4 ms·

A lot of numbers have representations in base-10. A fact about the base-10 digits is a fact about the base-10 representation of the number, which is also a fac

A lot of numbers have representations in base-10. A fact about the base-10 digits is a fact about the base-10 representation of the number, which is also a fact about the number. You might be able to satisfy yourself by replacing "the digits of" with "the decimal digits in the base-10 representation of".

codeflo 3y ago
The point is that most of the time when digits are mentioned, it’s only a coincidental fact about one inelegant representation of the number — and often people are insufficiently aware of that.
- bee_rider 3y ago
  Base-10 isn’t inelegant, is it? I mean there are good arguments for 12 being better but it isn’t like 10 is prime or anything. Happened across a neat comment yesterday that presents a defense of ten. Not 100% convinced but it is interesting to see pushback. https://news.ycombinator.com/item?id=39000882 https://news.ycombinator.com/item?id=39000882
  - dr_dshiv 3y ago
    1+2+3+4=10 And you can swear by that, if you know what I mean. * * * * * * * * * *
  - spenczar5 3y ago
    Rational fractions will terminate only if the denominator’s prime factors are the base’s factors. So for example, 1/2 = 0.5 and 1/5 = 0.2, but 1/3 = 0.333… and 1/7 = 0.142857…. 1/4 = 0.25 works because the prime factors of 4 are 2 and 2… but 1/6 fails because 6 is infected by 3. Now, base 12 has 2 prime factors (2 and 3) so it much any better than 10 really. But may I introduce base 30 (235)? Or perhaps base 210 will strike your fancy?
- kelnos 3y ago
  It's only a coincidence if you ignore the fact that "digit" first and foremost refers to the things sticking out of your hands, and so was repurposed to talk about numbers because we have 10 digits on our hands.
  - codeflo 3y ago
    That's the word's origin, not its current mathematical meaning. Also, number systems that are not base IIIIIIIIII have been used historically. That humans can only compute in a base that matches the number of fingers we have is a relatively recent myth.
shermantanktop 3y ago
In some cases, the fact in base-10 has analogous facts in other bases. A trivial example that adding N-1 to any base-N number yields a value with the same digit sum. That makes it a bit more interesting. But I can't think of an example that doesn't pivot on the representation rather than something more fundamental.
- toxik 3y ago
  1+9=10
  - ChainOfFools 3y ago
    i'm not sure what you are demonstrating? 1 sums digitwise to 1 1 + (10-1) = 10 which also sums to 1 in the same way
  - semiquaver 3y ago
    1 = 1+0
  - shermantanktop 3y ago
    Right, the digit sum of 10 is 1...perhaps I should have said "final digit sum." Same for 10000, or 1 with any number of zeroes after it. The point of this trickery is that N-1 added to any number is really adding N (which adds 1 to the second position, by definition) and adding -1 (which subtracts 1 from the first position). In base 10, this is the adding 9 trick. It can be extended by using any multiple of 9. That applies to the N-1 version, so that adding M*(N-1) to a base N number yields the same digit sum. 1+9 = 10 = 1 1 + 27 = 28 = 10 = 1 In hex: 1 + F = 10 = 1 1 + 2D = 2E = 10 = 1

codeflo 3y ago

The point is that most of the time when digits are mentioned, it’s only a coincidental fact about one inelegant representation of the number — and often people are insufficiently aware of that.

bee_rider 3y ago
Base-10 isn’t inelegant, is it? I mean there are good arguments for 12 being better but it isn’t like 10 is prime or anything. Happened across a neat comment yesterday that presents a defense of ten. Not 100% convinced but it is interesting to see pushback. https://news.ycombinator.com/item?id=39000882 https://news.ycombinator.com/item?id=39000882
- dr_dshiv 3y ago
  1+2+3+4=10 And you can swear by that, if you know what I mean. * * * * * * * * * *
- spenczar5 3y ago
  Rational fractions will terminate only if the denominator’s prime factors are the base’s factors. So for example, 1/2 = 0.5 and 1/5 = 0.2, but 1/3 = 0.333… and 1/7 = 0.142857…. 1/4 = 0.25 works because the prime factors of 4 are 2 and 2… but 1/6 fails because 6 is infected by 3. Now, base 12 has 2 prime factors (2 and 3) so it much any better than 10 really. But may I introduce base 30 (235)? Or perhaps base 210 will strike your fancy?
kelnos 3y ago
It's only a coincidence if you ignore the fact that "digit" first and foremost refers to the things sticking out of your hands, and so was repurposed to talk about numbers because we have 10 digits on our hands.
- codeflo 3y ago
  That's the word's origin, not its current mathematical meaning. Also, number systems that are not base IIIIIIIIII have been used historically. That humans can only compute in a base that matches the number of fingers we have is a relatively recent myth.

shermantanktop 3y ago

In some cases, the fact in base-10 has analogous facts in other bases. A trivial example that adding N-1 to any base-N number yields a value with the same digit sum. That makes it a bit more interesting. But I can't think of an example that doesn't pivot on the representation rather than something more fundamental.

toxik 3y ago
1+9=10
- ChainOfFools 3y ago
  i'm not sure what you are demonstrating? 1 sums digitwise to 1 1 + (10-1) = 10 which also sums to 1 in the same way
- semiquaver 3y ago
  1 = 1+0
- shermantanktop 3y ago
  Right, the digit sum of 10 is 1...perhaps I should have said "final digit sum." Same for 10000, or 1 with any number of zeroes after it. The point of this trickery is that N-1 added to any number is really adding N (which adds 1 to the second position, by definition) and adding -1 (which subtracts 1 from the first position). In base 10, this is the adding 9 trick. It can be extended by using any multiple of 9. That applies to the N-1 version, so that adding M*(N-1) to a base N number yields the same digit sum. 1+9 = 10 = 1 1 + 27 = 28 = 10 = 1 In hex: 1 + F = 10 = 1 1 + 2D = 2E = 10 = 1