Factional ValuesWhat is 0.75 in binary?

How could we represent fractions?

• In decimal:– As fractions : 1/5

How could we represent fractions?

• In decimal:– As fractions : 1/5 – As decimals : 0.2

hundreds102

tens101

ones100

tenths10-1

hundredths

10-2

0 2 0

Column Pattern

• What goes to the right of 1’s column?23

822

421

220

1

Column Pattern

• Negative powers of two:

2-3 = = = 0.125

23

822

421

220

12-1

0.52-2

0.252-3

0.1252-4

0.0625

Number

• Binary decimal:

10.112 = 2 + 0.5 + 0.25 = 2.7510

23

822

421

220

12-1

0.52-2

0.252-3

0.1252-4

0.0625

1 0 1 1

Number

• Binary decimal:

0.10012 = 0.5 + 0.0625 = 0.562510

23

822

421

220

12-1

0.52-2

0.252-3

0.1252-4

0.0625

0 1 0 0 1

Number

• Binary decimal:

0.10102 = 0.5 + 0.125 = 0.62510

23

822

421

220

12-1

0.52-2

0.252-3

0.1252-4

0.0625

0 1 0 1 0

Missing Numbers

• Where is 0.6?

0.10012 = 0.5 + 0.0625 = 0.562510

0.10102 = 0.5 + 0.125 = 0.62510

• An attempt with 11 decimal digits:

• Can only approximate

More Digits

Binary Decimal Place 2-1 2-2 2-3 2-4 2-5 2-6 2-7 2-8 2-9 2-10 2-11

Bit 1 0 0 1 1 0 0 1 1 0 1

Value 0.5 0 0 0.0625 0.03125 0 0 0.003906 0.001953 0 0.000488

Total 0.600098

Decimals are an Approximation

• Decimal has same issue:– What is 1/3 ?? – What is 2/7 ??

More Digits

• Given limited digits, must allocate them– To left of decimal:

Bigger values

– To right of decimal:More accurate values

25

3224

1623

822

421

220

12-1

0.52-2

0.25

21

220

12-1

0.52-2

0.252-3

0.1252-4

0.06252-5

0.031252-6

0.015625

Fixed Decimal Problems

• Fixed decimal points waste space:

400,000,000,000,000,000 0.000000000000025

Fixed Decimal Problems

• Fixed decimal points waste space:

400,000,000,000,000,000 vs 4.0 x 1017

0.000000000000025 vs 2.5 x 10-14

• In computers, space is precious– Computers use a floating decimal point

(Like scientific notation)

Floating Point

• Bits used to represent 3 parts:– Sign – Exponent – Fraction (or Mantissa)

Sign Exponent Mantissa

0 1 0 0 1 0 0 0

Sign

• 0 = positive, 1 = negative

Sign Exponent Mantissa

0 1 0 0 1 0 0 0

Exponent

• Binary integer in excess notation– Gives power of 2 to multiply by

100 = 0So 20

Sign Exponent Mantissa

0 1 0 0 1 0 0 0

Binary Value000 -4001 -3010 -2011 -1100 0101 1110 2111 3

Mantissa

• Fractional Value– Always a decimal

1000 = 0.5

Sign Exponent Mantissa

0 1 0 0 1 0 0 0

2-1

0.52-2

0.252-3

0.1252-4

0.0625

1 0 0 0

Examples

+ 20 x 0.5 = + 1 x 0.5 = + 0.5

Sign Exponent Mantissa

0 1 0 0 1 0 0 0+ 0 so 20 0.5

2-1

0.52-2

0.252-3

0.1252-4

0.0625

1 0 0 0

Examples

- 23 x 0.5625 = - 8 x 0.5625 = -4.5

Sign Exponent Mantissa

1 1 1 1 1 0 0 1- 3 so 23 0.5625

2-1

0.52-2

0.252-3

0.1252-4

0.0625

1 0 0 1

Examples

+ 2-4 x 0.25 = + 0.0625 x 0.25 = +0.015625

Sign Exponent Mantissa

0 0 0 0 0 1 0 0+ -4 so 2-4 or 0.25

2-1

0.52-2

0.252-3

0.1252-4

0.0625

0 1 0 0

Floating Point Math is HARD

• XOR the signs• Calculate new exponent– Excess Notation – different math rules!

• Calculate new mantissa– Normal binary multiplication

Sign Exponent Mantissa

0 1 0 0 1 0 0 0

Choices, choices

• Choices– More bits to exponent– More bits to mantissa– How we represent each• What is our excess start point?• Do mantissas have to start with 1?

32 Bit Floating Point

• PC processors are 32 or 64 bit– Size of data they can work on at one time

• IEEE specifies conventions for floating points:

Remember…

• Decimals are approximate

• 32 bit float gives 6-7 significant decimal digits• 64 bit float gives 15-16 significant decimal

digits