A Short Lesson about Triangular Numbers

Someone asked, "How can I add 1+2+3+4+5+6+7......+999+1000 in short way and short time? "

There is a really easy way to do that. It involves using "triangular numbers."

Certain numbers are known as triangular numbers because of the shape they would make if each number was represented by the same amount of objects, and the objects were stacked beneath or on top of each other.

There is a simple formula for figuring the total of all the objects (or numbers) of any given triangular number. For example, let's take the triangular number for 1000, which would be
1 +2 +3 +...+ 1000.
There are a few ways to calculate it. The simplest way, I think, is
(1000^2 + 1000)/2.*

To understand how this works, let's take a simpler example (1+ 2+ 3 + 4 ) which is the fourth triangular number. (The first triangular number is considered to be 1, then second is 1+2, the third is 1+2+3).
You can call 10 a triangular number in the same way as you could call 16 a square number. The triangular root of 10 is 4. (Don't confuse that with the third root of a number, that is different.)

Another way to think about triangular numbers is this:
If you wrote the all the numbers you needed to add up to find the triangular number for the root (say) 6, you would get :
1 + 2 + 3 + 4 + 5 + 6.
Now, if you used the same numbers over again, but in reverse order, you would get:
6 + 5 + 4 + 3 + 2 + 1.
If you stacked one group above the other, and added them together, you would get:

So in the end you get ( 7 + 7 + 7 + 7 + 7 + 7 ) which, you can see, is 6 x 7, which is the same thing as 6x6 +6, which is also 6^2 +6.
Then you have to make up for the doubling that you did in order to stack the two rows of numbers, by dividing by 2.

A note about triangular numbers:
Triangular numbers belong to a class of numbers known as figurate numbers. To those also belong square numbers, cubed numbers, tetrahedral numbers, etc. Tetrahedrals are to triangles what cubes are to squares. They take them into the third dimension. Lunatics like Pythagoras and his minions liked to find mysticism in figurate numbers. Go figure.

For a very good explanation of figurate numbers, check out the section on them in the book "Mathematics for the Million." It is a great book for all-around math knowledge.

Eventually, we will go into more detail about triangular numbers on this site, too.

* The ^ sign means "to the power of..." The reason you will see it used on the internet, instead of the typical little number raised a little higher than the normal text, is simply that it is a pain to format it (for the web) the other way. In "real life" it should be written the normal way, when possible.

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