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

Learn about number systems, including the number line and representation of various types of numbers. This chapter covers the basics of number systems, including positive directions and number lines.