Community Messageboard: The "leading zero" thing IS a bit misleading. by Joel

Back to Messageboard

Post New Thread

Help

Stats

Rules

The "leading zero" thing IS a bit misleading. Joel Send a noteboard - 26/05/2010 11:06:59 AM

And you're doing a good job being helpful in that regard, thanks.

Always good to think of the pythagorean thereom as a way to do subtraction in more than one dimension

Yeah, that I can handle. Maybe I should delve a bit more to see what's actually going on but the notation is truly daunting for me. It's a bit like philosophy where you have to take a couple courses to define the terms before you can being the real study.

'Daunting Notation' in my experience stops being daunting as soon as people stop trying to memorize and absorb the notation and instead stop and ask 'why do they use this notation?', which is often where knowing the history of the field, especially that specific problem, tends to help - along with a vague knowledge of latin and greek usually. You're almost always, when learning a new math or physics tool, better off looking up the original problem it was meant to solve and trying to work it out yourself for a bit, until even though you haven't solved it you have a strong familiarity with what actually is the problem, then you look at the solution and suddenly the notation makes obvious sense. Take a previous modular arithmetic example. the whole 'modular' and 'modulus' bit in there is just using 'module' in the sense of individual piece with its own distinct importance. That modulus is all that really matters, it's distinct and relevant existence to the problem at hand is why you are even using the method. The notation is just an easier and clear way of writing it in general form. 14 = 2 (mod 12) is simply how you'd write in math notation 'The fourteenth hour of the day looks the same on a clock as the second hour, or the 26th or 38th or 50th.' Of course it would be kind of case specific to just write 14=2 (clock) or 17=10=3 (week) or 367=2 (year) all of which everyone would recognize instantly but would instead be written as 14=2 (Mod 12), 17=3 (Mod 7), and 367=2 (Mod 365)... most notation is like that, just common sense terminology agreed on to be easier to read, not confusable in context with something else, and general enough to be usable for non obvious examples.

Lot to be said for that; it also helps to avoid pitfalls that have become common place due to the prevalence of people promoting things they don't properly grasp (e.g. the original logical positivists took God off the table as unverifiable, not nonexistent, but many modern heirs, or errs, opt to "simplify" that to the latter case. ) I should've made better use of that decade of free time with which I indulged myself, but had I done so I wouldn't have had a job. Something of a Catch 22, I suppose.

Well, most people pick up areas of knowledge as time and interest permit, but the problem with math is it requires a fairly intense initial study that's hard to do on your own, you can self-learn pretty well once you have down basics things like calculus but it's very hard to learn things with out a personal mental connection, I find the history and background of a given problem make the best route, it's sort of like learning how to use basic mechanical tools, a lot of people have problems figuring out which tool is which but if you stop and show them what is for and explain why it is called that, they'll never forget, whereas pointing and calling it a quarter inch socket wrench five or six times will just help them kinda remember for a little while. People do the same thing with math, science, and legal terms.

Well, they're taking the position that 142857 is the only non-trivial case of a decimal without leading zeroes that repeats. Out of an infinite series of integers it seems to be in a class by itself.

That 'no leading zeroes' thing is sort of rubbish though. The reason nothing else has a cyclic behavior without a leading zero is because in base 10 it is impossible to divide 1 by any number bigger than 10 without getting 0.0XXXX, you can't get less than .1 without having a zero, and you obviously can't get a number bigger than .1 when your divisor is greater than 10. Plenty of numbers cycle, 1/81 cycles, it's just that 81 is bigger than 10 so in base 10 it has to have a zero following the radix point. (Decimal point is just the name for the base 10 radix point), so it's absurd to even say that out of an infinite series of integers none else cycle without a zero because only the 0-9 in decimal can produce a number whose reciprocal is greater than .1, it's unique value in that respect is purely an artifact of base 10

It's less that only 7 produces such a value than it is that no other value does that without leading zeroes in base ten. Understanding why does take out some of the mystery, but it still seems odd that one and only one of eight digits (since the reciprocal of one is one and zero doesn't have one, being more the absence of value than a value itself) produces such a number. Maybe we should look for multiple non-trivial cyclic numbers in hex.

Well, again, how could any number over ten not have a leading zero? You can only not have a leading zero if the result is more than a tenth and there are only 9 numbers that can do that, 1-9. Each of those 9 values has something kinda interesting about it. 3 and nine are the only number giving a single repeater, .33333... and .1111111... and of course that's not a coincidence that 9 should do that since 3 does. Powers of 2 only have a set number of decimals and there are three of them in 1-9, 1's not really a very interesting reciprocal, that only leaves 5, 6 and 7 to expect anything interesting on, 5 is boring, just .2, so only 6 and 7, and 6 endlessly repeats after the leading 1, as .16666666666. So it's not that weird of a coincidence that only one number does this.

