1

Data..Representation & storage

Storing Real NumbersFixed-point Notation

Floating-Point (Scientific) Notation

2

Fractions in the Decimal System(Fixed-Point notation)

• We represent fractions using the idea of a radix point (called decimal point here), for example 3445.6701;

• The fraction part of the number is represented by digits to the right of the radix (in the example these are .6701);

• Positions to the right of the radix point carry weights, these weights are negative powers of 10

in the example above

.6701 = 6 x 10-1 + 7 x 10-2 + 0 x 10-3 + 1 x 10-4;• Lets copy the same ideas into the binary system

How…?

3

Fractions in Binary System(Fixed-Point notation)

• We represent fractions using the idea of a radix point for (can be called binary point here) example 101.1011;

• The fraction part of the number is represented by binary digits to the right of the radix (in the example these are .1011);

• Positions to the right of the radix point carry weights, these weights are negative powers of 2

in the example above .1011 = 1 x 2-1 + 0 x 2-2 + 1 x 2-3 + 1 x 2-4

= 1 x ½ + 0 x ¼ + 1 x 1/8 + 1 x 1/16

In decimal system this is = 11/16 = (0.6875)10;

• The above example illustrates how to convert a binary fraction to an equivalent decimal fraction;

• How do we convert a decimal fraction to a binary fraction ?

4

Fractions in Binary Systemconverting Decimal to Binary

• Lets consider a simple example (0.5)10 = (?)2

• We all know this will be (.1)2

• Lets arrive at it in a different way:– Multiply 0.5 by 2 = 1.0– Ignore the fraction of the result (here it is 0) and take the digit

to the left of the decimal point as the digit in position 2-1 in the binary representation;

• Another simple example (0.25)10 = (?)2

– 0.25 x 2 = 0.50– .5 x 2 = 1.0 (since fraction in the result = 0 stop)– The binary equivalent of (0.25)10 = (.01)2

• Another example (0.3)10 = (?)2 …

5

Fractions in Binary Systemconverting Decimal to Binary cont’d

• Another example (0.3)10 = (?)2 …. (0.01001)2

– 0.3 x 2 = 0.6– 0.6 x 2 = 1.2– 0.2 x 2 = 0.4– 0.4 x 2 = 0.8– 0.8 x 2 = 1.6– 0.6 x 2 … oops we have a cycle (0.6 appeared

earlier), what does that mean ?

• (0.3)10 can not be represented by a finite number of bits in the binary system!

6

Floating-Point NotationHow can we represent very small fractions…

• Consider that we have a total of 8 bit to store a number in the binary system; we use 2 bits to store the fraction part, and 6 bits to store the whole part.

Ex 101101.10 is a number that can be represented

• Question: can we represent 3.125 in that system ?– For the whole part (3) the representation in 6 bits is simple; 000011– For the fraction part (0.125), lets convert to a binary fraction:.125 x 2 = 0.250.250 x 2 = 0.5

.5 x 2 = 1.0 the fraction should be (.001)2

• Because we use 2 bits for the fraction part the correct result (000011.001) will be represented as

(000011.00); an error occurs• The fraction .001 is too small to be represented in just 2 bits, the

solution?

7

Floating-point notationPosition of the Radix point

• The problem with the previous example is that the first 1 in the fraction part occurs after a number of leading zeros;

• In the decimal system Floating-point notation is used to efficiently represent such fractions;

• Examples:– 0.00003 is represented as .3 x 10-4 (also written .3 E–4)– 0.0000 0000 625 is represented by .625 E–8;

• The Floating-point notation is a way to record the “true” place of the decimal point;

• For example, .3E-4 means the decimal point is really 4 places to the left of the digit 3 , i.e. .00003;

8

Components of a number represented in Floating-point Notation in decimal system

• Consider the number 0.625E-4, the components of this number are:– A sign (here the number is positive the sign is + but is

omitted);

– A Mantissa field; this represents the value of the fraction part (in this case it is .625);

– An Exponent; this is how we record the true position of the decimal point (here the exponent is –4, and the true position of the decimal point is thus 4 positions to the left of the most significant digit in the mantissa, i.e. .0000626;

• Important: in Floating-point notation the mantissa is made to be always a fraction, for example– 133.625 is represented as .133625E+3.

9

Floating-Point Notation in the binary system

• A number is represented using the same components we discussed , a sign , a mantissa, and an exponent;

• Numbers are represented using a fixed number of bits (e.g. 8, 16, 24, etc..);

• Bits are used as follows:– The sign is always represented by the most significant bit (0

means positive, 1 means negative);

– A number of bits to the right of the sign bit are used to represent the exponent in excess notation;

– The remaining bits represent the mantissa as a binary fraction;