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Greatest Integer Function and its Applications
by Total Gadha - Monday, 26 February 2007, 03:27 AM
 

The greatest integer function, denoted by [x], gives the greatest integer less than or equal to the given number x.

To put it simply, if the given number is an integer, then the greatest integer gives the number itself, otherwise it gives the first integer towards the left of the number of x on the number line.

For example,

[1.4] = 1

[4]= 4

[3.4] = 3

[ - 2.3] = - 3

[ - 5.6] = - 6, and so on.

NOTE: We can see that [1.4] = 1 + 0.4 or x = [x] + {x}, where {x} is the fractional part of x. For x = - 2.3, [x] = - 3 and {x} = 0.7

In the following figure, we can see that the greatest integer function gives the number itself (when the given number is an integer) or the first integer to the left of the number on the number line.

                          integer function

The graph of greatest integer function is given below. Note that the red dot indicates that integer value on the number line is not included while the green dot indicates that the integer value is included.

                          greatest integer

greatest integer

Re: Greatest Integer Function and its Applications
by Acer Josh - Saturday, 2 June 2007, 01:34 AM
  Might not be my place to correct but I think the result of the first problem is 212 and not 217. 
Re: Greatest Integer Function and its Applications
by Tempo # 911 - Saturday, 2 June 2007, 03:08 AM
 

Avinash...

217 is correct.

Re: Greatest Integer Function and its Applications
by MANISH TALWAR - Saturday, 2 June 2007, 11:35 PM
  yes, 217 is right,but i m interested in knowing how r u getting 212?
Re: Greatest Integer Function and its Applications
by Acer Josh - Monday, 4 June 2007, 12:23 AM
  Sorry my mistake , a case of incorrect addition smile

217 is correct
Re: Greatest Integer Function and its Applications
by ashish tyagi - Monday, 6 August 2007, 07:38 AM
  nice work sir , actually i joined total gadha recently ,n now im reading all of ur quant articles ,can u tell me sth more abt t g , do u have some rticles for lr nd di , if yes then where i can get .....pls reply ...........
Re: Greatest Integer Function and its Applications
by we have worked together - Saturday, 15 September 2007, 05:57 PM
  well i  must say that this site is quite useful....total gadha just to tell u, we have wrked together for a few months and u do know me...good wrk....
Re: Greatest Integer Function and its Applications
by surya pratap - Thursday, 27 September 2007, 08:28 PM
  i a really affected by this site... i am loving itsmile
Re: Greatest Integer Function and its Applications
by nirmesh sinha - Monday, 1 October 2007, 09:36 AM
 

Hi TG

plz. take some time fro ur busy schedule and come out a paneca for Modulus functions.

like plotting of  ||x|-6|+|x|+ ||X-2|+4| function and finding area ,absolute value.

Re: Greatest Integer Function and its Applications
by jitendra havaldar - Monday, 1 October 2007, 10:47 AM
 

Hi TG,

Have 1 Q. on this greatest integer function :-

if [a] denotes the sum of m real positive nos. and [b] denotes the sum of n positive real numbers as given below(where [x] denotes the greatest integer function less than equal to x ) :

[a] = [s1] +[s2] + [s3] + [s4] + ..... [sm] + 4 ;

[b] = [t1] + [t2] + [t3] + [t4] + ..... [tn] + 3 ;

then what can be the min. possible value of [a] + [b] ?

Options: 4/3/7/9/cannot be determined

Thanks

Jitendra

 

Re: Greatest Integer Function and its Applications
by Natasha Sachdeva - Friday, 19 October 2007, 02:19 AM
 

hi Jitendra,

ans. 7

+ve real no. means > 0

hence its []=0 to be min (take .1 eg)

4+0=4

3+0=3

4+3=7

 

Re: Greatest Integer Function and its Applications
by jitendra havaldar - Monday, 22 October 2007, 10:56 AM
 

Hi Natasha,

Thanks for your "late-night-early morning "  wink soln.