1/11 repeats, it just repeats .09090909090909, so in hex that becomes .1745D1745D1745, a constant repetition of 1745D (95,325 in base ten), no leading zero, since in hex any number 0-15 has a reciprocal greater than .1

There really isn't much use of 'decimals' in other bases, because you'd get something that looks like Pi = 3.243F6A8885, 1/7 is actually easier to remember in base 16 because 1/7=.249249249...249... etc

Like I said, cyclic decimals of prime numbers is very common, you just won't get them without a leading zero unless you're in a base bigger than they are. 1/13 = 0.0769230, that repeats as 13B in hex, .13B13B13B etc

There's nothing special about 7 doing this, just numerological straw-grasping.

If you note, the original article isn't treating with values <1 at all, but integers; it's only when we consider the matter in terms of reciprocals that the mystery of why 142857 repeats when multiplied by any other number and no other "non-leading zero" number does. Until then it DOES seem a little disturbing that any integer will produce a different number when multiplied by another--EXCEPT 142857, which ALWAYS produces a variation of itself. On the other hand, you could say that the product of any number and another is "a variation" of both.

Honorbound and honored to be Bonded to Mahtaliel Sedai
Last First in wotmania Chat
Slightly better than chocolate.

Love still can't be coerced.
Please Don't Eat the Newbies!

LoL. Be well, RAFOlk.

Reply to message

Continuing the Math Theme, WTF Is Up with the Seven? Joel - 25/05/2010 02:12:09 AM 650 Views

Base 10 number system Birdeye - 25/05/2010 02:16:49 AM 487 Views

I Kinda Figured, but Why Does 7 Conflict with Base 10 So Much? Joel - 25/05/2010 02:27:51 AM 478 Views

Number theory holds a concept called "cyclic numbers." Dreaded Anomaly - 25/05/2010 06:04:43 AM 540 Views

"If leading zeros are not permitted on numerals, then 142857 is the only cyclic number in decimal. " Joel - 25/05/2010 06:27:36 AM 545 Views

Maria - 25/05/2010 08:19:12 AM 481 Views

Happy Birthday...?

Joel - 25/05/2010 08:56:59 AM 415 Views

You're a day early

Maria - 25/05/2010 09:10:30 AM 430 Views

Got me on a technicality. Joel - 25/05/2010 09:19:54 AM 331 Views

Re: Got me on a technicality. Maria - 25/05/2010 09:27:59 AM 370 Views

But we're AWESOME! Joel - 25/05/2010 09:47:30 AM 471 Views

Maria - 25/05/2010 09:50:43 AM 348 Views

Number THEORY is great fun, but too many folks make math too tedious, I'm afraid.

Joel - 25/05/2010 09:53:21 AM 386 Views

You tend to get cyclic repeats when dividing by primes Isaac - 25/05/2010 11:58:34 AM 564 Views

I actually DO feel like taking a course on number theory. Joel - 25/05/2010 12:12:10 PM 523 Views

Re: I actually DO feel like taking a course on number theory. Isaac - 25/05/2010 01:46:10 PM 365 Views

Well, I'm not vouching for Wikipedias claim, just reiterating it. Joel - 25/05/2010 02:35:23 PM 581 Views

It's usually right but I wouldn't but much value on the implied importance Isaac - 25/05/2010 04:22:52 PM 426 Views

If you say so; I really try hard not to channel Pythagoras.

Joel - 26/05/2010 09:26:22 AM 595 Views

Re: If you say so; I really try hard not to channel Pythagoras.

Isaac - 26/05/2010 10:00:37 AM 542 Views

The "leading zero" thing IS a bit misleading. Joel - 26/05/2010 11:06:59 AM 675 Views

Re: The "leading zero" thing IS a bit misleading. Isaac - 26/05/2010 12:33:08 PM 398 Views

i have a book you're welcome to try reading.... moondog - 26/05/2010 07:30:46 AM 357 Views

Yeah, I keep getting much that impression. Joel - 26/05/2010 09:07:41 AM 339 Views

Why is 6 afraid of 7?

