CS 312, Spring 2009
:
Some Notes on Floating Point Arithmetic.
[email protected]
1
Introduction
Blah, blah, say some words to introduce the topic, maybe provide an example, whatever. On second
thought, let’s just dive headfirst into things and see where the current takes us.
2
Normalized Real Numbers in Decimal
Before we hit the meat of the topic too terribly hard, let’s briefly review scientific notation, signifi
cant places, and the process for normalizing real numbers in the base that we’re familiar with: base
10. Additionally, we say real number to mean any number
x
in
R
, the set of real numbers.
Scientific notation is a way of representing a real number that would otherwise be cumbersome
to be written conventionally. We’ve all seen this several thousand times already, so I’ll save a lot
of the detail except for the form of the representation and what it means for a real number to be
normalized.
That said, a normalized real number
x
in scientific notation has the form
x
=
a
·
10
b
,
(1)
where 1
≤ 
a

<
10. We call the coefficient
a
one of either the
significand
,
mantissa
, or
fraction
and
b
the
characteristic
or
exponent
. In 312, we used the terms
fraction
and
exponent
.
The process of normalization itself is braindead simple to the point that we can probably do it
in our sleep. Informally, we just shift the decimal place as appropriate to ensure 1
≤ 
a

<
10 and
keep count of how many places we shift it. The count of places is, hence,
b
.
3
Normalized Real Numbers in Some Base
β
Now that we’ve hit the material with which we’re all familiar, we can finally abstract the select few
parts so that we can glean more insight as to how this mess works as a whole.
Really, the changes to the definition we discussed earlier are minimal. A normalized real number
x
in a base
β
in floatingpoint notation has the form
x
β
=
a
·
β
b
,
(2)
where 1
≤ 
a

< β
. The same vocabulary applies, and so on. The process of normalization is even
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 Spring '08
 Tornaritis,S
 emin, FLOATING POINT ARITHMETIC, Normalized Real Numbers

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