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Distribution of numbers in the multiplication table
#1
Distribution of numbers in the multiplication table
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.
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#2
RE: Distribution of numbers in the multiplication table
It's the nature of arithmetic.  Not all numbers from 1-100 are the product of integers from 1-10.

I'm not sure why this is even a question.

Boru
‘Let me never fall into the vulgar mistake of dreaming that I am persecuted whenever I am contradicted.’ Ralph Waldo Emerson
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#3
RE: Distribution of numbers in the multiplication table
Aliens.
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#4
RE: Distribution of numbers in the multiplication table
(June 8, 2020 at 5:06 am)BrianSoddingBoru4 Wrote: It's the nature of arithmetic.  Not all numbers from 1-100 are the product of integers from 1-10.

I'm not sure why this is even a question.

Boru

Of course not all numbers from 1-100 are the product of integers from 1-10, but why aren't those numbers that are products evenly distributed? Furthermore, why is the distribution function mostly falling, but also has short intervals of growth?
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#5
RE: Distribution of numbers in the multiplication table
(June 8, 2020 at 5:49 am)FlatAssembler Wrote:
(June 8, 2020 at 5:06 am)BrianSoddingBoru4 Wrote: It's the nature of arithmetic.  Not all numbers from 1-100 are the product of integers from 1-10.

I'm not sure why this is even a question.

Boru

Of course not all numbers from 1-100 are the product of integers from 1-10, but why aren't those numbers that are products evenly distributed? Furthermore, why is the distribution function mostly falling, but also has short intervals of growth?

Why SHOULD they be evenly distributed? As you get to larger and larger numbers, it takes larger and larger integers to multiply to produce those numbers. But you're  only working with factors of 1-10. So, the higher the product, the less likely it is to be produced by multiplying just 1-10 by other integers from 1-10.

For example, lets look at two numbers, 30 and 94.  The factors of 30 (that is, those integers that you can multiply to get 20) on a 10x10 multiplication table are 3,5,6, and 10.  But for 94, the only factor on the same table is 2.

Boru
‘Let me never fall into the vulgar mistake of dreaming that I am persecuted whenever I am contradicted.’ Ralph Waldo Emerson
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#6
RE: Distribution of numbers in the multiplication table
(June 8, 2020 at 6:01 am)BrianSoddingBoru4 Wrote:
(June 8, 2020 at 5:49 am)FlatAssembler Wrote: Of course not all numbers from 1-100 are the product of integers from 1-10, but why aren't those numbers that are products evenly distributed? Furthermore, why is the distribution function mostly falling, but also has short intervals of growth?

Why SHOULD they be evenly distributed? As you get to larger and larger numbers, it takes larger and larger integers to multiply to produce those numbers. But you're  only working with factors of 1-10. So, the higher the product, the less likely it is to be produced by multiplying just 1-10 by other integers from 1-10.

For example, lets look at two numbers, 30 and 94.  The factors of 30 (that is, those integers that you can multiply to get 20) on a 10x10 multiplication table are 3,5,6, and 10.  But for 94, the only factor on the same table is 2.

Boru

OK, well, prime numbers become more and more rare the numbers you are looking at get bigger. And, since prime numbers can't be found in an n*n multiplication table except in the interval <1,n], one could as well expect there to be more numbers in the multiplication table in higher intervals, rather than in lower intervals (there will be more prime numbers in lower intervals).
Furthermore, why would somebody expect there to be short intervals of growth in the distribution function, ones that my web-app ( https://flatassembler.github.io/multiplication.html ) draws red?
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#7
RE: Distribution of numbers in the multiplication table
(June 8, 2020 at 6:51 am)FlatAssembler Wrote:
(June 8, 2020 at 6:01 am)BrianSoddingBoru4 Wrote: Why SHOULD they be evenly distributed? As you get to larger and larger numbers, it takes larger and larger integers to multiply to produce those numbers. But you're  only working with factors of 1-10. So, the higher the product, the less likely it is to be produced by multiplying just 1-10 by other integers from 1-10.

For example, lets look at two numbers, 30 and 94.  The factors of 30 (that is, those integers that you can multiply to get 20) on a 10x10 multiplication table are 3,5,6, and 10.  But for 94, the only factor on the same table is 2.

Boru

OK, well, prime numbers become more and more rare the numbers you are looking at get bigger. And, since prime numbers can't be found in an n*n multiplication table except in the interval <1,n], one could as well expect there to be more numbers in the multiplication table in higher intervals, rather than in lower intervals (there will be more prime numbers in lower intervals).
Furthermore, why would somebody expect there to be short intervals of growth in the distribution function, ones that my web-app ( https://flatassembler.github.io/multiplication.html ) draws red?

It’s got nothing to do with prime numbers (well...a little bit to do with prime numbers). It has to with there being fewer factors of larger products in the 1-10 integer range.

Let’s look at the number 60. Factors for this are 1,2,3,4,5,6,10,12,15,20,30 and 60. But on a 10x10 table, you can only include 6 and 10 (the others are disqualified because you need a multiplier greater than ten to reach 60). This necessarily result in fewer products in the high end of the table.

Boru
‘Let me never fall into the vulgar mistake of dreaming that I am persecuted whenever I am contradicted.’ Ralph Waldo Emerson
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#8
RE: Distribution of numbers in the multiplication table
I blame Trump.


Signed


The Left
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#9
RE: Distribution of numbers in the multiplication table
I blame Obama.




Signed




The Right.

Boru
‘Let me never fall into the vulgar mistake of dreaming that I am persecuted whenever I am contradicted.’ Ralph Waldo Emerson
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#10
RE: Distribution of numbers in the multiplication table
Exactly
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