Jitendra

Re: Greatest Integer Function and its Applications
by Anubhav Jain - Sunday, 28 October 2007, 12:22 AM
  sirji ur gr888

i mean every thread is sooo usefulll
Re: Greatest Integer Function and its Applications
by dheeraj sunkavalli - Thursday, 1 November 2007, 12:51 PM
  WoW!!! Im simply enjoyin this TG stuff.......It surely helps to increase the confidence levels when CAT is just round the corner!!
Re: Greatest Integer Function and its Applications
by anuj hembrom - Saturday, 31 May 2008, 06:16 PM
  hi TG great work... kindly write an article about function & it's application 
Re: Greatest Integer Function and its Applications
by ATOM ANT - Sunday, 1 June 2008, 07:20 AM
 

NOTE: We can see that [1.4] = 1 + 0.4 or x = [x] + {x}, where {x} is the fractional part of x. For x = - 2.3, [x] = - 3 and {x} = 0.7

can the fractional part of a number be negative...?

 

Re: Greatest Integer Function and its Applications
by Aditya Zutshi - Monday, 2 June 2008, 11:28 AM
 

Hi Atom,

Fractional part can never be negative. It can lie only between 0 and 1 (excluding 0 and 1).

Re: Greatest Integer Function and its Applications
by Aditya Zutshi - Monday, 2 June 2008, 11:50 AM
 

Hey i knw its a dumb ques... but is thr a direct way to find the sum of this...

3x1 + 5x2 +.....+ 13x6.... i.e a series of product of 2 or more AP's... Thanks smile

Re: Greatest Integer Function and its Applications
by ATOM ANT - Monday, 2 June 2008, 01:18 PM
  Thanks.smile
Re: Greatest Integer Function and its Applications
by ishan ishu - Monday, 18 August 2008, 01:40 PM
 

Hi All,

Can anyone explain me this question:

If [x]^2 <= 16

what should be the value of x?

Please reply, I am not able to get the explanation for this question.

Thanks in advance

 

Re: Greatest Integer Function and its Applications
by ankit mahajan - Tuesday, 19 August 2008, 11:20 AM
 

[x]^2<=16

=> ([x]-4)([x]+4)<=0

=>[x] belongs to {-4,4}

=> x belongs to {-4,4.99}

please somebody correct me if i am wrong.

Re: Greatest Integer Function and its Applications
by Total Gadha - Tuesday, 19 August 2008, 02:33 PM
  Hi Ishan and Ankit

–4 ≤ x < 5

Total Gadha
Re: Greatest Integer Function and its Applications
by ankit mahajan - Tuesday, 19 August 2008, 05:13 PM
 

thanks for the post TG..

that was precise..

I am posted at singapore these days.. and I was looking for some material online which could resolve all my quant related issues as i havent brought any material over here..

and I came across this site..

I think this is probably the best site i have come across in terms of quality of content especially for Quant..and all the topics explained the way they ought to be..

I am really happy to be part of TG family.. Hope to contribute some useful posts too..

 

 

Re: Greatest Integer Function and its Applications
by ishan ishu - Wednesday, 20 August 2008, 01:58 PM
 

Thanks TG

Re: Greatest Integer Function and its Applications
by Amit Chakraborty - Thursday, 21 August 2008, 10:52 AM
 

Hi Aditya,

surely there is a direct way to find out the sum of the series like:-

3X1 + 5X2 + 7X3+......+ 13X6 ....

Here we go:-

if you notice the series well you can understand the nth term of the series will be (2n+1) X n

Now expanding we get :- (2n+1)xn = 2n2 + n

so rewriting the series we can get:-

   3X1 + 5X2 + 7X3 + .... + 13X6...+ (2n+1)Xn 

= (2x12+1) + (2x22+2) +(2x32+3) +........+(2x62+6) +....+(2xn2+n)

= 2(12+22+32+....+62+....+n2 ) + ( 1+2+3+....+6+....n)

= 2{n(n+1)(2n+1)/6} + {n(n+1)/2}

= n(n+1)(4n+5)/6

smile

Re: Greatest Integer Function and its Applications
by ravi mr. - Thursday, 16 October 2008, 09:44 AM
  Hi Aditya,

Sn = 3x1 + 5x2 +.....+ 13x6....


