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Exponential form

When numbers are very large or very small they are written in Scientific Notation<pre>
     n
a x 10       1<a<10
<
Write the following in Scientific Notation:
58 000 000				.035
48600					.00478
1 430 000 000				.000 000 000 893
228 000 000				.00064

Rewrite the above putting the decimal point directly in front of the first non zero digit:
58 000 000				.035
48600					.00478
1 430 000 000				.000 000 000 893
228 000 000				.00064</pre>

This is called exponential form: The number is called the MANTISSA.

The power is called the EXPONENT.

Binary Exponential Form:

In binary exponential form we use powers of 2 instead of powers of 10.

Examples of binary numbers written in binary exponential form with exactly 5 bits.
<pre>
Binary number	Exponential form	Mantissa	Exponent/power

1010.1
0.001111
-111
0.1
-0.01010101	0.10101 x 24
0.11110 x 2-2
-0.11100 x 23
0.100000 x 20
-0.10101 x 2-1	0.10101
0.11110
-0.11100
0.10000
0.10101	4
-2
3
0
-1

</pre>
Binary numbers as above are stored in the computer in their binary exponential form.
Each number has 3 pieces of information to be stored –

•	the sign of the number

•	the mantissa

•	the power or exponent

To store this number the computer has 3 blocks of storage, eg, with a 32 bit memory we could store a binary exponential as follows
<pre>
1 bit for sign
0= +
1= -	7 bits for the exponent	24 bits to store the mantissa</pre>

This is called FLOATING POINT REPRESENTATION.
The sign bit:
This is 0 for a positive number and 1 for a negative number

The mantissa:
Convert a decimal number to binary by dividing the part before the decimal point repeatedly by 2 and writing the remainders from the bottom up.  Multiply the decimal part repeatedly by 2 and write the carrys from the top down.

The exponent or power:
This can be stored in two ways:-
Firstly it can be stored as a positive number if it is positive and store as its 2’s complement if it is negative.

Or

Store as its characteristic.
The characteristic is calculated by  n + 2t-1 , t is the number of bits for storing the exponent.
Most computers use the characteristic for storing powers/exponents.

Eg to store the number 
-110111100.011101

A; Convert to exponential form

= - 0.110111100011101 x 29

B:-

The sign bit is 1 as the number is negative

C:-

The mantissa is

110111100011101

As we want the mantissa to be 24 bits we add zeros after to make up the correct number of bits

D:-

The characteristic is:  n +  2t-1

We have 7 bits for the exponent so t = 7

n + 2t-1 =

9 + 27-1 =

9 + 26 =

9 + 64 = 73 = 1001001 in binary

So, the floating point representation of my number is:-
<pre>
1	1001001	110111100011101000000000
Sign	characteristic		mantissa
</pre>