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'See' the Numbers!
by Quant Ghost - Thursday, 11 January 2007, 12:30 AM

Quant Ghost

All you guys and gals, you are warned not to repeat what you have done till now, and for which you have repented and cried your hearts out in the past. It would not be respecting yourself and giving recognition to the gift lying within you. So you better pull your socks up and pledge to yourself that from now on you shall daringly explore the hidden facts embedded in mathematics. Before moving forward, I must say that I assume the following about you:

  1. You are aware of basic family tree of natural numbers, whole numbers, integers, fractions, perfect numbers, real numbers and complex numbers.
  2. You are aware of simple equations such as linear and quadratic and the basic nuisances encountered in it.
  3. You don’t get scared when you see triangles, circles, lines and four sided figures.
  4. You don’t mind if the side of any triangle becomes an equation or some equations vomit some numbers as their solutions.
  5. You give “zero” the right status just like you give to other numbers though I would never like to see this number in your report card.
  6. You never lose the confidence and become the victim of “imagining yourself to be a failure.”
  7. You will respect every element of the mathematical fact coming across to you as much as you respect your parents and friends.
  8. You are not scared of ghosts, at least the one who teaches you mathematics.

Ok, let us start from the building blocks of mathematics i.e. the natural numbers, or the numbers which we use for counting. My main emphasis here would be tell you how closely you have to look at numbers

What happens when we add 1 + 3? The addition gives the number 4 which is a square. What happens when we add 1 + 3 + 5? Again, our addition gives us a perfect square- the number 9. If I take 4 balls and arrange them, I can easily make a square like figure. Same goes with 9 and with all the numbers which are perfect squares

Similarly, there are some numbers which when arranged geometrically can give a triangular figure. These numbers are 1, 3, 6, 10, 15...etc. If you are not getting me clearly, following figures would definitely help you out


Observe the triangular behavior of the numbers given as follows


So what’s my point? It’s this- use pictures as much as you can because your imagination is your best friend. What else can you observe here? You added 1 + 3 in the figure and got 4, you added 1 + 3 + 5 and got 9. Therefore, when you add 1 + 3 + 5 + 7 +…up to n numbers in all, you would get n2. Moreover if you find the area of the square having dimensions n, it gives you an area n2.

What else can be derived from these figures? If we put two triangular numbers of the side n together, they form a rectangle, n + 1 by n, whose area is n(n+1). What? What did you don’t agree with me? Ok, let’s check it out; just observe the following figures.


If you just look at the rectangle at the extreme right corner you would see that its area is n multiplied by n + 1. See how easily we can derive these relations from the numbers diagram? Also, you can say easily that


I shall cover some more extension of this concept on the next page

What do you get if you add two consecutive triangular numbers?


That’s a perfect square!!

So we saw that numbers gain their importance in our mind only when we explore how they can help us in understanding the relation between them. The ghost shall be happy if the reader understands what he has tried and responds to it. The reader should bear in mind that the ghost is really concerned about them. The reader is advised to explore more relations as it would help him know the numbers in a better way!!

Quant Ghost

Re: 'See' the Numbers!
by Choco Pie - Thursday, 11 January 2007, 04:29 PM

This is really awesome man ... enjoyed a lott all through your article!!!

Waiting eagerly for some more good mathematics !! wide eyes

Re: 'See' the Numbers!
by Chirag Taneja - Thursday, 11 January 2007, 11:59 PM
  I must tel u this is awsome stuff....whosoever is behind this is doing a great job....this is wat we lack....application .....KUDOS to Quantghost
Re: 'See' the Numbers!
by Quant Ghost - Tuesday, 16 January 2007, 04:30 AM
  Quantghost bows his head in acknowledgement for choco pie and Mr.TanejaK
Re: 'See' the Numbers!
by santo john - Sunday, 4 March 2007, 10:50 AM
  I wish you could go and teach every student with the same patience as we would have expected from our are indeed an expert..My friends visited this website and were taken aback to see your section..your story is truly inspiring and an eye-opener to us all
Re: 'See' the Numbers!
by vikram singh - Monday, 19 March 2007, 02:21 PM


The post is really helpful in understanding the way quant is to be looked at. This also boosts the confidence in staying calm and downsizing the problem when confronted with a question that seems to be a monster for a while.

