ELEC3020 - Lecture 2

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ELEC3020 - Lecture 1
Number Representations - Slides

Fixed Point Numbers

For a number in base $n$:
$i_2{i}_1{i}_0.i_{-1}{i}_{-2}{i}_{-3}$

We can easily convert to base 10:
$i\ast n^2+i\ast n^1+i\ast n^0+i\ast n^{-1}+i\ast n^{-2}+i\ast n^{-3}$

Signed Binary Numbers

Two’s Complement Representation

• 2’s complement of a binary number is formed by taking 1’s complement of the number and adding 1 to the least significant bit
  • Alternate way: complement bits to the left of the rightmost 1
• 1’s complement negates numbers by complementing each bit of the number

2’s complement = 1’s complement +1

Taking the 2’s complement of a binary number negates it (changes its sign)

• Summing the complement of a number with itself should equal 0

Converting 2’s complement binary to decimal:

• Same process as earlier, but minus away the sign bit

Link to original

Signed Numbers

Rule of thumb -> the signed negative number for any binary or hexadecimal number when added to itself, equals zero

Floating Point Numbers

• IEEE standard for FP numbers

• 32 bits, single precision:

  • 1 bit sign, 8 bits exponent, 23 bits fraction

  
  ${\text{Number}}_{10}=(-1{)}^{\text{sign}}\ast 2^{(\text{exponent}-127)}\ast 1.{\text{fraction}}_2$

Special Cases

Exponent = 0

• Fraction = 0 -> result = 0
• Fraction $\ne$ 0 -> result = $(-1{)}^{\text{sign}}\ast 2^{(-126)}\ast 0.{\text{fraction}}_2$

Exponent = 255

• Fraction = 0 -> result is $\pm$ infinity
• Fraction $\ne$ 0 -> result is NaN

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ELEC3020 - Lecture 3