13.3 Floating-point numbers, representation and manipulation

Mantissa

• Stores the actual digits of the number
• Increasing the number of bits in mantissa increases accuracy

Exponent

• Tells you where the decimal point is placed by shifting
• Increasing the number of bits in exponent increases range

Floating Point Number Representation

• ± mantissa × 2exponent
• Can represent fractions and negative numbers
• In scientific notation a number is represented in the following way: 2.15 x 104
• Here 2.15 acts like mantissa (the significant digit) while 104 acts like the exponent (shifts decimal point 4 places to the right)
• The decimal point in binary numbers may be termed as binary point

Converting binary to floating-point real number (Method 1)

1. You will be given the value of mantissa and exponent
   Mantissa: 01110100 Exponent: 0010
2. Write the binary number with decimal point: by default consider that the decimal point is after the first digit of mantissa
   0.1110100
3. Evaluate the value of exponent
   0010 = 2
4. Shift the decimal point to the right the number of times equaling the value of the exponent
   011.10100
5. Evaluate the value

Value: 2 + 1 + (½) + (⅛) = 3.625

Converting binary to floating-point real number (Method 2)

1. You will be given the value of mantissa and exponent
   Mantissa: 01110100 Exponent: 0010
2. Answer will be mantissa × 2exponent

= 0.1110100 x 20010

= (½ + ¼ + ⅛ + 1/32 ) x 22 = 3.625

Converting denary to binary floating-point number (example)

1. You will be given a denary number and the number of bits for mantissa and exponent

1. Convert to binary putting the decimal point in the correct place

1. Move the decimal point until the number is normalized (The value represented is the mantissa)

1. Calculate the exponent.
   Since the decimal point moved by 7 places, the value of exponent will be 7: 0111
   Mantissa: 1001100101 Exponent: 000111

Note: If a given number cannot be represented fully, represent the closest number possible (Precision may be lost)

Normalization

• Normalization is done to ensure uniqueness of number representation
• Makes calculation more straight forward
• Stores maximum range of numbers in minimum number of bits
• Enables very large/small number to be stored with accuracy
• During normalization a positive number must start with 01 from the left and negative number starts with 10
• Normalization Steps:
  • Example: Normalize 0010110110 0010
  • Step 1: Adjust the decimal point by shifting it left or right until the number begins with 01 or 10
    • 0.010110110 becomes 0.101101100
  • Step 2: Adjust the exponent as you shift the decimal point: moving it to the left increases the exponent, while moving it to the right decreases the exponent.
    • Decimal point moves 1 to the right, so 1 is deducted from the exponent. Answer becomes 0101101100 0001
• Normalizing Negative Floating Point Values
  • Example: 11110010 0111
  • By default the number is in the format
    1**.1110010 x 20111 = 1.**1110010 x 27
  • Move the decimal point until normalized
    1111**.**0010000
  • Since the decimal point moved by 3 places, deduct 3 from exponent
    = 1**.**0010000 x 24
    = 10010000 0100

Overflow

• Happens when a number is too large to be represented within the available number of bits.
• The exponent exceeds its maximum value.
• Precision may be lost
• Excess values cut off
• Example: Trying to store 220 with only a 4-bit exponent.

Underflow

• Happens when a number is too small (close to 0) to be represented within the available bits.
• The exponent is too negative (less than the minimum allowed).
• Result: The number may be stored as zero, losing precision.
• Example: 0.0000001 with limited mantissa and exponent bits.

Rounding Error

• When a number cannot be represented exactly in binary, it is approximated to the closest possible value.
• This small difference can cause inaccuracies, especially when such numbers are used in multiple calculations.
• Over time, these tiny errors accumulate and become noticeable
• Example: 0.1 + 0.7 might give 0.7999999999999 instead of 0.8.

Important values

Consider for the following 8 bit mantissa and 4 bit exponent