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Unsensible Rules/Laws of negative number operations 
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#19
May1614, 09:27 PM

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#20
May1614, 09:34 PM

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#21
May1614, 09:36 PM

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#22
May1614, 09:43 PM

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That is a silly argument, people at the bank have knowledge of algebra! Who would dig out an algebra book in the middle of a banking transaction? 


#23
May1614, 09:54 PM

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What I don't understand is why you would get this book, and then pretty much ignore what it says, substituting your own "rules" in place of how the book is presenting them. The rules for addition, subtraction, multiplication, and division (the four arithmetic operations), work the same whether you're working with natural numbers, integers, rational numbers, and real numbers. If you're seeing inconsistencies, it's because of problems in your understanding. 


#24
May1614, 09:55 PM

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Take Femat's Last Theorem. The final proof takes well over 100 pages and will require at least half a decade of formal study to be able to parse. 


#25
May1614, 10:04 PM

P: 50

Axioms, correct me If I am wrong is not absolutely true but highly likely that is true. Well mostly because it is self evident. And also I must correct my disdain on the number line. I recognize the importance of it just this day so thanks.



#26
May1614, 10:10 PM

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The arithmetic of the integers does indeed say things about negative integers. You seem to have this strange idea that "arithmetic = natural numbers" and "algebra = everything else". We are telling you that is not true. Algebra has two meanings in mathematics: elementary algebra is the study of the properties of polynomials, abstract algebra is the study of sets with nary operations. Neither is being talked here. 


