Limitations of storing numbers

Integer Representation

• Definition: Whole numbers without fractional components (e.g., 120, 10, 0, -20).

• Binary Representation:

  • Uses bits to store integers.

  • 4-bit Example: 0 (sign) 111 (value) represents 7.

  • 6-bit Example: 0 (sign) 11111 (value) represents 31.

Overflow

• Definition: Exceeding the maximum storage capacity of a system.

• Example: In a 4-bit system, adding 1 to 7 causes overflow.

• 6-bit System Overflow:

  • Operations causing overflow: 3 + 29, 24 + 6, 15 + 15.

Floating-Point Representation

• Definition: Represents non-integers using base 2 (similar to scientific notation).

• 64-bit System:

  • 1 bit for sign, 11 bits for exponent, 52 bits for the number.

  • Example: 0.375 in binary floating-point.

Roundoff Errors

• Definition: Precision loss due to inability to fully represent certain numbers in binary.

• Example: Adding 0.1 three times may not equal 0.3 exactly.

Summary

• Integers: Stored using bits, watch for overflow.

• Floating-Point Numbers: Used for non-integers, limited by precision.

• Modern Systems: 64-bit architecture provides high precision for critical applications.