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Distribution of numbers in the multiplication table
#11
RE: Distribution of numbers in the multiplication table
(June 8, 2020 at 4:53 am)FlatAssembler Wrote: So, what do you guys here think, what would be the distribution of the numbers in the multiplication table? That is, in the 10x10 multiplication table, why are there 6 numbers between 10 and 20 (12, 14, 15, 16, 18), 5 numbers between 20 and 30 (21, 24, 25, 27 and 28), but no numbers between 90 and 100 and only one number between 80 and 90 (81)? I've made a web-app (works only in modern browsers, not even in Internet Explorer 11) that calculates the properties of that distribution, but I can't figure out if it's some distribution that's already been described.

Well, think of it like this.

Suppose you look instead look at products of numbers up to 100. The products can be up to 10,000, but the next highest product is 9900 (twice) and the next below that is 9801 with 9800 (twice) just below that. That means there are six products in the 2% interval from 9800 to 10,000.

So, part of your difficulty is that the 'gap' between factors is 10% of your largest factor. In the case where you multiply up to 100, the gap is a mere 1%.

And, as you increase the largest number available, you get a more and more continuous distribution.

This suggests that we actually want to look at products from the unit interval [0,1] to the unit interval [0,1].

So, what is the probability that a product xy where 0<= x,y <=1 is between the values s and t? This corresponds to the area between the curves y=t/x and y=s/x inside the unit square.

This is actually not too bad to calculate and the answer is (t-s)+s ln(s)-tln(t).

The corresponding distribution function is -ln(t).

This says that products close to 0 are much more likely and those close to 1 much less likely. The distribution itself is fairly well known.
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#12
RE: Distribution of numbers in the multiplication table
polymath257 Wrote:The corresponding distribution function is -ln(t).
I am not sure what you mean. Distribution function has to be bound by 0 and 1, right? While -ln(1)=0, -ln(0)=infinity.
Besides, -ln(t) would be a homogenous function, always falling. The distribution of numbers in the multiplication table has short intervals of growth (that my web-app draws red).
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#13
RE: Distribution of numbers in the multiplication table
(June 9, 2020 at 1:49 am)FlatAssembler Wrote:
polymath257 Wrote:The corresponding distribution function is -ln(t).
I am not sure what you mean. Distribution function has to be bound by 0 and 1, right? While -ln(1)=0, -ln(0)=infinity.
Besides, -ln(t) would be a homogenous function, always falling. The distribution of numbers in the multiplication table has short intervals of growth (that my web-app draws red).

I went for the derivative, which is the density function. The distribution function you are thinking of will be t(1-ln(t)). At t=0, the logarithm goes to -infty, but at a rate which makes the product go to 0.

The growth that you see in the tables is because of the granularity (large percentage jump between the numbers you use). In the limit, that goes away.
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#14
RE: Distribution of numbers in the multiplication table
polymath257 Wrote:The growth that you see in the tables is because of the granularity (large percentage jump between the numbers you use). In the limit, that goes away.
But intervals of growth only start to appear in 10x10 multiplication table, they don't appear in 9x9 multiplication table or less. And they become more and more common the bigger the multiplication table is.
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#15
RE: Distribution of numbers in the multiplication table
(June 10, 2020 at 3:15 pm)FlatAssembler Wrote:
polymath257 Wrote:The growth that you see in the tables is because of the granularity (large percentage jump between the numbers you use). In the limit, that goes away.
But intervals of growth only start to appear in 10x10 multiplication table, they don't appear in 9x9 multiplication table or less. And they become more and more common the bigger the multiplication table is.

Again, a very coarse case. Take something like the 1000x1000 table and see what happens.
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#16
RE: Distribution of numbers in the multiplication table
(June 10, 2020 at 4:20 pm)polymath257 Wrote:
(June 10, 2020 at 3:15 pm)FlatAssembler Wrote: But intervals of growth only start to appear in 10x10 multiplication table, they don't appear in 9x9 multiplication table or less. And they become more and more common the bigger the multiplication table is.

Again, a very coarse case. Take something like the 1000x1000 table and see what happens.

In fact, in the 1000x1000 table, there are more than 300 intervals of growth. Here is an optimized web-app written in C++ that calculates that, the source code is here.
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#17
RE: Distribution of numbers in the multiplication table
(June 11, 2020 at 7:03 am)FlatAssembler Wrote:
(June 10, 2020 at 4:20 pm)polymath257 Wrote: Again, a very coarse case. Take something like the 1000x1000 table and see what happens.

In fact, in the 1000x1000 table, there are more than 300 intervals of growth. Here is an optimized web-app written in C++ that calculates that, the source code is here.

What is the percentage fluctuation?
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#18
RE: Distribution of numbers in the multiplication table
All I know is that when I turn 42 I can claim to be the answer to life.
[Image: mtfbwyf.jpg]
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#19
RE: Distribution of numbers in the multiplication table
(June 11, 2020 at 9:42 am)polymath257 Wrote:
(June 11, 2020 at 7:03 am)FlatAssembler Wrote: In fact, in the 1000x1000 table, there are more than 300 intervals of growth. Here is an optimized web-app written in C++ that calculates that, the source code is here.

What is the percentage fluctuation?

What's that? How do you calculate that? I have only taken Linear Algebra, Mathematics 1, Mathematics 2, Mathematics 3 and now I am taking Signals and Systems and Probability and Statistics classes, you know, I am not really an expert at mathematics.

(June 11, 2020 at 9:58 am)Eleven Wrote: All I know is that when I turn 42 I can claim to be the answer to life.

I don't know what you mean.
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#20
RE: Distribution of numbers in the multiplication table
(June 11, 2020 at 9:59 am)FlatAssembler Wrote:
(June 11, 2020 at 9:42 am)polymath257 Wrote: What is the percentage fluctuation?

What's that? How do you calculate that? I have only taken Linear Algebra, Mathematics 1, Mathematics 2, Mathematics 3 and now I am taking Signals and Systems and Probability and Statistics classes, you know, I am not really an expert at mathematics.

(June 11, 2020 at 9:58 am)Eleven Wrote: All I know is that when I turn 42 I can claim to be the answer to life.

I don't know what you mean.

OK, break it up into intervals of the same *relative* size. So, you broke up the 100x100 into intervals of size 200 and the 200x200 into intervals of size 200. So, the first gives 100 intervals and the second gives 200.

You should first break the results into the same number of intervals to get a fair comparison.

Second,I think it is clear that the numbers have an overall downward trend with small upward fluctuations. But if you consider comparable intervals, the percentage of those increases will be less for larger squares and will go to decreases when you get large enough.

In regard to your second question, 42 is the answer to life the universe and everything in the Hitchhiker's Guide to the Galaxy (humor)
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