#27
May1614, 10:17 PM

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#28
May1714, 12:06 AM

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The great mathematician Gelfand has written an algebra book accessible for high schoolers even. In there, he explains why we adopt these rules. So I'm going to steal a bit of his argumentation here, I do highly encourage everybody to check out his wonderful book!
Firstly, math is usually presented by the use of axioms. Axioms are always true statements for the purpose of the theory. However, they might not be applicable in real life. For example, you might easily adopt the axiom that ##(15)\cdot (3) =  45##. This is a perfectly allowed axiom, but it will be totally useless. In math, you do get to make up whatever rule you wish, but if the rule is not applicable to real life, then it's a useless rule that nobody is going to adopt. Of course, this means that we know have the burden to explain to you why ##(15)\cdot (3) = 45## is such a useful thing. There are various answers. The answers you have been getting here are all coming from abstract algebra. There you have a set of axioms (which I will call the field axioms) and then you derive from there that a negative times a negative is a positive. Of course, this is not entirely satisfactory because then we must give some argumentation of why the field axioms are so useful. Well, on the one hand the field axioms are a really natural extension of the arithmetic of natural numbers and they are also really elegant. Any other axiom system would violate the field axioms and would be far less natural! Of course, this is not the explanation that was given by mathematicians back in time because they knew nothing of field axioms. Let's look at some other explanations. First on the addition problem. We can see 3+5=8 as an abstraction of the following problem: Yesterday it was 3 degrees and today it is 5 degrees warmer, thus today it is 8 degrees. So what is (3) + 5? Well, Yesterday it was 3 degrees and today it is 5 degrees warmer, if you look at a temperature scale, then you will easily see that today must be 2 degrees. So what about 3 + (5)? Yesterday it was 3 degrees and today it is 5 degrees warmer, meaning it is 5 degrees colder. Thus it must be 2 degrees now. And then finally (3) + (5)? Yesterday it was 3 degrees and today it is 5 degrees colder, thus it must be 8 degrees now. Curiously enough, there is also an experimental way to check addition with negative numbers. Nature has provided us with such a tool called antiparticles. So adding up two particles/antiparticles is just regular addition. So we get 3 protons combined with 5 protons is 8 protons. 3 protons combined with 5 antiprotons yields 2 antiprotons (+ gamma radiation which we will ignore) 5 protons combined with 3 antiprotons yields 2 protons and 5 antiprotons combined with 3 antiprotons yields 8 antiprotons. So we see that our use of addition and our rule of ##3+(5) = 2## is actually applicable in physics even. This applicability should give us the information that our rule is useful and thus the axioms are valid. Now what about multiplication? Things like ##3\cdot 5## can be seen as repeated addition. So we have ##5+5+5## or ##3+3+3+3+3##. If we accept this rule of repeated addition to be true for negative numbers also, then we will get [tex](3)\cdot 5 = (3) + (3) + (3) + (3) + (3) = 15[/tex] and we will also get [tex]3\cdot (5) = (5) + (5) + (5) = 15[/tex] which is forced upon us by the repeated addition. Now, what about ##(3)\cdot (5)##? The repeated addition analogy will fail this time because I have no idea how to express addition of ##3## a ##5## times. Here are some arguments for why it must be ##15##. First, a very weak argument. But we have ##3\cdot 5 = 15##, ##(3)\cdot 5 = 15## and ##3\cdot (5) = 15##. So we already had two negative outcomes and one positive outcome. If mathematics were politically correct, then the outcome should be ##15##. Of course, why should mathematics be politically correct? Second, an argument by Euler in his book "Elements of Algebra". He considers it to be clear that ##(3)\cdot (5)## must be either ##15## or ##15##. Now, if we were to accept the answer of ##15##, then ##(3)\cdot (5) = (3)\cdot 5##. So we are in the situation ##5x = 5x##. The rules of arithmetic should give us ##5 = 5## which is insane. Of course, this approach is just a veiled use of the field axioms. Third. You might like the following analogy: Getting ##5## dollars ##3## times is the same as getting ##15## dollars. Thus ##5\cdot 3 = 15##. Paying a ##5## dollar fine (= getting ##5## dollars) ##3##times is the same as paying ##15## dollars. Thus ##(5)\cdot 3 = 15##. Not getting ##5## dollars ##3## times is not getting ##15## dollars. Thus ##5\cdot (3) = 15##. Finally, the case we're interested in. Not getting ##5## dollars (= not paying a ##5## dollar fine) ##3## times is the same as you getting ##15## dollars. Thus ##(5)\cdot (3) = 15##. Finally, let us write multiplication by ##3## in the following way: [tex]\begin{array}{cccccc} 1 & 2 & 3 & 4 & 5 & ...\\ 3 & 6 & 9 & 12 & 15 & ...\end{array}[/tex] We already agreed that the use of repeated addition forces upon us the rule ##3\cdot (5) = 15##. So we can extend the previous: [tex]\begin{array}{ccccccccccc} ... & 3 & 2 & 1 & 0 & 1 & 2 & 3 & 4 & 5 & ...\\ ... & 9 & 6 & 3 & 0 & 3 & 6 & 9 & 12 & 15 & ...\end{array}[/tex] So we see that multiplication behaves rather nicely. To get the next number in the sequence, we just add up ##3##, to get the previous number, we subtract ##3##. Let us look at multiplication by ##3##: [tex]\begin{array}{cccccc} 1 & 2 & 3 & 4 & 5 & ...\\ 3 & 6 & 9 & 12 & 15 & ...\end{array}[/tex] So in this case, we just get multiplication by ##3## is mirrored. To get the next number in the sequence, we add up ##3##, to get the previous number, we subtract ##3##. So continuing this sequence would give us: [tex]\begin{array}{ccccccccccc} ... & 3 & 2 & 1 & 0 & 1 & 2 & 3 & 4 & 5 & ...\\ ... & 9 & 6 & 3 & 0 & 3 & 6 & 9 & 12 & 15 & ...\end{array}[/tex] So we see that the rule ##(3)\cdot (5) = 15## is also forced upon us. But again, making a number system which does have ##(3)\cdot (5) = 15## is perfectly possible. But then you would have the previous points to be wrong which would make the number system pretty useless. Also, you would not have things like complex numbers which are very useful for physics. The way that physics, engineering, biology, chemistry, etc. makes use of our laws of mathematics and it yields good theories which are correct should tell you something. It should tell you that our rules are good rules. 


#29
May1714, 01:05 PM

P: 50

Very insightful post Micromass. I. M. Gelfand and A. Shen, Algebra is this the book you are talking about? And by the way I see your point on now on why axioms are sufficient to prove mathematical theories since it works on nature itself. What book can anyone suggest for abstract algebra?



#30
May1714, 01:24 PM

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P: 18,277

The second stage is deducing theorems from the axioms. Now we are completely surrendered to deductive reasoning. The only results we now accept as true are the axioms and everything we can logically deduce from the axioms. The axioms might not be true in the real world however, but we don't care in the second stage. We are working in an alternate universe where the axioms are true. There is a lot of freedom in mathematics in the sense that we are free to choose the axioms and definitions. But once we have done this, we must play by the rules. Finally, notice that the previous two stages are not exactly the way mathematics is done professionally. The way it's done is not first selecting axioms and then seeing what we can possibly deduce from them. What we have in mind is first the results we want to prove, and then we find an axiom system that makes it true. An example will make it clear. Addition on the natural numbers can be axiomatized by demanding things like ##a+b=b+a##. But what we actually want to do is to find axioms which give our usual sense of numbers and of addition like ##10+2 = 12##. That's what our goal is. The axioms are chosen so that we reach our goal. If the axioms turn out to give results like ##1+1=3##, then we would reject the axiom system (it might still be interesting to study it, but it wouldn't reflect how we think of reality). But again, as soon as the axiom system is fixed, the results that we obtain are fixed too. 


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