[kenberg]

These things can be subtle. Some 45 years ago when I was writing my thesis I cam up with a method that helped me prove a result I believed to be true. Then I noticed that this method also quickly established another result. And then another. Then I got suspicious. Then I found my error. It was more subtle than the product of (-2)X(-3). Same general idea though: Stay calm, think it through, then you get it right.

[Winstonm]

It should be a shock to no one that I bareley made it though algebra with my skin intact, so I studied zero mathematical theory. But I have a question.

It appears to me that there are a number of apparently reasonable ways to think about the basic problem presented of (-2)x(-3), including Ken's correct method of removing two -3 debits to create +6 increase in wealth, but it also seems to me just as valid with this same method that Ken used to come to the conclusion that by eliminating the debits, you have arrived at zero, as you had to be at -6 to start in order to have two -3 debts in the first place.

Now, it does make sense from an accounts payable basis, that if business x pays business y by transfering two -3 debts due that business y has gained +6, but why wouldn't that be expressed as 2x(-3)? for business x, and (-2)x3 for business y?

So, to my question - is it pure reasoning that leads to the right method of thinking about the equation, or is correct thinking a product of axiomatic stipulations?

Thanks to anyone with the patience to answer.

[Echognome]

Winston - How about this? You have two line items under accounts payable of $3 each. You then remove them with (-2)x(-$3). What happens to your ending balance?

[wyman]

Yes, Winston, if you're going to be thinking about numbers as being debts you owe and debts you are owed, you should think about the numbers not as absolutes; rather you should consider them as operators on your total balance.

So (-2) x (-3) is the number which operates by removing 2 debts of 3 dollars from your balance. That is, your balance will go from B to B+6. (this is the same effect as if we had just received a +6 to begin with; hence these are the same "number" or "operator on balance")

To someone's other point, that this clearing of debt should net 0. Your reasoning is slightly bad, since you may have started not with a debt of 6 but with a debt of 100. This only serves to show that -2x-3 needn't be 0 (that is, that the logic by which someone arrived there is bad). That -2 x -3 actually equals 6 I claim is evident by the above.

-----------------

Also, if we believe that:
(a) multiplication of whole numbers commutes (that is, a x b = b x a for every pair a, b );
(b )multiplication is associative (order does not matter: (a x b ) x c = a x (b x c))
(c )-1 x a = -a for every positive a
(d) -1 x -1 = 1

then we can show that -2 x -3 = 6 formally, without considering numbers as debts or balances.

Then, -2 x -3 = (-1 x 2) x (-1 x 3) = [since (a) and (b ) allow us to regroup and swap as we wish]
(-1 x -1) x (2 x 3) =
1 x 6 =
6

At the very least, we've reduced the problem to convincing ourselves that -1 x -1 = 1.

edit: freakin' smilies

[Winstonm]

Quote

(a) multiplication of whole numbers commutes (that is, a x b = b x a for every pair a, b );
(b )multiplication is associative (order does not matter: (a x b ) x c = a x (b x c))
(c )-1 x a = -a for every positive a
(d) -1 x -1 = 1

then we can show that -2 x -3 = 6 formally, without considering numbers as debts or balances.

Then, -2 x -3 = (-1 x 2) x (-1 x 3) = [since (a) and (b ) allow us to regroup and swap as we wish]
(-1 x -1) x (2 x 3) =
1 x 6 =
6

Wyman,

Thanks for the explanation - this last makes sense to me, but I note that it is axiomatically-driven.

In other words, to me it appears that the correct answer cannot be - for want of a better word - reasoned out without knowledge of the axiomatic bases. Or perhaps it can be, but it is easier to come to incorrect conclusions if you try to "reason it out" instead of following the logical path provide by the axioms.

[Winstonm]

 Echognome, on 2011-February-09, 16:16, said:

Winston - How about this? You have two line items under accounts payable of $3 each. You then remove them with (-2)x(-$3). What happens to your ending balance?

The ending balance is improved by +6. Let's see, my accounts payable is a negative to my bottom line, so I have two -3's. (O.K. so far.) I remove them...

Hmmm. That language gets a bit subjective. How do I remove them - cook the books and simply erase them, or take in +6 from accounts receivable to negate them? If I negate them with income from accounts receivable, my net profit is zero on that transaction. It is only a +6 profit if my accountant can "make them disappear", wink, wink, nudge, nudge.

I'm kidding - kind of. I see what you are saying, that if the two -3's are eliminated I have gained +6 to my bottom line. At the same time, note the confusion from simply attempting to grasp what exactly is meant by "taking away".

[Echognome]

Winston - The "taking away" part is easy. We suppose that the parties that issued us invoices (and triggered the accounts payable) have canceled those invoices. I don't think you can extend anything to accounts receivable, because those are derived from different underlying transactions (such as our company issuing invoices to a completely different party). We are talking only about a change in AP, not a change in AR.

So let's take this example to your specific questions and continue our hypothetical.