Re: 'See' the Numbers!
by Abha Soni - Tuesday, 3 April 2007, 02:18 PM

Dear Quant ghost

It is a  really good stuff one taught me maths like this ......

but I am not able to understand one thing that how we will come to know that which number is a triangle number...........

If you flash some more light on it then it vl be much more helping for all of us....




Re: 'See' the Numbers!
by Quant Ghost - Wednesday, 4 April 2007, 01:51 AM

Thank you for the compliments Abha and Vikram.

Hi Abha, every triangular number is expressed as triangular numbers.You keep putting N = 1, 2, 3, 4… etc in this expression and you will get as many triangular numbers as you want. And if you want to recognize them in a second, keep this expression always with you .

Hope quant ghost has made it clear to you. If not, do get back to me.

Quant Ghost

Re: 'See' the Numbers!
by manish sharma - Thursday, 12 April 2007, 12:06 PM

i have just started prep. for cat.. i will be appearing in next year.. can u plz tell me what's the use of triangular numbers.

Re: 'See' the Numbers!
by Quant Ghost - Thursday, 19 April 2007, 02:20 AM

Dear Manish,

Your question is very similar to a query which asks about the use of whole numbers and natural numbers. Just as you find whole numbers present everywhere in the nature, so you discover the triangular numbers. Moreover, recognizing triangular numbers in a sequence or a series helps a lot. Also, they keep showing their presence in the reasoning types of questions.

I hope quant ghost has answered the questions arising in your mind…

numbers quant triangular

Re: 'See' the Numbers!
by paras udani - Friday, 20 April 2007, 01:50 AM
  hallo..this is paras frm mumbai..actually sir this is a very typical type of question but its very imp for me to get ur guidance..actually i m very weak in quants section and thts y i m not getting interest into it...evryday i first study the basic concepts and thn start with the sums but i really dnt know tht i m not able to solve it..evn the simple sums is taking a lot of my time....thn wat shud i der any book of quants tht i can refer to..??/
Re: 'See' the Numbers!
by Quant Ghost - Friday, 20 April 2007, 08:38 AM
  Dear Paras,

This is EXACTLY the reason TG created this website, that students like you can come and recieve help. For study purpose, refer to TG's lessons and NCERT books for now. Start with simple problems so that you become comfortable with Math. But more than that, start asking. Start posting your problems in the forums and we will post the answers.

number theory problems CAT MBA
hello QG
by sudama baraily - Tuesday, 24 April 2007, 12:20 AM



Re: hello QG
by Quant Ghost - Tuesday, 24 April 2007, 02:11 AM
  The ghost smiles at Sudama...

Re: hello QG
by quaint lee - Saturday, 28 April 2007, 10:07 PM

Hi QG,
Neat stuff. Thanks!

So triangular numbers are essentially the series 1+2, 1+2+3,1+2+3+4, 1+2+3+4+5 ....

Is 1 also a triangular number? It does fulfil the condition n(n+1)/2 for n =1

Re: hello QG
by Quant Ghost - Sunday, 29 April 2007, 05:05 AM
  Yes. 1 is also a triangular number.

numbers triangular numbers
Re: 'See' the Numbers!
by shruti a - Monday, 25 June 2007, 09:43 PM

hi QG....

great article....can u please answer two questions....

1.n(n+1)/2 is also the sum of the first n natural any relation of triangle number with that?

2.this question has already been asked before but can u plz illustrate with an example how the concept of triangle numbers will help in solving a problem in a more efficeint manner?

thank you

Re: 'See' the Numbers!
by shruti a - Monday, 25 June 2007, 09:44 PM

hi QG....

great article....can u please answer two questions....

1.n(n+1)/2 is also the sum of the first n natural any relation of triangle number with that?

2.this question has already been asked before but can u plz illustrate with an example how the concept of triangle numbers will help in solving a problem in a more efficeint manner?

thank you

Re: 'See' the Numbers!
by jo paix - Wednesday, 25 July 2007, 09:32 AM
  a triangular number is of the form  n(n+1)/2.
Re: 'See' the Numbers!
by arun r - Wednesday, 15 August 2007, 08:54 AM

Hi quant ghost

                  am new to this site.... this article s simply amazin.....

