Floating Point Numbers

Overview

• Introduction to floating point numbers and their significance in computing.
• Comparison between fixed point numbers and floating point numbers.
• Examples of large and small numbers in computing (e.g., mass of planets, Avogadro's number, Planck's constant).

Fixed Point Numbers

• Definition: Numbers where the position of the radix point (decimal point) is fixed.
• Examples: Integers and real numbers.
  • Integers: No fractional part; assumed decimal point at the end.
  • Real Numbers: Decimal point is placed before the fractional part (e.g., 11.75).

Storage Issues

• Stored in binary format using a fixed number of bits (e.g., 10-bit format).
• Range Limitations:
  • Unsigned integers: Range from 0 to 1023 (for 10 bits).
  • Signed integers: Range from -512 to +511.
• Real Numbers:
  • Max representable value when reserving bits for integer and fractional parts.
  • Example: 6 bits for integer, 4 bits for fraction results in max 63.9375 and min 0.0625.

Precision and Range Trade-off

• Fixed point representation has a fixed radix point.
  • Adjusting bits for precision affects the number range.
  • Example: Allocating more bits for fractional part reduces integer range.

Floating Point Numbers

• Definition: Allows dynamic shifting of the radix point, enhancing range and precision.
• Comparison: Capable of representing very large and very small numbers effectively.

Representation of Floating Point Numbers

• Similar to scientific notation (one significant digit before the decimal point).
• Structure:
  • Consists of three parts: sign, fraction (mantissa), and exponent.
  • The base of the exponent is 2 in binary representation.
• Normalization:
  • Example of binary normalization: shifting the binary point to ensure only one significant digit before the point.

Storage Format

• Memory Storage:
  • 1 bit for the sign (0 for positive, 1 for negative).
  • Bits reserved for exponent and fraction.
• Mantissa: The integer part is always 1 and is not stored explicitly.

Standardization

• Common standards for storage defined (e.g., IEEE 754).
• Next video will cover the IEEE format in detail.

Conclusion

• Key differences between fixed and floating point representation highlighted.
• Importance of floating point for representing large and small values with precision.
• Invitation for questions and subscription reminders.