Q: Do I remove them by taking in +6 from accounts receivable?
A: No. This was a cancellation of an invoice issued to our company and does not affect our AR. The party that canceled the invoices is one of our vendors and therefore we do not issue invoices to this party. We have no AR with this vendor at all, so there is nothing to net or offset.

Q: If I negate them with income from accounts receivable, my net profit on that transaction is zero, right?
A: No. They are from completely separate transactions. In addition, any discussion of "profit" should be referred to my profit and loss statement rather than my balance sheet. The only netting that occurs is when I am determining what my current balance is.

Edit: I just thought of a way to summarize. We state the question as:

"What happens to our balance sheet when a vendor cancels two $3 invoices issued to us for payment?"

Answer: Our balance sheet gains $6 as (-2)x(-$3) = $6.

[BunnyGo]

 Winstonm, on 2011-February-09, 18:22, said:

Wyman,

Thanks for the explanation - this last makes sense to me, but I note that it is axiomatically-driven.

In other words, to me it appears that the correct answer cannot be - for want of a better word - reasoned out without knowledge of the axiomatic bases. Or perhaps it can be, but it is easier to come to incorrect conclusions if you try to "reason it out" instead of following the logical path provide by the axioms.

You suggest that the axioms are not "reasonable" (something which can be "reasoned out") in that you have been mentioning them in contrast to each other. Isn't it the case that a student does (or can if given the chance) reason out these "rules" (in fact, isn't that how people found them in the first place?). The axioms of arithmetic are simply what many people have found reasonable and "discovered" and so wrote them down. Likewise the "laws" of physics are simple patterns and mathematics that people have found reasonable.

One reasons out that 2+3 = 3+2 or that 2*3 = 3*2. After enough examples where this pattern emerges, one may find their own reason for it: My favorite is that 2+2+2 = 3+3 by taking one from each 2 for the first 3, and one from each 2 for the second 3.

At the end of the day, I think any explanation which makes sense to you is just as valid and reasoned out as another.

I also think wyman usually explains these things better than I do...

[Winstonm]

 Echognome, on 2011-February-09, 18:57, said:

Winston - The "taking away" part is easy. We suppose that the parties that issued us invoices (and triggered the accounts payable) have canceled those invoices. I don't think you can extend anything to accounts receivable, because those are derived from different underlying transactions (such as our company issuing invoices to a completely different party). We are talking only about a change in AP, not a change in AR.

So let's take this example to your specific questions and continue our hypothetical.

Q: Do I remove them by taking in +6 from accounts receivable?
A: No. This was a cancellation of an invoice issued to our company and does not affect our AR. The party that canceled the invoices is one of our vendors and therefore we do not issue invoices to this party. We have no AR with this vendor at all, so there is nothing to net or offset.

Q: If I negate them with income from accounts receivable, my net profit on that transaction is zero, right?
A: No. They are from completely separate transactions. In addition, any discussion of "profit" should be referred to my profit and loss statement rather than my balance sheet. The only netting that occurs is when I am determining what my current balance is.

Edit: I just thought of a way to summarize. We state the question as:

"What happens to our balance sheet when a vendor cancels two $3 invoices issued to us for payment?"

Answer: Our balance sheet gains $6 as (-2)x(-$3) = $6.

O.K. If I have 6 apples set aside to pay back my apple-debt to Bill Jones, and Bill Jones cancels two 3-apple debts, then those 6 apples are now mine again, and I have a positive 6 apples.

The key here seems to be remembering (if a child) that the (-2) of the (-)x(-) equation represents the number of times an elimination of debt occurs while the (-3) represents the amount of each debt.

[Winstonm]

 BunnyGo, on 2011-February-09, 20:13, said:

You suggest that the axioms are not "reasonable" (something which can be "reasoned out") in that you have been mentioning them in contrast to each other. Isn't it the case that a student does (or can if given the chance) reason out these "rules" (in fact, isn't that how people found them in the first place?). The axioms of arithmetic are simply what many people have found reasonable and "discovered" and so wrote them down. Likewise the "laws" of physics are simple patterns and mathematics that people have found reasonable.

One reasons out that 2+3 = 3+2 or that 2*3 = 3*2. After enough examples where this pattern emerges, one may find their own reason for it: My favorite is that 2+2+2 = 3+3 by taking one from each 2 for the first 3, and one from each 2 for the second 3.

At the end of the day, I think any explanation which makes sense to you is just as valid and reasoned out as another.

I also think wyman usually explains these things better than I do...

I did not mean to imply that rules are not reasonable. I only meant to say that if one relies on trying to figure out for oneself what seems to make sense, then there are many ways to go wrong - it is indeed logical and reasonable to follow the axiomatic pathway.

