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Number Representations

Short Notes

Number systems define how values are stored in binary. GATE focuses heavily on signed integers and floating-point representations.

Integer Representations (\(n\) bits)

  • Unsigned: Range \([0, 2^n - 1]\).
  • Signed Magnitude: Range \([-(2^{n-1} - 1), 2^{n-1} - 1]\). Two representations for zero (+0 and -0).
  • 1's Complement: Range \([-(2^{n-1} - 1), 2^{n-1} - 1]\). Two representations for zero.
  • 2's Complement: Range \([-2^{n-1}, 2^{n-1} - 1]\). Unique representation for zero. Most efficient for hardware (subtraction is just addition).

Key Theories & Formulas

1. Overflow Detection in 2's Complement

Overflow occurs when adding two numbers of the same sign results in a different sign.

  • Hardware Logic: \(Overflow = C_{n} \oplus C_{n-1}\) (Carry into MSB XOR Carry out of MSB).

2. IEEE 754 Floating Point (Single Precision - 32 bits)

  • Sign Bit (1 bit): 0 for +, 1 for -.
  • Exponent (8 bits): Biased by 127. Store \(E + 127\).
  • Mantissa (23 bits): Normalized form \(1.M\). Only the fractional part \(M\) is stored (Hidden bit '1').
  • Value: \((-1)^S \times 1.M \times 2^{E-127}\).

Example Problems

Problem: Represent -5 in 8-bit 2's complement.

  1. Binary of +5: 0000 0101
  2. 1's complement: 1111 1010
  3. Add 1 for 2's complement: 1111 1011 Result: 0xFB

Problem: What is the range of 8-bit 2's complement?

  • Min: \(-2^7 = -128\)
  • Max: \(2^7 - 1 = 127\)

Hardest GATE Questions

Topic: IEEE 754 Precision and Min/Max values Tricky Question (GATE 2018): Consider a floating-point system with 1 bit sign, 3 bit biased exponent (bias=3), and 2 bit mantissa. What is the smallest positive normalized number?

  • Analysis:
  • Smallest normalized number has Exponent = 1 (actual exponent \(E = 1 - 3 = -2\)).
  • Mantissa is \(1.00_b\) (hidden bit is 1, fraction is 00).
  • Value = \(+1.00 \times 2^{-2} = 0.25\).
  • The "Trap": Exponent bits 000 and 111 are reserved for Denormalized numbers and NaN/Infinity respectively. Using \(E_{stored}=0\) would lead to a denormalized number, which is a different category.
  • Hard Aspect: Differentiating between the "Smallest Normalized" and "Smallest Denormalized" value.
  • Smallest Denormalized (for same system): \(0.01_b \times 2^{1-3} = 0.25 \times 2^{-2} = 0.0625\)

References