Re: 'See' the Numbers!
by Vinod Paramasivan - Monday, 17 September 2007, 07:56 AM
  Great article QG..

It would be even great if you could provide some examples where this concept of triangular numbers is used.
Re: 'See' the Numbers!
by upadrasta jagadish - Sunday, 14 October 2007, 03:41 PM

Dear Quant Ghost,

Its a great support that we are recieving from you.

Thanks for it.

In this article I am unable to understand how " twice the triangular number is the product of two consecutive integers". Please explain.




Re: 'See' the Numbers!
by shivendu singh - Monday, 15 October 2007, 12:41 PM


Any triangular no will be of form--N(N+1)/2   i.e.(1+2+3.......+N)

twice the triangular no = 2 * N(N+1)/2 = N(N+1), which is product of two consecutive nos.

you can take one example , 10 is a triangle no.

so 2*10= 20 = 4 * 5...


I hope this will clear ur doubt

Re: 'See' the Numbers!
by out onalimb - Saturday, 12 January 2008, 02:59 AM

Dunno if you're still around but damn, I wish we taught this way to kids. I finally understand why it's said there is an elegance to maths.

I wish you'd post some more stuff. I also wish you'd do some teaching/consulting in the school/service I'm going to build one day. big grin

Re: 'See' the Numbers!
by kamal mehta - Thursday, 17 January 2008, 01:27 PM

keep posting such they are  really of immense importance to us . i have even done coaching for cat but still i was not aware of these things please keep posting . i also want to know 1 thing that does u people also offer solution to the problems that u r providing ex:number system problems , geometry problems that u  r providing in the main forum. plaese let me know as soon as possible.

Re: 'See' the Numbers!
by Utkarsh Parhad - Wednesday, 9 April 2008, 01:08 PM


Can anybody please tell me how to get the hard copy of TG's book on number systems

Re: 'See' the Numbers!
by sandeep chincholkar - Saturday, 27 September 2008, 11:40 PM

dats stupendous.....n of great help to all...


Re: 'See' the Numbers!
by Venkata Narsi Reddy G - Tuesday, 16 June 2009, 01:57 PM

Dear Jagadish,

I hope you've observed the figure above the statement saying, "twice the triangular number is the product of two consecutive integers". Imagine the triangles(same right triangles) being joined together with the 2nd triangle inverted. Now, I hope you know dat no. of balls in any figure is the number it represents. So, when the triangles are joined, the last line having n balls of the 1st triangle joins with the first line of the 2nd triangle having 1 ball in the first line to make the total balls in the line to n+1. Similarly, the 2nd line of the 1st triangle having n-1 balls joins with the 2nd line of the 2nd triangle having 2 balls to make the total no. of balls in that line to n+1. In the same way, it repeats for all the n lines of the two triangles.

Now, a rectangular shape is forned with (n+1) balls along the length and n balls along the breadth. So, the no. this figure represents is the no. of balls, i.e., n*(n+1).

I hope this explains your question.

Re: 'See' the Numbers!
by anish bansal - Monday, 2 November 2009, 10:48 AM
  Hi Quant Ghost!!

This article is simply brilliant. smile smile

Re: 'See' the Numbers!
by Saurabh - confusion fused - Tuesday, 3 November 2009, 12:44 AM
  Dear ghost,

The article is nice in a sense that it created a different perspective to visualize mathematics. But I think you should explain the application of triangular numbers. We are anxiously waiting for it.

Thanks for a nice article.
Re: 'See' the Numbers!
by prasanth warrier - Sunday, 17 January 2010, 10:28 AM
  fantabulous way of teaching...sir iam prashant from mumbai...i wil be givin cat this november..i dint get one thing in this segment....where do we exactly use it...can u give some examples of its actual use

Re: 'See' the Numbers!
by deepti anand - Tuesday, 25 May 2010, 04:01 PM
  hii ghost..
m new 2 dis site...ur article is can u tell hw dis triangular concept can b used in ques...i mean in wat type of ques can v use dis concept..??
Re: 'See' the Numbers!
by Nikhil Khnadelwal - Sunday, 22 May 2011, 01:27 PM
  wonderful article............ thanks all concerned
Re: 'See' the Numbers!
by sumit garg - Thursday, 4 June 2015, 11:55 AM
  Awesome Sir

very new concept