I said the clause was an unnecessary inclusion, the ellipses are there specifically to highlight that. The rest of the article is about the cyclicity of the digit.
Your ellipses completely change the meaning. The clause may be unnecessary (as is the entire article) but the bit you complain about is useful if you don't chop things out first.
Well, not sure it was a complaint, I was just lightheartedly commenting that it was a bit redundant, but ok, umbrage seems a reasonable response.
And yeah, that's pretty much the point of eliding bits of text, to put things together that weren't together before; I'm almost positive that's why we invented that specific punctuation mark.
It's like saying: "The 35th president of the United States, JFK, who was widely known for the space race and the bay of pigs, was elected after the 34 presidents before him."
And my comment is like: "The 35th president of the United States ... was elected after the 34 presidents before him."
My point is that ellipses are for when you're excising something that is redundant to your purposes and whose omission doesn't change the overall meaning.
Your example with JFK is fine. Here's a counterexample where it doesn't work: "JFK had an affair with Marilyn Monroe, who was a famous Hollywood actress." -> "JFK... was a famous Hollywood actress."
I maintain that your shortened quote is more like this. The article says that 142857, when multiplied, produces numbers which correspond to the digits of 2/7, 3/7, 4/7, etc.
Your version of the quote omits the words "correspond to," changing it from an observation about how an integer matches the decimal expansion of a fraction to an observation about how multiplying a number gets you a multiple of that number.
Ask a random person if 1 and 0.999999... are the same number and most people will tell you they aren't the same number. Yet: 1 = 3 * (1/3) = 3 * 0.33333... = 0.99999... (they're also epsilon close for every epsilon larger than zero). So yes, this kind of stuff is surprising for a lot of people.
I'd like to have everyone in the world give me their money and I will return that money times 1.0000000001 about a week later, after making bank off the interest. Sadly there seems to be no general enthusiasm for this plan. :)
Also: Tell you what... you give me your account details, and I'll send you (1 - 0.9999...) times my money. Deal?
It's also my favorite number, but I don't think any of us likes it more than the guy in the video :) Thanks for sharing.
I learned about 142857 when reading The Man Who Counted as a kid. I remember getting obsessed with it and discovering more properties (much fewer than the ones shown on the video), e.g.:
Conversely, if you have any number whose digits added together sum to 9 (or a multiple of 9), the original number is a multiple of 9.
So, all of your original sum numbers (999, 99, 27, 1485, 2871, and 42858) are themselves multiples of 9, which will be the case for any number obtained by adding a set of numbers which together contain all and only the digits 142857. That means a lot of other "also" additions will work out too! You can even change the order, like 578 + 214 = 792 (a multiple of 9, and hence the digital root will end up being 9). Any order and any choice of how to break the numbers will work, because of the digital root property (and, importantly, the rule that "The digital root of a + b is congruent with the sum of the digital root of a and the digital root of b modulo 9").
Try this fun Python program to see how multiples of 9 are always generated no matter how you combine the digits:
#!/usr/bin/env python
import random
digits = [1, 4, 2, 8, 5, 7]
for times in range(20):
digits_left = digits[:]
nums = []
while digits_left:
this_num = 0
for i in range(random.randint(1, len(digits_left))):
digit = random.choice(digits_left)
this_num *= 10
this_num += digit
digits_left.remove(digit)
nums.append(this_num)
print " + ".join(map(str, nums)), "=", sum(nums), "(a multiple of 9)"
Sample output:
4271 + 58 = 4329 (a multiple of 9)
7412 + 58 = 7470 (a multiple of 9)
8 + 124 + 5 + 7 = 144 (a multiple of 9)
845127 = 845127 (a multiple of 9)
147285 = 147285 (a multiple of 9)
1 + 824 + 75 = 900 (a multiple of 9)
5 + 2 + 481 + 7 = 495 (a multiple of 9)
758 + 41 + 2 = 801 (a multiple of 9)
47125 + 8 = 47133 (a multiple of 9)
7584 + 12 = 7596 (a multiple of 9)
1275 + 8 + 4 = 1287 (a multiple of 9)
475 + 12 + 8 = 495 (a multiple of 9)
12 + 5 + 478 = 495 (a multiple of 9)
4251 + 7 + 8 = 4266 (a multiple of 9)
185 + 274 = 459 (a multiple of 9)
28514 + 7 = 28521 (a multiple of 9)
41278 + 5 = 41283 (a multiple of 9)
457182 = 457182 (a multiple of 9)
845712 = 845712 (a multiple of 9)
2718 + 54 = 2772 (a multiple of 9)
> multiples of 9, which will be the case for any number obtained by adding a set of numbers which together contain all and only the digits 142857
So that, then, is the fascinating root observation: that any series of decimal number made from these digits is a multiple of nine.
