbers make up the collection of real numbers