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Given that we cannot perfectly represent real numbers on digital computers, we must come up with a compromise that allows us to approximate real numbers. Interestingly, analog computers have an easier time representing real numbers. Imagine a “water- adding” analog computer which consists of two glasses of water and an empty glass. The amount of water in the two glasses are perfectly represented real numbers. By pouring the two glasses into a third, we are adding the two real numbers perfectly (unless we spill some), and we wind up with a real number amount of water in the third glass. The problem with analog computers is knowing just how much water is in the glasses when we are all done. It is also problematic to perform 600 million additions per second using this technique without getting pretty wet. Try to resist the temptation to start an argument over whether quantum mechanics would cause the real numbers to be rational numbers. And don’t point out the fact that even digital computers are really analog computers at their core. I am trying to keep the focus on floating-point values, and you keep drifting away! There are a number of different ways that have been used to represent real numbers. The challenge in selecting a representation is the trade-off between space and accuracy and the tradeoff between speed and accuracy. In the field of high performance computing we generally expect our processors to produce a floating- point result every 600-MHz clock cycle. It is pretty clear that in most applications we aren’t willing to drop this by a factor of 100 just for a little more accuracy. Before we discuss the format used by most high performance computers, we discuss some alternative (albeit slower) techniques for representing real numbers.

Binary coded decimal

In the earliest computers, one technique was to use binary coded decimal (BCD). In BCD, each base-10 digit was stored in four bits. Numbers could be arbitrarily long with as much precision as there was memory:

123.45 0001 0010 0011 0100 0101

This format allows the programmer to choose the precision required for each variable. Unfortunately, it is difficult to build extremely high-speed hardware to perform arithmetic operations on these numbers. Because each number may be far longer than 32 or 64 bits, they did not fit nicely in a register. Much of the floating- point operations for BCD were done using loops in microcode. Even with the flexibility of accuracy on BCD representation, there was still a need to round real numbers to fit into a limited amount of space.

Another limitation of the BCD approach is that we store a value from 0–9 in a four-bit field. This field is capable of storing values from 0–15 so some of the space is wasted.

Rational numbers

One intriguing method of storing real numbers is to store them as rational numbers. To briefly review mathematics, rational numbers are the subset of real numbers that can be expressed as a ratio of integer numbers. For example, 22/7 and 1/2 are rational numbers. Some rational numbers, such as 1/2 and 1/10, have perfect representation as base-10 decimals, and others, such as 1/3 and 22/7, can only be expressed as infinite-length base-10 decimals. When using rational numbers, each real number is stored as two integer numbers representing the numerator and denominator. The basic fractional arithmetic operations are used for addition, subtraction, multiplication, and division, as shown in [link] .

Questions & Answers

what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Cómputo de alto rendimiento. OpenStax CNX. Sep 02, 2011 Download for free at http://cnx.org/content/col11356/1.2
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