What this means is that we can choose six random powers of 10 between 100 and 105 and make vector out of them, for instance <1, 100, 10, 10, 100000, 1000>. Then we do a dot-product between this and the vector <1, 4, 2, 8, 5, 7>. The result will be a multiple of 9.
If we choose the vector as <1, 1, 1, 1, 1, 1> we get the straight sum of the digits. If we choose <100000, 10000, 1000, 100, 10, 1> we get 142857, and so on.
Here is why it works:
10**<whatever> x == x (mod 9).
That is to say, any integer x is congruent, modulo 9, to a power of 10 times that integer.
For instance 4 mod 9 == 4. 40 mod 9 == 4. 400 mod 9 == 4.
And the reason for that is that 10 is 9 + 1; i.e. 10 is congruent to 1 modulo 9. So we are really multiplying by 1 under the congruence.
So the choices of powers of ten in the coefficient vector do not matter.
It works in other bases. For instance if any integer x is multiplied by a power of 6, that is congruent to x, modulo 5:
Okay, I now wrapped my hackerly head around this enough that I can garbage collect it away as fairly uninteresting.
:)
But, one more thing: what is special about 1, 4, 2, 8, 5, 7 in connection to 9? Why, they are relatively prime to 9. The remaining three positive residues in the mod 9 congruence are not: 0, 3 and 6.
Euler's totient function phi counts the number of such integers. phi(9) == 6 (there are six of these numbers for 9). The above page even uses this very example.
1, 4, 2, 8, 5, 7 comprise the multiplicative group of integers modulo 9.
The "other bases" issue, which you come to right at the end, fascinated me in middle school: for the reasons you describe, digital root tests for divisibility by a digit d work in any base b if d divides (b-1). For example, in hexadecimal there is a digital root test for divisibility by 1 (trivially), 3, 5, or 15 in hexadecimal.
For instance, D+E+A+D+B+E+E+F=104₁₀ which is not divisible by 5, but that indicates that (DEADBEEF+1) will be divisible by 5, which is correct.
We're also used to the "final digit test" for divisibility in base 10, which works for 1 (trivially), 2, 5, and 10; and indeed it works for a digit d in a base b if d divides b. Ternary has a digital-root test to determine whether a number is even (as in all odd bases, you can't use the final digit alone to answer that question). For example, 1202112₃ is odd because 1+2+0+2+1+1+2 is odd.
(Edit: deleted a spurious claim about hexadecimal divisibility tests.)
24 comments
[ 5.0 ms ] story [ 43.4 ms ] threadWas it surprising to multiply 1/7 by two and get 2/7? Has science gone too far?
And yeah, that's pretty much the point of eliding bits of text, to put things together that weren't together before; I'm almost positive that's why we invented that specific punctuation mark.
It's like saying: "The 35th president of the United States, JFK, who was widely known for the space race and the bay of pigs, was elected after the 34 presidents before him."
And my comment is like: "The 35th president of the United States ... was elected after the 34 presidents before him."
Your example with JFK is fine. Here's a counterexample where it doesn't work: "JFK had an affair with Marilyn Monroe, who was a famous Hollywood actress." -> "JFK... was a famous Hollywood actress."
I maintain that your shortened quote is more like this. The article says that 142857, when multiplied, produces numbers which correspond to the digits of 2/7, 3/7, 4/7, etc.
Your version of the quote omits the words "correspond to," changing it from an observation about how an integer matches the decimal expansion of a fraction to an observation about how multiplying a number gets you a multiple of that number.
Also: Tell you what... you give me your account details, and I'll send you (1 - 0.9999...) times my money. Deal?
