# If 4! = 1x2x3x4 and is called 4 factorial, what is the sum of the numbers e.g. 4(?)= (1+2+3+4) called?

Asked by LuckyGuy (34723) January 21st, 2013

I know how to easily calculate the answer. Is there an operator name for it?
Example:
6 factorial is 6! = 1×2x3×4x5×6 = 720
6 summit is 6| = 1+2+3+4+5+6 = 21
Obviously I made up the name “summit” and symbol ”|”

Is there a commonly used term and symbol for this? I can use the summation symbol but that is unwieldy.

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What’s wrong with the summation notation? It does just that. I don’t think there’s any other way of expressing it other than writing it out.
[addition]: After more research, I found a formula where 1+2+3+4….+n= n*(n+1)/2
http://en.wikipedia.org/wiki/Triangular_number

dxs (14503)

It’s the 4th triangular number, for reasons explained at the link (think bowling pins!). The usual notation is Σn with n=1 below the sigma and (in this case) 4 above to indicate the maximum.

The explicit formula is S = n*(n+1) / 2

There a famous story about the great mathematican Carl Friedrich Gauss discovering the formula as a school boy when asked to add up the numbers from 1 to 100.

gasman (11264)

Because the factors are changeable, use Sigma as the operator, and then standard notation for the terms. Factorial is nice and convenient, but it’s limited to integers. You can’t use factorial for squares or only odd numbers, or other sequences.

Sigma has the convenience of being adaptable.

zenvelo (30989)

Those are called triangular numbers, because they can be arranged to form a triangle.

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If f(x) = the sum of all natural numbers less than or equal to x, then f(x) = (x^2 + x)/2. This can also be notated using the sigma as @gasman described.

PhiNotPi (12647)

I am unsure there is a formal operand for this. I understand the concept of needing factorial when doing statistics. What is your usage of this for?

blueiiznh (16658)

I remember in middle school, I used to write things like 10? to represent 10+9+8…etc. 5? = 15, 4? = 10.

I assume it doesn’t have an actual dedicated symbol because it’s not as important as having the factorial. Nonetheless, I like the question mark. :)

DominicX (28762)

@blueiiznh This is work related problem. I’m using very fast timing circuits that continuously count up and there is an interrupt signal that needs to be recorded for every one of 250 events while the counter is running up.

I was hoping there was a quick and easy description that would simplify the system mechanization description.
I like @DominicX ‘s 10? to represent the sum of 1=2+...10 . I might even do that with a brief definition of triangular number. Then I could use the shorthand for the rest of the long description.
I have to explain something very complicated in words that anyone can understand it. Ugh!

LuckyGuy (34723)

I was also given this problem in grade school and came up with a formula that worked but was not the same. where x

x=a*.5+.5*a
x=100*.5+.5*100
x=50+.5*100
x=50.5*100
x=5050

DrBill (16046)

If you do enough math, that equation will end up to be n(n+1)/2.

n*½+½ = n/2+½ Then multiply it by n to make it (n/1)* n/2+n/2 which equals n(n+1)/2

dxs (14503)

@dxs yes, I know, but that is what I came up with in grade school with no knowledge of the original. Considering I came up with it in the fifth grade, at a time when algebra was not taught till college.

DrBill (16046)
Response moderated (Spam)

One nice thing about the formula n(n+1)/2 is that it makes intuitive sense. The average of the numbers from 1 to n is just the average of the first and last, (n+1)/2. To find the sum, multiply the average times the number of terms n, to get n(n+1)/2.

There should be an operator or symbol for this. Or at least a name. Jeez! Saying the 100th Triangular number (5050) just doesn’t cut it. It needs further explanation which I am trying to avoid. 100!, 100 factorial, everyone knows. It’s ginormous.

@DrBill Clearly you were listening in school. Around the same time and age my teacher told us about square roots and how you can’t find the answer to the sqrt of 10 exactly. I set out to try. I discovered that If you made a guess and divided that into the number and averaged your guess and the result, you will get a much closer answer. You repeat the process and you get the answer quickly. (Like you, I did this before algebra and calculators) For example: let’s say you want to know the square root of 25 (We know the answer but let’s say you didn’t.) You guess 4.5 . 25/ 4.5 = 5.555555~ Average 4.5 and 4.5555555~ to get 5.027777~ Shorten it to 5.027778. Now use that as your next guess. 25/ 5.027778 = 4.972373 Average the two and get 5.0000765. Use that number as the next guess and…..
I filled pages of loose leaf paper with my sqrt of 10 long division. First guess =3 Oh well, it was good practice.
@LostInParadise That is a neat, intuitive way of understanding the formula.

LuckyGuy (34723)

@LuckyGuy I believe your square roots algorithm is known as “Newton’s method.” You’re in good company!

gasman (11264)

I actually figured that if I used enough digits and was very close, it would hit exactly if I did the problem enough times. 10 times maybe. 12 times? Of course now I know better and understand calc and limits. But that exercise/quest sure helped me appreciate math at an early age. Who knows, maybe that’s part of why I became an engineer.
Or, more likely, I was already an engineer even as a child.

So I guess I will just suck it up and figure there is no neat name for this operator. I will define it using the Sum symbol with i=1 to n. Darn it.

LuckyGuy (34723)

@LuckyGuy I think I might be a bit older than you, I was not told about square roots till high school.

DrBill (16046)

It seems useful to me to need to express a set of numbers in such a way. @LuckyGuy If you make up a symbol, I’ll use it too and maybe we can start a trend.

dxs (14503)

I have always used delta after the number i.e. 100∆

DrBill (16046)

or