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In this module, we shall study a family of functions which return integers based on certain rule, corresponding to a real number. Greatest integer function (floor), least integer function (ceiling) and nearest integer function form part of this family.

Greatest integer function (floor function)

Greatest integer function returns the greatest integer less than or equal to a real number. In other words, we can say that greatest integer function rounds “down” any number to the nearest integer. This function is also known by the names of “floor” or “step” function. The greatest integer function (GIF) is denoted by the symbol “[x]” .

Interpretation of Greatest integer function is straight forward for positive number. Consider the values “0.23” and “1.7”. The greatest integers for two numbers are “0” and “1”. Now, consider a negative number “-0.54” and “-2.34”. The greatest integers less than these negative numbers are “-1” and “-3” respectively.

We can observe here that greater integer function is actually a function that returns the integral part of a positive real number. This interpretation is clear for positive number. Interpretation for negative numbers needs some explanation. We interpret these values in the context of the fact that every real number can be decomposed to have two parts (i) integral and (ii) fractional part. From this point of view, the negative number can be thought as :

-0.54 (real number) = -1 (integral part) + 0.36 (fraction part)

-2.34 (real number) = -3 (integral part) + 0.66 (fraction part)

We may be tempted to disagree (why not -2 + -0.34 = -2.34?). But, we should know that this is how greatest integer function (GIF) treats a negative number. It returns "-3" for "-2.34" - not "-2". Subsequently, we shall define a function called fraction part function (FPF) that returns fraction part of real number. We shall find that the function exactly returns the same fraction for negative number as has been worked out. The fraction part function (FPF) returns a fraction, which is always positive. It is denoted as {x}. Because of these aspects of GIF and FPF, we can understand the reason why negative number is treated the way it has been presented above. In terms of integral and fraction parts, we write a real number "x" as :

x = [ x ] + { x }

In the nutshell, we can use any of the following interpretations of greatest integer function :

  • [x] = Greatest integer less than equal to “x”
  • [x] = Greatest integer not greater than “x”
  • [x] = Integral part of “x”

The value of "[x]" is an integer (n) such that :

f x = [ x ] = n ; if n x < n + 1 n Z

Working rules for evaluating greatest integer function are two step process :

  • If “x” is an integer, then [x] = x.
  • If “x” is not an integer, then [x] evaluates to greatest integer less than “x”.

Graph of greatest integer function

Few initial function values are :

F o r - 2 x < - 1, f x = [ x ] = - 2

F o r - 1 x < 0, f x = [ x ] = - 1

F o r 0 x < 1, f x = [ x ] = 0

F o r 1 x < 2, f x = [ x ] = 1

F o r 2 x < 3, f x = [ x ] = 2

The graph of the function is shown here :

Greatest integer function

The domain of the function is R.

Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
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Crow Reply
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RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
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Mahi
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
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if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
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what is Nano technology ?
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write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
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what king of growth are you checking .?
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What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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biomolecules are e building blocks of every organics and inorganic materials.
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research.net
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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Period of sin^6 3x+ cos^6 3x
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Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
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