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Rational number mathematics

This figure shows three rows of equations. The first is one third times thirty sevenths equals thirty twenty-firsts, which equals ten sevenths. The second is one sixth plus one fifth equals five thirtieths plus six thirtieths, which equals eleven thirtieths. The third is the quantity fourteen thousand one hundred seventy three divided by twenty-one thousand two-hundred twenty four times the quantity seventy seven thousand two hundred thirty four divided by two thousand one hundred twenty one, which equals the quantity one billion, ninety-four million, six hundred thirty seven thousand, four hundred eighty two divided by forty-five million, sixteen thousand, one hundred four, which equals the quantity five hundred forty seven million, three hundred eighteen thousand, seven hundred forty one divided by twenty-two million, five hundred eight thousand fifty two.

The limitation that occurs when using rational numbers to represent real numbers is that the size of the numerators and denominators tends to grow. For each addition, a common denominator must be found. To keep the numbers from becoming extremely large, during each operation, it is important to find the greatest common divisor (GCD) to reduce fractions to their most compact representation. When the values grow and there are no common divisors, either the large integer values must be stored using dynamic memory or some form of approximation must be used, thus losing the primary advantage of rational numbers.

For mathematical packages such as Maple or Mathematica that need to produce exact results on smaller data sets, the use of rational numbers to represent real numbers is at times a useful technique. The performance and storage cost is less significant than the need to produce exact results in some instances.

Fixed point

If the desired number of decimal places is known in advance, it’s possible to use fixed-point representation. Using this technique, each real number is stored as a scaled integer. This solves the problem that base-10 fractions such as 0.1 or 0.01 cannot be perfectly represented as a base-2 fraction. If you multiply 110.77 by 100 and store it as a scaled integer 11077, you can perfectly represent the base-10 fractional part (0.77). This approach can be used for values such as money, where the number of digits past the decimal point is small and known.

However, just because all numbers can be accurately represented it doesn’t mean there are not errors with this format. When multiplying a fixed-point number by a fraction, you get digits that can’t be represented in a fixed-point format, so some form of rounding must be used. For example, if you have $125.87 in the bank at 4% interest, your interest amount would be $5.0348. However, because your bank balance only has two digits of accuracy, they only give you $5.03, resulting in a balance of $130.90. Of course you probably have heard many stories of programmers getting rich depositing many of the remaining 0.0048 amounts into their own account. My guess is that banks have probably figured that one out by now, and the bank keeps the money for itself. But it does make one wonder if they round or truncate in this type of calculation. Perhaps banks round this instead of truncating, knowing that they will always make it up in teller machine fees.


The floating-point format that is most prevalent in high performance computing is a variation on scientific notation. In scientific notation the real number is represented using a mantissa, base, and exponent: 6.02 × 10 23 .

The mantissa typically has some fixed number of places of accuracy. The mantissa can be represented in base 2, base 16, or BCD. There is generally a limited range of exponents, and the exponent can be expressed as a power of 2, 10, or 16.

The primary advantage of this representation is that it provides a wide overall range of values while using a fixed-length storage representation. The primary limitation of this format is that the difference between two successive values is not uniform. For example, assume that you can represent three base-10 digits, and your exponent can range from –10 to 10. For numbers close to zero, the “distance” between successive numbers is very small. For the number 1.72 × 10 -10 , the next larger number is 1.73 × 10 -10 . The distance between these two “close” small numbers is 0.000000000001. For the number 6.33 × 10 10 , the next larger number is 6.34 × 10 10 . The distance between these “close” large numbers is 100 million.

In [link] , we use two base-2 digits with an exponent ranging from –1 to 1.

Distance between successive floating-point numbers

This figure is a horizontal line with labeled hash-marks at various distances From left to right, the hash marks read 0.0 times 2^-1, 0.1 times 2^-1, 1.0 times 2^-1, 1.1 times 2^-1, 1.0 times 2^0, 1.1 times 2^0, 1.0 times 2^1, and 1.1 times 2^1.

There are multiple equivalent representations of a number when using scientific notation:

6.00 × 10 5
0.60 × 10 6
0.06 × 10 7

By convention, we shift the mantissa (adjust the exponent) until there is exactly one nonzero digit to the left of the decimal point. When a number is expressed this way, it is said to be “normalized.” In the above list, only 6.00 × 10 5 is normalized. [link] shows how some of the floating-point numbers from [link] are not normalized.

While the mantissa/exponent has been the dominant floating-point approach for high performance computing, there were a wide variety of specific formats in use by computer vendors. Historically, each computer vendor had their own particular format for floating-point numbers. Because of this, a program executed on several different brands of computer would generally produce different answers. This invariably led to heated discussions about which system provided the right answer and which system(s) were generating meaningless results. Interestingly, there was an easy answer to the question for many programmers. Generally they trusted the results from the computer they used to debug the code and dismissed the results from other computers as garbage.

Normalized floating-point numbers

This figure is a horizontal line with labeled hash-marks at various distances From left to right, the hash marks read 0.0 times 2^-1, 0.1 times 2^-1, 1.0 times 2^-1, 1.1 times 2^-1, 1.0 times 2^0, 1.1 times 2^0, 1.0 times 2^1, and 1.1 times 2^1. Pointing at the first two hash marks with two arrows is the label, not normalized.

When storing floating-point numbers in digital computers, typically the mantissa is normalized, and then the mantissa and exponent are converted to base-2 and packed into a 32- or 64-bit word. If more bits were allocated to the exponent, the overall range of the format would be increased, and the number of digits of accuracy would be decreased. Also the base of the exponent could be base-2 or base-16. Using 16 as the base for the exponent increases the overall range of exponents, but because normalization must occur on four-bit boundaries, the available digits of accuracy are reduced on the average. Later we will see how the IEEE 754 standard for floating-point format represents numbers.

Questions & Answers

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
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
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Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, High performance computing. OpenStax CNX. Aug 25, 2010 Download for free at http://cnx.org/content/col11136/1.5
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