Lecture Notes: Representing Real Numbers in Binary

Introduction

• Focus on representing real numbers with binary, particularly those with fractional parts (decimal parts).
• Explore the concept of binary representation of partial numbers.

Place Value System

• Decimal System:
  • Base-10 place value.
  • Example: 3725 = 3 thousands, 7 hundreds, 2 tens, 5 ones.
• Binary System:
  • Base-2 place value.
  • Example: 0101 in binary = 1 eight, 0 fours, 1 two, and 0 ones; equals 10 in decimal.

Extending Place Value with Fractions

• Decimal Fractions:
  • Tenths: 10^-1 = 0.1
  • Hundredths: 10^-2 = 0.01
• Binary Fractions:
  • Halves: 2^-1 = 0.5
  • Quarters: 2^-2 = 0.25
  • Example: 10.75 in decimal can be represented as 10.11 in binary (1 half + 1 quarter).

Fixed Point Representation

• Fixed point representation is impractical for precision beyond set fractional bits.
• Example: Representing "one and a 20th" becomes challenging.
• Human conventions like repeating decimals (e.g., 0.33 repeating) can't be replicated in binary.

Floating Point Representation

• Scientific Notation in Computing:
  • Mantissa (significand) and exponent used to store numbers.
  • Similar to scientific notation but uses base 2.
• Floating Point Notation:
  • Allows moving the "decimal" point to represent different degrees of precision.
  • Limitation: Cannot represent infinite numbers between 0 and 1.

IEEE Standards

• Single Precision (Floats):
  • 32-bit representation.
• Double Precision (Doubles):
  • 64-bit representation.

Hardware Considerations

• Floating point operations require different circuitry than integer operations.
• Integer circuitry is essential for non-math operations, notably in memory management.

Key Takeaways

• Understanding of single precision and double precision.
• Knowledge of floating point as a form of scientific notation in computing.
• Future focus: Memory operations in upcoming course modules.