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Equation of the Day #10: Triangular Numbers


Akano

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I like triangles. I like numbers. So what could possibly be better than having BOTH AT THE SAME TIME?! The answer is nothing! 8D

 

The triangular numbers are the numbers of objects one can use to form an equilateral triangle.

 

500px-First_six_triangular_numbers.svg.png

Anyone up for billiards? Or bowling? (Image: Wikimedia Commons)

 

Pretty straightforward, right? To get the number, we just add up the total number of things, which is equal to adding up the number of objects in each row. For a triangle with n rows, this is equivalent to

 

4njEwfo.png

 

This means that the triangular numbers are just sums from 1 to some number n. This gives us a good definition, but is rather impractical for a quick calculation. How do we get a nice, shorthand formula? Well, let's first add sequential triangular numbers together. If we add the first two triangular numbers together, we get 1 + 3 = 4. The next two triangular numbers are 3 + 6 = 9. The next pair is 6 + 10 = 16. Do you see the pattern? These sums are all square numbers. We can see this visually using our triangles of objects.

 

200px-Square_number_16_as_sum_of_two_triangular_numbers.svg.png

(Image: Wikimedia Commons)

 

You can do this for any two sequential triangular numbers. This gives us the formula

 

hebS7pK.png

 

We also know that two sequential triangular numbers differ by a new row, or n. Using this information, we get that

 

W1EWM1n.png

 

Now we finally have an equation to quickly calculate any triangular number. The far right of the final line is known as a binomial coefficient, read "n plus one choose two." It is defined as the number of ways to pick two objects out of a group of n + 1 objects.

 

For example, what is the 100th triangular number? Well, we just plug in n = 100.

 

T100 = (100)(101)/2 = 10100/2 = 5050

 

We just summed up all the numbers from 1 to 100 without breaking a sweat. You may be thinking, "Well, that's cool and all, but are there any applications of this?" Well, yes, there are. The triangular numbers give us a way of figuring out how many elements are in each row of the periodic table. Each row is determined by what is called the principal quantum number, which is called n. This number can be any integer from 1 to infinity. The energy corresponding to n has n angular momentum values which the electron can possess, and each of these angular momentum quanta have 2n - 1 orbitals for an electron to inhabit, and two electrons can inhabit a given orbital. Summing up all the places an electron can be in for a given n involves summing up all these possible orbitals, which takes on the form of a triangular number.

 

gqyRQJf.png

 

The end result of this calculation is that there are n2 orbitals for a given n, and two electrons can occupy each orbital; this leads to each row of the periodic table having 2⌈(n+1)/2⌉2elements in the nth row, where ⌈x⌉ is the ceiling function. They also crop up in quantum mechanics again in the quantization of angular momentum for a spherically symmetric potential (a potential that is determined only by the distance between two objects). The total angular momentum for such a particle is given by

 

hYs3EPJ.png

 

What I find fascinating is that this connection is almost never mentioned in physics courses on quantum mechanics, and I find that kind of sad. The mathematical significance of the triangular numbers in quantum mechanics is, at the very least, cute, and I wish it would just be mentioned in passing for those of us who enjoy these little hidden mathematical gems.

 

There are more cool properties of triangular numbers, which I encourage you to read about, and other so-called "figurate numbers," like hexagonal numbers, tetrahedral numbers, pyramidal numbers, and so on, which have really cool properties as well.

 

TmFf04p.png

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I saw the (n+1)choose(2) pattern immediately (that's a nice way of writing it, I realize). :P This is music to my ears. Anyway, do you study physics, sir?

 

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I do study physics! I'm currently in grad school en route to my Ph.D. :)

 

akanohi.png

Nice! :) My brother studies physics as well. I study math, which is why I said what you wrote is music to my ears. :P We're both in the fourth semester currently but we intend to study Computer Science together when we're done. ^_^

 

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My version of numbers mixed with triangles: Triforce.

 

But, your thing is totally not overly complicated for someone at my level and did't go totally over my head too.

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Nice! :) My brother studies physics as well. I study math, which is why I said what you wrote is music to my ears. :P We're both in the fourth semester currently but we intend to study Computer Science together when we're done. ^_^

 

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Very cool. Part of me always wishes that I had double majored in math and physics, but, alas, I did not. Glad to see another face on the forums that appreciates my equations of the day. ^_^

 

My version of numbers mixed with triangles: Triforce.

 

But, your thing is totally not overly complicated for someone at my level and did't go totally over my head too.

Yay! 8D

 

akanohi.png

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Huh, for all the math quizes I've ever taken that required knowledge of triangular numbers, I don't think I've ever been introduced to that equation. Or if I have, I never bothered to memorize it, but the logic is simple enough that it doesn't seem like something I'd blow off as too complicated, so I'm sticking with my "never heard it that way" theory. :P

 

Great entry, thanks!

 

:music:

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I know Taka Nuvia also studies physics. Maybe if she saw your entries, they'd appreciate them as well. :P

 

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She has definitely seen these entries before and has told me she's a fan. :) I don't know if she's seen this one yet, though.

 

Huh, for all the math quizes I've ever taken that required knowledge of triangular numbers, I don't think I've ever been introduced to that equation. Or if I have, I never bothered to memorize it, but the logic is simple enough that it doesn't seem like something I'd blow off as too complicated, so I'm sticking with my "never heard it that way" theory. :P

 

Great entry, thanks!

 

:music:

I honestly can't remember if we ever talked about triangular numbers in any of my classes, but I think they're really cool, and they aren't overly complicated.

 

akanohi.png

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I know Taka Nuvia also studies physics. Maybe if she saw your entries, they'd appreciate them as well. :P

 

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She has definitely seen these entries before and has told me she's a fan. :) I don't know if she's seen this one yet, though.

[...]

akanohi.png

 

 

Sadly only just now, but whaaaat now the binomial coefficient finally makes sense/I have a way to memorize it! Thanks a lot. =D

 

(previously: "now where do I write which number again...")

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