Notes on Number Systems

Overview of Number Systems

• Started with natural numbers (N) and integers (Z).
• Progressed to rational numbers (Q): defined as P/Q where P and Q are integers.

Properties of Rational Numbers

• Rational numbers are dense on the number line.
  • Between any two rational numbers, you can find another rational number.
• Integers are a subset of rational numbers.
  • Example: 7 can be written as 7/1.

Exploring Non-Rational Numbers

• Question: Do all rational numbers fill the number line?
• Not all numbers on the number line are rational.

Square and Square Roots

• Definition of square:
  • If m is a number, m squared (m^2) is m * m.
• Definition of square root:
  • Square root of a number m is a number r such that r * r = m.
• Perfect squares:
  • Examples: 1 (1^2), 4 (2^2), 9 (3^2), etc.
• Non-perfect squares like 10 have non-integer square roots (between 3 and 4).

The Case of Square Root of 2

• Square root of 2 is a classic example of an irrational number.
  • Cannot be expressed as P/Q (known since ancient Greek times).
  • Can be represented geometrically (hypotenuse of a square with sides of length 1).
• Irrational numbers exist on the number line but aren’t rational.

Real Numbers

• Real numbers (R) consist of both rational and irrational numbers.
  • Denoted by the double line R.
• Examples of irrational numbers:
  • Pi (π): ratio of circumference to diameter of a circle.
  • Euler's number (e): approximately 2.718.
• Square roots of non-perfect squares (like 2, 3, 5, 6) are irrational.

Denseness of Real Numbers

• Real numbers are also dense:
  • Between any two real numbers, there exists another real number.
  • Average of any two real numbers is a real number.

Complex Numbers

• Square roots of negative numbers lead to complex numbers.
• Complex numbers extend real numbers.
• Not covered in this course.

Summary

• Sequence of number types:
  • Natural numbers (N) → Integers (Z) → Rational numbers (Q) → Real numbers (R) → Complex numbers (not covered).
• Important note: Only perfect squares have rational square roots.