When you multiply whole numbers, you never get a number like: 64.786…
When you divide numbers, sometimes the answer is something like: 0.5 or 1.3…
0.5 terminates, 1.3… terminates into infinite regress (keeping it rational)
There’s an even finer point to this mystery that is often overlooked. When you place a bar over the 3 to signify its rational repetition, the threes actually regress, technically, at no point are those 3’s actually repeating!!
You seem to think that just because you have learned how to divide into decimals that you actually understand why some numbers terminate and others infinitely regress. I assure you, you have no clue.
You can multiply any two natural numbers but you cannot divide any two natural numbers.
Operations have limits. They cannot operate on any kind of input you feed into them.
You cannot, for example, take (3) apples and divide them in (5) sets of equal number of apples. (\frac{3}{5}) is an impossible operation. But you can say that this impossible operation is equal to some other impossible operation such as (\frac{6}{10}).
When we say (\frac{3}{5}=0.6) that’s all we’re saying: that (\frac{3}{5}=\frac{6}{10}).
Decimal number is simply a fraction of the form (\frac{a}{10^n}).
(\frac{1}{3}) gives you a non-terminating decimal when you try to represent it in decimal form because there is no natural number that you can multiply by (3) and get some power of ten.
Yes, you are right, we are doing division wrong. That must be it. It’s not because division simply does not operate on certain values. No, it’s that we’re doing division wrong because every operation must operate on any kind of value we feed it into it. Nothing should have limits, everything should be without limits, so that when morons such as Ecmandu come along and do whatever they want to do they can never be accused of doing things the wrong way.
I proved that occasionally some odd numbers cause infinite regress in this system, evens never do.
It’s obvious that until someone figures out why these exceptions occur in this system, not just that they do, that these proofs one way or the other are incoherent.
Sure, if you change the definition of ‘even’, then you could get any result you want. But that’s not usually how we think of it (5 isn’t even in base 10, even though it’s also a prime factor of 10).
I pointed out earlier that addition and multiplication, even subtraction of wholes never yield regresses.
3*3 never equals 999999999…
So the fact that you wouldn’t find this issue in factors and such is no surprise - division is inverse to multiplication, yet has very strange properties relative to the inverse of addition and subtraction.
So, my question is… why?
I was trying to point to a clue, since we brought up bases… if base is number, it sets up its lowest common denominator as the even. 2 is the lowest of ten, so no matter what you divide in base 10, that is a multiple of two, will terminate.
In base 3, 2 and 2 add to higher than base, 2 cannot be the lowest common denominator, it has to be 3 or multiples of 3
I don’t disagree, but that isn’t what “even” means. 2 is still even in base 3.
You’re looking at “whole” numbers, by which I take you to mean either integers. And you’re right that there’s an asymmetry between multiplication and division there: multiply two integers and you always get an integer, divide one integer by another and you don’t necessarily get another integer. But you can look at larger and smaller sets of numbers. If you look at just the natural numbers, you see a similar asymmetry between addition and subtraction: add two natural numbers and you always get a natural number, but 3-5 is not a natural number.
Conversely, the rational numbers are closed on division (except for zero), as are the real numbers.
I’m not sure if there are any sets that are closed under division but not multiplication, or subtraction but not addition.
There are as many naturals as negatives, you’re just using an extra operator to separate the two.
There are also as many positive AND negative integers as positive or negative integers…, just depends how you explicate the operators.
But that’s not the point of this discussion.
The big question, and I still maintain that when you switch base, an entirely different psychology is needed about what counts as even, and you divide, that infinite regress kicks in, for some instances of the odd numbers in that base.
This isn’t, nor was it ever a discussion of whether rationals are ordered or not…
This is a discussion about why this method doesn’t terminate if you keep running the method of dividing integers, such that one has to take a step back, convert the decimal to a fraction that it really isn’t anymore, and then add the fractions together and magically call it an equality
You’re not making any sense. In what way do the 3’s at no point ever repeat? You mean on paper or in concept? Of course, they don’t repeat on paper–at least not infinitely–but in concept, of course they do!
I have not just learned to divide, I have understood the reason why division churns out the (rational) number that it does. Take 1 and divide it by 2. You get .5. But this is not just a rule I have learned to blindly follow. I understand that dividing one thing into two equal halves gives you some something that we represent as “0.5”. I know what a “half” is. Just as in long division, every digit you get in the decimal expansion, you can understand why you got that digit. With irrational numbers, I can understand your point (sort of)–you don’t actually know what the full number is (what series of digits the decimal expansions is represented with), but with decimal expansions that terminate or with decimal expansions that repeat (rational numbers), you can understand how each digit is derived.
You’re saying that you understand it because there’s a process that consistently executes it, and then you take a leap, you then say, without any support, that you actually understand WHY this process sometimes causes infinite regress… !!!
Actually, and I made this finer point before, BECAUSE it is a regress issue, technically, the number is irrational!! You’re not actually repeating any of those 3’s!!!
Think of it this way!!
What does 999999999… eventually equal??
But!! Somehow, it makes perfect sense to you that 0.999999… equals 1???
99999 makes sense. 99999… somehow makes sense to you as a rational number???
It’s the same argument behind the decimal point as well.