*NM* Dark Knight - 25/05/2010 02:15:59 PM 187 Views

There it is again! Because 7 is a gluttonous beast. Joel - 25/05/2010 02:27:58 PM 401 Views

Back to Messageboard

Post New Thread

Help

Stats

Rules

Reply to Message

Subject

Formatting

Smiley Codes

Body

<quote id="111220">
<quote id="111214"><quote id="111204">And you're doing a good job being helpful in that regard, thanks. :)</quote>

Always good to think of the pythagorean thereom as a way to do subtraction in more than one dimension ;)

<quote id="111204"><quote id="110927"><quote id="110891">Yeah, that I can handle.  Maybe I should delve a bit more to see what's actually going on but the notation is truly daunting for me.  It's a bit like philosophy where you have to take a couple courses to define the terms before you can being the real study.</quote></quote>
<quote id="111204">'Daunting Notation' in my experience stops being daunting as soon as people stop trying to memorize and absorb the notation and instead stop and ask 'why do they use this notation?', which is often where knowing the history of the field, especially that specific problem, tends to help - along with a vague knowledge of latin and greek usually. You're almost always, when learning a new math or physics tool, better off looking up the original problem it was meant to solve and trying to work it out yourself for a bit, until even though you haven't solved it you have a strong familiarity with what actually is the problem, then you look at the solution and suddenly the notation makes obvious sense. Take a previous modular arithmetic example. the whole 'modular' and 'modulus' bit in there is just using 'module' in the sense of individual piece with its own distinct importance. That modulus is all that really matters, it's distinct and relevant existence to the problem at hand is why you are even using the method. The notation is just an easier and clear way of writing it in general form. 14 = 2 (mod 12) is simply how you'd write in math notation 'The fourteenth hour of the day looks the same on a clock as the second hour, or the 26th or 38th or 50th.' Of course it would be kind of case specific to just write 14=2 (clock) or 17=10=3 (week) or 367=2 (year) all of which everyone would recognize instantly but would instead be written as 14=2 (Mod 12), 17=3 (Mod 7), and 367=2 (Mod 365)... most notation is like that, just common sense terminology agreed on to be easier to read, not confusable in context with something else, and general enough to be usable for non obvious examples. </quote></quote>
<quote id="111204">Lot to be said for that; it also helps to avoid pitfalls that have become common place due to the prevalence of people promoting things they don't properly grasp (e.g. the original logical positivists took God off the table as unverifiable, not nonexistent, but many modern heirs, or errs, opt to "simplify" that to the latter case. ) I should've made better use of that decade of free time with which I indulged myself, but had I done so I wouldn't have had a job.  Something of a Catch 22, I suppose. :P</quote>

<quote id="111204"><quote id="110927"><quote id="110891">Well, they're taking the position that 142857 is the only non-trivial case of a decimal without leading zeroes that repeats.  Out of an infinite series of integers it seems to be in a class by itself.</quote></quote>
<quote id="111204">That 'no leading zeroes' thing is sort of rubbish though. The reason nothing else has a cyclic behavior without a leading zero is because in base 10 it is impossible to divide 1 by any number bigger than 10 without getting 0.0XXXX, you can't get less than .1 without having a zero, and you obviously can't get a number bigger than .1 when your divisor is greater than 10. Plenty of numbers cycle, 1/81 cycles, it's just that 81 is bigger than 10 so in base  10 it has to have a zero following the radix point. (Decimal point is just the name for the base 10 radix point), so it's absurd to even say that out of an infinite series of integers none else cycle without a zero because only the 0-9 in decimal can produce a number whose reciprocal is greater than .1, it's unique value in that respect is purely an artifact of base 10</quote></quote>
<quote id="111204">It's less that only 7 produces such a value than it is that no other value does that without leading zeroes in base ten.  Understanding why does take out some of the mystery, but it still seems odd that one and only one of eight digits (since the reciprocal of one is one and zero doesn't have one, being more the absence of value than a value itself) produces such a number.  Maybe we should look for multiple non-trivial cyclic numbers in hex. :P</quote>

1/11 repeats, it just repeats .09090909090909, so in hex that becomes .1745D1745D1745, a constant repetition of 1745D (95,325 in base ten), no leading zero, since in hex any number 0-15 has a reciprocal greater than .1

There really isn't much use of 'decimals' in other bases, because you'd get something that looks like Pi = 3.243F6A8885, 1/7 is actually easier to remember in base 16 because 1/7=.249249249...249... etc

Like I said, cyclic decimals of prime numbers is very common, you just won't get them without a leading zero unless you're in a base bigger than they are. 1/13 = 0.0<u>769230</u>, that repeats as 13B in hex, .13B13B13B etc

There's nothing special about 7 doing this, just numerological straw-grasping.</quote>
If you note, the original article isn't treating with values <1 at all, but integers; it's only when we consider the matter in terms of reciprocals that the mystery of why 142857 repeats when multiplied by any other number and no other "non-leading zero" number does.  Until then it DOES seem a little disturbing that any integer will produce a different number when multiplied by another--EXCEPT 142857, which ALWAYS produces a variation of itself.  On the other hand, you could say that the product of any number and another is "a variation" of both.
</quote>

Link title

Link url

Send notification emails?

Username

Password