[gwnn]

3x(-6)=(-6)+(-6)+(-6)
2x(-6)=(-6)+(-6)
1x(-6)=(-6)
0x(-6)=0 (no -6's at all)
So you get from one number of -6's to the previous number of -6's by eliminating them. When you get into the negative numbers domain, you have to start substracting them, because there were none added to dispose of.
(-1)x(-6)=-(-6)
(-2)x(-6)=-(-6)-(-6)
(-3)x(-6)=-(-6)-(-6)-(-6)

[manudude03]

A lot of the understanding in Mathematics as others have said is that you need to go on gradually. I remember in my final year of school, I was in a class of 4 doing Further Mathematics and we were doing De Moivre's Theorem. My teacher also had to cover a class of 4th years so we were all in the same room. That class would have been lucky if they had even heard of complex numbers (not taught until final year), let alone some of the theory behind it. About 15 minutes into the class, one of the fourth years asked "Sir, what on earth is all this on the board?". Rather than try to explain it, the teacher just replied "Only the hardest stuff you'll learn in school". Of course, that is just another way of saying "If you don't understand it, don't worry about it "

One thing I have always been critical of is that schools generally teach you how to pass exams instead of how to think. That's probably why my final year was 10 times better than any of my other years since we would continuously be asked to prove stuff without the teacher's help, and their job was basically to teach us any new theories and address any other problems we had. Myself and 2 of my classmates tried for the British Maths Olympiad in our final year, all of us failing miserably. It was probably my favourite "exam" ever since the conditions were so relaxed, we were allowed to chat to each other (but not about the questions ), have food+drink etc. The first question, which was supposed to be a warmup was:

"Find 4 prime numbers less than 100 which are factors of 3^32 - 2^32."

If you know how to do this, it takes less than 5 minutes to get all 4 numbers. One of my classmates hadn't realised the significance of 32 and ended up doing 3x3x3x3x3x3x3x3x3x3x3x3x3x3x3x3x3x3x3x3x3x3x3x3x3x3.........., might have been 9x9x9x9x9.... to make it faster, can't remember. 2 hours later (no calculator allowed), we just said to him "you do know you're doing this the hard way, right?" 30 mins after that, it clicked and he found 3 of the numbers. If you really care, 3^32 - 2^32 = 1,853,015,893,884,545.

A few weeks later, that problem was given to the 4th year class mentioned above. To make it "easier" we let them use calculators and let them discuss how to go about doing it. Collectively they didn't get close, they guessed the lowest. I specifically told them it was just a topic they would have done already in disguise, that didn't help.

I really only the latter story because it is a pretty basic problem in disguise, yet unless you recognise it and know how to do it, you will be struggling. The fact that the three of us were the best Maths students in the school and I was the only one to get all 4 numbers (one of them I believe is doing a postgraduate Maths degree at Cambridge) just shows how difficult the easy stuff can be.


A couple of off-topic points:

1. Would anyone laugh if a first grade (or first year of primary) when being taught how to count asked their teacher to prove that is the order?

2. I was thinking about statistics earlier and have come to the conclusion that statistically statistics are statistically irrelevent.

[gwnn]

is it like

3**32-2**32=(3**16+2**16)(3**8+2**8)(3**4+2**4)(3**2+2**2)(3+2)=(big number)(big number)*97*13*5

I suppose the second big number is not so big.

2**8= 256
3**8=6561
---------
2**8+3**8=6817

aha 6817=17*401!

so 5, 13, 17, 97!

This post has been edited by gwnn: 2011-February-10, 05:29

[manudude03]

Yep, that's the proper way (there is some other theorem by Fermat to get 17 too)

[gwnn]

Yes so the more rules and theorems you know, the easier maths problems become. Fair enough and true as well, but it seems to be the opposite of winston's point applying new and nifty formulas is not my idea of thinking for yourself.

[gwnn]

I think 1, 2, 3, 4, etc each are just defined to be the successor of the number one less than them, therefore you don't have to prove they are in this order. So yes, I would LOL right in their face!

[manudude03]

New and nifty formulas? It's just basic algebra ( x^2-y^2=(x+y)(x-y) ). When I was doing it, I just did the 3^8+2^8 myself, only learnt of the other theorem afterwards, and still haven't found another use for it

[gwnn]

I meant to say that it's way too advanced for winstonm

[Winstonm]

 gwnn, on 2011-February-10, 03:35, said:

3x(-6)=(-6)+(-6)+(-6)
2x(-6)=(-6)+(-6)
1x(-6)=(-6)
0x(-6)=0 (no -6's at all)
So you get from one number of -6's to the previous number of -6's by eliminating them. When you get into the negative numbers domain, you have to start substracting them, because there were none added to dispose of.
(-1)x(-6)=-(-6)
(-2)x(-6)=-(-6)-(-6)
(-3)x(-6)=-(-6)-(-6)-(-6)

Finally, someone who talks sense.

[Winstonm]

 manudude03, on 2011-February-10, 06:06, said:

New and nifty formulas? It's just basic algebra ( x^2-y^2=(x+y)(x-y) ). When I was doing it, I just did the 3^8+2^8 myself, only learnt of the other theorem afterwards, and still haven't found another use for it

To some of us, basic algebra is an oxymoron.