# floating point question..

Discussion in 'Homework Help' started by transgalactic, May 20, 2008.

1. ### transgalactic Thread Starter Active Member

Apr 29, 2008
74
0
a number of 36 bits with the representation of a floating point has 8 bits plus the sign bit for the exponent and 26 bits plus a sign bit for "mantissa"(what is that??)
the mantissa is normalized(whats normalized mantissa??)
the exponent and the mantissa numbers are represented by a sign bit
and the absolute value.

what is the biggest and the smallest number we can represent in this way excluding zero??

i am new to this consept
????

2. ### Papabravo Expert

Feb 24, 2006
10,157
1,796
Any number can be expressed as a fraction(aka mantissa) f, where 0 <= f < 1 , times the base of the number system raised to some power(exponent).

http://en.wikipedia.org/wiki/Mantissa

3. ### transgalactic Thread Starter Active Member

Apr 29, 2008
74
0
i understand that there is the integer part the mantissa part which is the fracture or all the letters before the point.

plus there is the exponent (10^x)which can move the point in our number.

i have trouble to image this number and how looks this exponent
how looks this whole thing
so i could see what its maximal range
???

4. ### Papabravo Expert

Feb 24, 2006
10,157
1,796
It seems that you read it, but I don't think you understood much.

Reread what I wrote. There is a mantissa which is a number greater than or eqaul to zero and strictly less than one. There is no integer part.

The same rules apply for base-10 and base-2

Example: 3-bit mantissa and three bit exponent
Code ( (Unknown Language)):
1.
2. Manitssa    Value        Detail
3. (Binary)    (Decimal)
4. 000         .000
5. 001         .125         0*(.5) + 0*(.25) + 1*(.125)
6. 010         .250         0*(.5) + 1*(.25) + 0*(.125)
7. 011         .375         0*(.5) + 1*(.25) + 1*(.125)
8. 100         .500         1*(.5) + 0*(.25) + 0*(.125)
9. 101         .625         1*(.5) + 0*(.25) + 1*(.125)
10. 110         .750         1*(.5) + 1*(.25) + 0*(.125)
11. 111         .875         1*(.5) + 1*(.25) + 1*(.125)
12.
13. NOTE:
14. 2^-1 = 0.500
15. 2^-2 = 0.250
16. 2^-3 = 0.125
17.
Now that we have the mantissa laid out you can see how the maximum value is determined. The exponent will be treated like a signed twos complement number so a three bit quantity will be in the range -4 to +3.

Some examples

111 exp 011 is (0.875) * (2 ^ 3) = 7 largest positive number
001 exp 100 is (0.125) * (2 ^ -4) = .0078125 smallest non-zero number
000 exp 000 is (0.000) * (2 ^ 0) = 0

Does that clarify things

5. ### Mark44 Well-Known Member

Nov 26, 2007
626
1
Floating point numbers as used in computers generally adhere to the IEEE 754 standard. As you mentioned in your first post, the three parts are the sign (1 bit), the mantissa (more frequently called the significand these days), and the exponent.

6.023 x 10$^{-23}$

Here the sign is positive, the significand is 6.023, and the exponent is -23.

Floating point numbers as they are used in computers are similar but not identical. Two significant differences are 1) they are stored using binary digits, not decimal digits, and 2) the radix is 2, not 10.

So a number such as 2.375 (base 10) would look like this as a binary number, including its part to the right of the "binary" point:
10.011 (base 2).

The part to the left of the "binary" point is 2 (base 10).
The part to the right of the "binary" point is 0*.5 + 1*.25 + 1*.125 (base 10).

Our binary number, 10.011 (base 2) is not normalized, since there is more than one digit to the left of the binary point. We can normalize it by moving the binary point and adjusting the exponent, like so:
1.0011 * 2$^{1}$.

So, in the context of your question, the sign is positive (usually represented with a 0 bit, with a 1 bit used for negative numbers), the normalized significand (aka mantissa) is 1.0011, and the exponent is 1.

You need to figure out how large a number and and how small a number you can represent with 8 bits for the exponent, 1 bit for the sign, and 27 bits for the significand.

Mark