Lecture Notes: Representing Real Numbers

Introduction to Real Numbers

• Definition: Real numbers are any numbers that can be represented as decimals.
  • Examples include negative numbers, fractions, irrational numbers, etc.
• Examples of Real Numbers:
  • Negative Numbers: -8
  • Whole Numbers: 0, 1.75
  • Fractions: 3/2
  • Irrational Numbers: √3, π (approximately 3.14)

Subsets of Real Numbers

Natural Numbers

• Also known as "counting numbers."
• Begin at 1 and continue infinitely (1, 2, 3, ...).

Whole Numbers

• Includes all natural numbers and adds 0.
• Starts from 0 and continues infinitely (0, 1, 2, 3, ...).

Integers

• Includes all whole numbers and their negative counterparts.
• Examples include: -2, -1, 0, 1, 2...

Rational Numbers

• Definition: Numbers that can be expressed as fractions.
  • Examples: 1/2, 3/4
  • Integers can be considered rational by placing them over 1 (e.g., -4 as -4/1).
• Important Rule: Denominator of a fraction cannot be 0.

Decimal Representation of Rational Numbers

• Terminating Decimals: Decimals that come to an end (e.g., 0.5, 0.75).
• Infinitely Repeating Decimals: Decimals that have a repeating sequence (e.g., 4/11 = 0.363636...)
  • Written with a bar over the repeating section (0.36̅)

Examples

• 1/16: 0.0625 (terminating decimal)
• 55/27: 2.037037... (repeating decimal, written as 2.0̅37)
• 1/17: An example where the repeating sequence is long.

Irrational Numbers

• Definition: Numbers that cannot be written as fractions.
  • Examples: π, √3
• Characteristics:
  • Decimal representation never terminates or repeats.
  • Often represented by symbols (e.g., π for pi).

Conclusion

• Real Numbers encompass several subsets, each with distinct characteristics.
• Understanding these subsets and their properties aids in mathematical representation and calculations.