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 April 28th, 2011, 06:45 AM #1 Newbie   Joined: Apr 2011 Posts: 19 Thanks: 0 Floating Point Notation... Hi All, I think I may be coming to terms with Floating Point Notation [Finally...] Example: [Using 1 Bit for the Sign, 3 Bits for the Exponent and 4 Bits for the Mantissa - This is how we were shown in class] Binary Number: 01001010 So, first we extract the Mantissa to obtain .1010 Then the Exponent to obtain 100 [Using 3 Bit Excess Notation we come to the conclusion that 100 is actually 0, so no need to move the Radix Point] Then the Sign Bit which is a 0, so the number will be non negative. So, to the Mantissa, which is actually 5/8? [To gain 5/8, the first number after the Radix is a 1, which is 1/2, then a 0, then a 1 [which is 1/8] and the last which is a 0] The answer is 5/8... Another example: Binary Number: 01101101 Mantissa = 1101 Exponent = 110 [Using 3 Bit Excess Notation we get +2] Sign = 0 [Non Negative] The Exponent then moves the Radix Point two places to the right of the Mantissa to get 11.01 The result = 3 1/4?? Regards, Daz.
April 28th, 2011, 06:57 AM   #2
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Re: Floating Point Notation...

Quote:
 Originally Posted by Primalscientist Example: [Using 1 Bit for the Sign, 3 Bits for the Exponent and 4 Bits for the Mantissa - This is how we were shown in class] Binary Number: 01001010 So, first we extract the Mantissa to obtain .1010 Then the Exponent to obtain 100 [Using 3 Bit Excess Notation we come to the conclusion that 100 is actually 0, so no need to move the Radix Point] Then the Sign Bit which is a 0, so the number will be non negative. So, to the Mantissa, which is actually 5/8? [To gain 5/8, the first number after the Radix is a 1, which is 1/2, then a 0, then a 1 [which is 1/8] and the last which is a 0]
Looks fine to me. I assume from what you write that the bias is 4 ("3 Bit Excess Notation"?) and that your notation does not use an implied 1.

I should also note that "mantissa" is out of favor; you should not use it except while in this class (obviously whatever the instructor says is right in the classroom). The proper/modern term is "significand".

Quote:
 Originally Posted by Primalscientist Binary Number: 01101101 Mantissa = 1101 Exponent = 110 [Using 3 Bit Excess Notation we get +2] Sign = 0 [Non Negative] The Exponent then moves the Radix Point two places to the right of the Mantissa to get 11.01 The result = 3 1/4?
Right. 11 = 3, .01 = 1/4.

 April 28th, 2011, 07:04 AM #3 Newbie   Joined: Apr 2011 Posts: 19 Thanks: 0 Re: Floating Point Notation... You dont know how pleased I am that I FINALLY got my head round this!! YEAH!!!! Yes, I can confirm that we are using 4 [3 Bit Excess Notation]. What is the implied 1? Is this when converting from base 10 to base 2? Thank you for clarifying for me!! Daz. I will change from Mantissa to Significand from now on then. Thanks again.
April 28th, 2011, 11:11 AM   #4
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Re: Floating Point Notation...

Quote:
 Originally Posted by Primalscientist You dont know how pleased I am that I FINALLY got my head round this!! YEAH!!!!
Good for you. It's remarkably difficult to understand considering how easy it is once you do.

Quote:
 Originally Posted by Primalscientist Yes, I can confirm that we are using 4 [3 Bit Excess Notation].
I don't know the term "X Bit Excess Notation" but your very clear explanation made me assume that it's the same as what I call "bias 4". I suspect in general that "X Bit Excess" means "bias 2^(X-1)".

Quote:
 Originally Posted by Primalscientist What is the implied 1?
It's a way to cheat more precision out of the notation. (Alternate name: hidden 1.)

You'll notice that, other than 0, pretty much all numbers start with 1. If the number is 0.01 you would represent it as 1.00 * 2^-2, etc. The basic idea of the implied 1 is that you write all your numbers as "1.---" where --- is the significand. So the 4-bit significand of 1.01 would be 1010 without the implied 1 or 0100 with the implied 1 -- giving us an effective precision of 5 bits even though we only store 4. This does mean that there's no way to directly represent 0, but special techniques are used that allow 0 and nearby numbers to be represented differently -- "subnormal" numbers. (Similarly, floating point often encodes infinities and NaNs which are special in the same way.) If you ignore those the concept of an implied 1 is easy.

 May 2nd, 2011, 07:52 AM #5 Newbie   Joined: Apr 2011 Posts: 19 Thanks: 0 Re: Floating Point Notation... Hi CRGreathouse, Thank you for the explanation there. One step at a time for me !!! Thanks again for helping me to understand it!! Although you mention about using 5 bits, is this where truncation error comes in? Or am I barking up the wrong tree so to speak? Best, Daz.
 May 2nd, 2011, 08:55 AM #6 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Floating Point Notation... You'll always have truncation error, but using the hidden/implied bit means that it's only half as large (on average).
 June 1st, 2011, 01:40 AM #7 Newbie   Joined: Apr 2011 Posts: 19 Thanks: 0 Re: Floating Point Notation... I have my exam today... I feel sick, I really do. anyway, there will be questions with regards to what we have discussed, and i would like to thank you for your assistance. with out your help, I dont know where I would be... Best, Darren.
 June 1st, 2011, 05:01 AM #8 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Floating Point Notation... Glad to have helped.
 June 1st, 2011, 11:23 PM #9 Newbie   Joined: Apr 2011 Posts: 19 Thanks: 0 Re: Floating Point Notation... I think I blew it... There was loads of questions, and not one on floating point representation which sort of surprised me...
 June 2nd, 2011, 06:29 AM #10 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Floating Point Notation... So what did it cover?

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