Tn = (2n+1) x n where n=1,2,3,4………………….

Now,
Sn = ∑ Tn

Sn = ∑ [(2n+1) x n]

Sn = ∑ [ 2n2 + n ]

Sn = 2∑n2 + ∑n

Sn = 2[n(n+1)(2n+1)]/6 + n(n+1)/2

Sn = n(n+1)/2 [ 2(2n+1)/3 + 1 ]

Sn = n(n+1)(4n+5)/6



Re: Greatest Integer Function and its Applications
by ravi mr. - Thursday, 16 October 2008, 09:45 AM
  Hi Aditya, Sn = 3x1 + 5x2 +.....+ 13x6.... Tn = (2n+1) x n where n=1,2,3,4…………………. Now, Sn = ∑ Tn Sn = ∑ [(2n+1) x n] Sn = ∑ [ 2n2 + n ] Sn = 2∑n2 + ∑n Sn = 2[n(n+1)(2n+1)]/6 + n(n+1)/2 Sn = n(n+1)/2 [ 2(2n+1)/3 + 1 ] Sn = n(n+1)(4n+5)/6 
Re: Greatest Integer Function and its Applications
by ravi mr. - Thursday, 16 October 2008, 09:46 AM
  Hi Aditya, Sn = 3x1 + 5x2 +.....+ 13x6.... Tn = (2n+1) x n where n=1,2,3,4…………………. Now, Sn = ∑ Tn Sn = ∑ [(2n+1) x n] Sn = ∑ [ 2n2 + n ] Sn = 2∑n2 + ∑n Sn = 2[n(n+1)(2n+1)]/6 + n(n+1)/2 Sn = n(n+1)/2 [ 2(2n+1)/3 + 1 ] Sn = n(n+1)(4n+5)/6 
Re: Greatest Integer Function and its Applications
by ravi mr. - Thursday, 16 October 2008, 09:46 AM
  Hi Aditya,

Sn = 3x1 + 5x2 +.....+ 13x6....


Tn = (2n+1) x n where n=1,2,3,4………………….

Now,
Sn = ∑ Tn

Sn = ∑ [(2n+1) x n]

Sn = ∑ [ 2n2 + n ]

Sn = 2∑n2 + ∑n

Sn = 2[n(n+1)(2n+1)]/6 + n(n+1)/2

Sn = n(n+1)/2 [ 2(2n+1)/3 + 1 ]

Sn = n(n+1)(4n+5)/6



Re: Greatest Integer Function and its Applications
by ravi mr. - Thursday, 16 October 2008, 10:09 AM
  Hi Aditya,

Sn = 3x1 + 5x2 +.....+ 13x6....


Tn = (2n+1) x n where n=1,2,3,4………………….

Now,
Sn = ∑ Tn

Sn = ∑ [(2n+1) x n]

Sn = ∑ [ 2n^2 + n ]

Sn = 2∑n^2 + ∑n

Sn = 2[n(n+1)(2n+1)]/6 + n(n+1)/2

Sn = n(n+1)/2 [ 2(2n+1)/3 + 1 ]

Sn = n(n+1)(4n+5)/6



Re: Greatest Integer Function and its Applications
by tarun jha - Tuesday, 11 May 2010, 05:27 PM
 

really very useful and explain very neatly,,,,

 

grt ,,,tarunj cool

Re: Greatest Integer Function and its Applications
by anuraag Burugu - Wednesday, 2 October 2013, 12:15 PM
  As ususal an simple and stuning article!!