5.4 - Binary Number System

Unsigned binary

• limited to nonnegative values
• 
• 
• Min and max values
  • range of numbers that can be encoded in unsigned binary depends upon no of bits allocated to represent the number
  • max: \(2^n - 1\)
  • min: 0
• Binary arithmetic
  • adding two unsigned binary integers
    • Just like adding in base 10, but carries when gets after 1 (aka would be 2)
    • 
    • basic rules:
      • \(0_2 + 0_2 = 0_2\).
      • \(0_2 + 1_2 = 1_2\).
      • \(1_2 + 0_2 = 1_2\).
      • \(1_2 + 1_2 = 0_2\), carry \(1_2\)
        • AKA: \(1_2 + 1_2 = 10_2​\)
  • Binary multiplaction
    • Exactly what you would expect (aka: same as denary, but the addition thing at the end is subject to rules above)
    • 
    • Note: if there is a situation where you have to do 1+1+1+1 in a column, you will have to carry to the next next column because 1+1+1+1 = 100

Two’s complement

• can represent negative integers
• 
• Facts independent of no of bits:
  • If most significant (first) bit is 1, then will be -ve, else +ve (or 0 if all 0).
  • lowest possible integer when most significant bit 1, and all else 0.
  • -1 is when all bits 1
  • highest possible integer is when most significant bit 0, and all else 1.
• Observations:
  • say we have binary number representing \(n_{10}\) of \(k_{10}\) digits. To make a signed integer of \((k+1)_{10}\) digits of value \(-n_{10}\), it is written as 1 prepended to \(2^k-n\) when it is written in binary with \(k\) digits
  • 
  • Above in other words: with \(k+1\) digits, when counting from \(00...00_2\) to \(01...11_2\), you count from \(0\) to \(+2^k -1\), then when counting from \(10...00_2\) to \(11...11_2\), you count from \(-2^k\) to \(-1\).
    • When I say ‘counting’ here, I mean counting the last \(k\) digits in the binary patterns
• The weight of each bit is pow of 2, except for most significant bit, whose weight is negative of corresponding power of two/
• Most significant bit sometimes AKA ‘sign bit’
• Converting:
  • Decimal to two’s complement
    • If non-negative, just convert as if unsigned, but pad begining
      • 
    • if negative:
      • Method 1:
        • convert magnitude to binary as if unsigned
        • prepend 0
        • flip bits
        • add \(1_2\)
        • 
      • Method 2:
        • convert magnitude to binary as if unsigned
        • prepend 0
        • flip all bits apart from the least significant bit
        • 
  • Two’s complement to decimal
    • same as converting binary to decimal, but remember that most significant bit’s value is \(-2^n\)
• Subtracting
  • can subtract B from A by negating B and adding to A.
  • \(A-B = A+(-B)\).
  • E.g: \(0101_2 - 0011_2 = 0101_2 + (-0011_2) = 0101_2 + 1101_2 = (1)0010_2\)
  • if this results in a carry, it is ignored because we restrict answer to same numer of bytes started with
• Why two’s complement good
  • don’t need seperate subtraction mechanism: just addition but with complement
    • the circuits for which are relatively simple
  • other representations have 2 different representations of 0, this only has 1. Useful because 0 is used heavily when checking for equality. I.e: \(A-B=0 \implies A=B\)
• Range
  • for \(n\) bits the range is \(-2^{n-1}\) to \(2^{n-1}-1\)
  • e.g for 8 bits: \(-2^7\) to \(2^7 - 1\) = \(-128\) to \(127\)

Fixed Point

• can represent numbers with fractional parts i.e: not whole numbers