CSCI130

1
Lecture 2 (1.1-2.3)

• CSCI130
• Instructor Dr. Imad Rahal

2
Real Number Representation

• We deal with a lot of real numbers in our lives
  (class average) and while using computers
• Fractions or numbers with decimal parts
• 3/100.3 or 82.34
• 82.34 810 21 31/10 41/100
• In Math to represent a real number in binary, we
  use a radix point instead of a decimal point
• 1101.11 8 4 1 ½ ¼ 13.75
• On Computers How can we represent the radix
  point? 0? 1?

3
Real Number Representation

• We resort to the floating-point representation
• We represent the number without the radix
  point!!!
• Based on a popular scientific notation that has
  three parts
• Sign
• Mantissa (one non-zero digit before the decimal
  point, and two or more after), and
• Exponent (written with an E following by a sign
  and an integer which represents what power to
  raise 10 to)
• 6.124E5
• 6.124 105 612,400
• Sign? Mantissa? Exponent?
• Scientific to decimal
• -9.14E-3 -9.14 10-3 -9.14/1000 - 0.00914

4
Real Number Representation

• Decimal to scientific
• Divide (or multiply) by 10 until you have only
  one digit (non-zero) before the decimal point
• Add (or subtract) one to the exponent every time
  you divide (or multiply)
• 123.8 ?
• 1.238E2
• 0.2348 ?
• 2.348E-1

5
Real Number Representation

• Floating-point numbers are very similar
• But use only 0s and 1s instead
• /- 1.xxxx2exponent
• For an eight bit number
• 0 000 0000
• Sign, exponent, mantissa (not the standard lab
  2)
• Sign 0 for positive and 1 for negative
• The exponent has three digits 0,7, but can be
  positive or negative
• shift by 4 ? -4,3
• 000?-4, 001?-3, 010?-2, , 111?3
• i.e. subtract 4 from the corresponding decimal
  value
• Use base two now (i.e. 2 raised to the power of
  the exponent value)

6
Real Number Representation

• The mantissa has one (non-zero) place before the
  decimal point and a number of places after it
• in binary our digits are 0 and 1 and so we always
  have 1.xxxx
• No need to represent the one (add one to the
  result)
• 11100010 ? 1 110 0010
• Negative
• 1106, -4 ? 2, exponent22
• Mantissa 0010 01/2 01/4 11/8 01/16
  1/8
• without the initial 1 which weve omitted ? 11/8
• Or 1 decimal equivalent/16 1 2/16
• - (22)(11/8) -4 1/2

7
Real Number Representation

• Floating point to decimal conversion
• 1 break bit pattern into sign (1 bit), exponent
  (3 bits), mantissa (4 bits)
• 2 sign is if 1 and otherwise
• 3 exponent decimal equivalent -4
• 4 mantissa 1 decimal equivalent/24
• 5 number (sign) mantissa2exponent
• Try 00101000

8
Real Number Representation

• What about from decimal to floating point?
• Format the number into the form 1.xxxx2exponent
• Multiply (or divide) by 2 until we have a 1
  before the decimal point
• Subtract (or add) 1 from (to) the exponent for
  every such multiplication (or division)
• Add 4 to result (in floating-to-decimal
  conversion we subtracted 4)
• Convert exponent to binary
• Subtract 1 from the resulting mantissa (in
  floating-to-decimal conversion we added 1)
• Multiply mantissa by 16 (in floating-to-decimal
  conversion we multiplied by 16)
• Round to the nearest integer
• Convert mantissa to binary

9
Real Number Representation

• -8.48
• Sign is negative ? 1
• 8.48 ? 4.24 ? 2.12? 1.06 ? 3 divisions
• exponent34 7 or 1112
• mantissa mantissa -1 ? .06
• multiply mantissa by 16 (write it in terms of
  16ths) ? 0.96 1 0r 0001
• 1 111 0001

10
Real Number Representation

• Decimal to floating-point conversion
• Sign bit 1 if negative and 0 otherwise
• Exponent 0, mantissa absolute(number)
• While mantissa lt 1 do
• mantissa mantissa 2
• exponent exponent -1
• While mantissa gt 2 do
• mantissa mantissa / 2
• exponent exponent 1
• Exponent exponent 4
• Change to binary
• Mantissa (mantissa -1)16
• Round off to nearest integer and change to binary
• Assemble number sign exponent mantissa

11
Real Number Representation

• 0.319
• Positive ? sign bit 02
• mantissa 0.319 2 0.638 ? exponent -1
• mantissa 0.638 2 1.276 ? exponent -2
• exponent -2 4 2 0102
• mantissa (mantissa -1)16 4.416 4 01002
• 0 010 0100
• - 0.319
• Negative ? sign bit 12
• exponent 0102
• mantissa 01002
• 1 010 0100
• 0
• Sign bit 02
• mantissa? Not going to change to 1.xxxx no matter
  what!

12
Real Number Representation

• No representation for 0
• 0 000 0000 1.0 2-4 (we assumed there is a 1
  before the mantissa)
• Assume this to be zero!
• Another issue, is the method exact?
• rounding? truncation (close numbers give same
  floating point values)
• Mantissa 4.416 or 4.0123 or 4.400 are all 4
  01002
• Problem is also due to using only 8 bits (4-bit
  mantissa)
• Floating-point numbers require 32 or even 64 bits
  usually
• But still we will have to round off and truncate
• That happens regularly with us even when not
  using computers
• PI 22/7
• 2/3 or 1/3
• We approximate a lot but we should know it !
• Higher precision applications use much larger
  size registers