I couldn’t figure out what the question was asking. Specifically, it asks “what numbers to the terms represent?” In fact, however, the answer indicates that the terms all represent the same number.
If the question had asked me to solve for x and y so that the three terms were equal, that would have been easy.
I’m confused (big shock, I know ;)) but why can’t y be 9 or 12 or 15…
For what it’s worth, I’ve never heard the phrase “counting numbers” before and had to look it up.
Because 3/9 is 1/3, 3/12 is 1/4, 3/15 is 1/5…The only way to get a positive integer (‘counting number’) from 3/y is for y to be 1 (so the answer is 3) or for y to be 3 (so the answer is 1). Anything else leads to a fraction (or negative integer).
Ah, I misread it as y/3 so that y=6 gives 2, y=9 gives 3, etc.
Thanks for the correction.
Millions of people reading their Sunday paper knew intuitively what it meant. It is “numbers that are used for counting objects”. Like 3, or 8, or 6,001. But not numbers like 5.38 or two-and-a-half or minus-3. The numbers children use when counting things.
Well, you didn’t have to look it up, because it explains it in the article, but I’ve only heard the phrase in passing too.
I concede it was more of a “Is there some aspect I’m missing here?” search.
I use intuition with numbers, as well. For instance, if I had .999999 (with an infinite number of 9s following the decimal point) of something, I would say: “I have almost 1 of those things. I’m damn close to having 1, but just not quite there yet.”
When spacecraft are launched, isn’t the standard countdown “T minus 30 seconds, T minus 20 seconds, T minus 10 seconds… <mumbo jumbo about main engine ignition or somesuch>…T plus 10 seconds…”?
0.999… is exactly equal to 1.
Well, okay. If Math is Fun - Maths Resources says so then it must be so. So how does someone count this quantity of tomatoes? Having been, until now, blissfully ignorant of “counting numbers”, I would have counted them as “two and a half tomatoes”. But thanks to what I have learned about the counting numbers, I could just mash the tomatoes and say “1 mess of ketchup”.
This makes no sense to me.
For instance, I do Kakuro puzzles, and unless I really, really get stuck, they’re strictly ‘if-then’ all the way through. I’m doing a proof, just not writing down the steps. If that ain’t logic, I don’t know what is.
And even when I get to a place where I’m stuck, a square might have two different possibilities. I can take one of those possibilities, and either finish the puzzle by if-then logic assuming that possibility, or if-then my way to a contradiction.
And one does this sort of thing in mathematical proofs all the time.
So to me, the implication of what you’re saying is that mathematics is no more than ‘deep guess and check.’ I think we have reductio ad absurdum at this point.
Counting numbers are also called whole numbers (OK, the set of whole numbers also includes zero, but aside from that), and you use them to count whole things. What you’re looking for is the thing you just used when you said ‘two and a half tomatoes’ - fractions. That’s what you use to count parts of whole things.
Or you can mash up the tomatoes, but you need to add a few things to get ketchup.
I think Chronos is specifically talking about Sudoku.
What I was saying applies equally well to sudoku or to kakaru. And it looks like RTFirefly is saying the same thing I did, so I’m not sure where the lack of understanding is.
OK, now I’m really confused about what you meant.
Because there’s little if any similarity in my mind between ‘deep guess and check’ and an if-then proof. In ‘deep guess and check,’ you make a guess and see where it leads you, until either it works out or you trip over your own trail. In the if-then proof, you’re not assuming anything except what’s known.
In a Kakuro puzzle, if I’ve got a two-square row that sums to 17, restricting the potential values of those two squares to 8 and 9 isn’t guessing; it’s a direct implication. And if I’ve got an 8 or a 9 in one of the crossing columns, then I can fill in those two squares without any guesswork.
And when I’m working on a Kakuro puzzle, that generally holds all the way through: no guessing, just ‘given what I already know, I can with certainty restrict the allowable values for this square in the following manner.’
But the question doesn’t say (x/2) and (3/y) are positive counting numbers. It says x and y are positive counting numbers. Therefore, the possibilities are endless and thus the terms represent???
Well, (x-y) represents the set of integers. X/2 represents positive multiples of 1/2, which could be useful in some cases in quantum mechanics. Not sure there’s much else to say about 3/y.