Here's an interesting video showing this stuff (And more) in a more user-friendly way https://www.youtube.com/watch?v=WUlaUalgxqI
I learned about 142857 when reading The Man Who Counted as a kid. I remember getting obsessed with it and discovering more properties (much fewer than the ones shown on the video), e.g.:
142 + 857 = 999; 9 + 9 + 9 = 27; 2 + 7 = 9
14 + 28 + 57 = 99; 9 + 9 = 18; 1 + 8 = 9
1 + 4 + 2 + 8 + 5 + 7 = 27; 2 + 7 = 9
Also,
1428 + 57 = 1485; 1 + 4 + 8 + 5 = 18; 1 + 8 = 9
14 + 2857 = 2871; 2 + 8 + 7 + 1 = 18; 1 + 8 = 9
1 + 42857 = 42858; 4 + 2 + 8 + 5 + 8 = 27; 2 + 7 = 9
1 + 42857 = 42858; 42 + 858 = 900; 9 + 0 + 0 = 9
This is kinda creepy. I wasn't really expecting the "also" additions to work.
https://en.wikipedia.org/wiki/Digital_root
Conversely, if you have any number whose digits added together sum to 9 (or a multiple of 9), the original number is a multiple of 9.
So, all of your original sum numbers (999, 99, 27, 1485, 2871, and 42858) are themselves multiples of 9, which will be the case for any number obtained by adding a set of numbers which together contain all and only the digits 142857. That means a lot of other "also" additions will work out too! You can even change the order, like 578 + 214 = 792 (a multiple of 9, and hence the digital root will end up being 9). Any order and any choice of how to break the numbers will work, because of the digital root property (and, importantly, the rule that "The digital root of a + b is congruent with the sum of the digital root of a and the digital root of b modulo 9").
So that, then, is the fascinating root observation: that any series of decimal number made from these digits is a multiple of nine.
What this means is that we can choose six random powers of 10 between 100 and 105 and make vector out of them, for instance <1, 100, 10, 10, 100000, 1000>. Then we do a dot-product between this and the vector <1, 4, 2, 8, 5, 7>. The result will be a multiple of 9.
If we choose the vector as <1, 1, 1, 1, 1, 1> we get the straight sum of the digits. If we choose <100000, 10000, 1000, 100, 10, 1> we get 142857, and so on.
Here is why it works:
That is to say, any integer x is congruent, modulo 9, to a power of 10 times that integer.For instance 4 mod 9 == 4. 40 mod 9 == 4. 400 mod 9 == 4.
And the reason for that is that 10 is 9 + 1; i.e. 10 is congruent to 1 modulo 9. So we are really multiplying by 1 under the congruence.
So the choices of powers of ten in the coefficient vector do not matter.
It works in other bases. For instance if any integer x is multiplied by a power of 6, that is congruent to x, modulo 5:
Elementary number theory, my dear Watson.Okay, I now wrapped my hackerly head around this enough that I can garbage collect it away as fairly uninteresting.
:)
But, one more thing: what is special about 1, 4, 2, 8, 5, 7 in connection to 9? Why, they are relatively prime to 9. The remaining three positive residues in the mod 9 congruence are not: 0, 3 and 6.
See here: https://en.wikipedia.org/wiki/Euler%27s_totient_function
Euler's totient function phi counts the number of such integers. phi(9) == 6 (there are six of these numbers for 9). The above page even uses this very example.
1, 4, 2, 8, 5, 7 comprise the multiplicative group of integers modulo 9.
https://en.wikipedia.org/wiki/Multiplicative_group_of_intege...
For instance, D+E+A+D+B+E+E+F=104₁₀ which is not divisible by 5, but that indicates that (DEADBEEF+1) will be divisible by 5, which is correct.
We're also used to the "final digit test" for divisibility in base 10, which works for 1 (trivially), 2, 5, and 10; and indeed it works for a digit d in a base b if d divides b. Ternary has a digital-root test to determine whether a number is even (as in all odd bases, you can't use the final digit alone to answer that question). For example, 1202112₃ is odd because 1+2+0+2+1+1+2 is odd.
(Edit: deleted a spurious claim about hexadecimal divisibility tests.)
14285 + 7 = 14292; 1+4+2+9+2 = 18; 1+8 = 9
14=2(7) 28=4(7) 56=8(7) 112=16(7) 224=32(7)
[0]: http://www.quora.com/Why-does-the-number-142857-have-such-in...