Unsensible Rules/Laws of negative number operations

[fde645]

First of all I must make a claim. Rules are shortcuts. Shortcuts that people use to not bother in understanding the complete picture of something. And this includes the rules of calculation between negative and positive numbers. Furthermore, I believe that these rules are counterintuitive or perhaps not counterintuitive, but it fails to explain what it means.
Rule 1: When adding numbers with unlike signs, add the absolute value of the two numbers and copy the sign of the larger number.

Okay, but I just can't explain to myself how 5-20 is the same as 5+ -20= -15.

Moreover, why is 5-(-4) equivalent to 5+4= 9

Furthermore, -5 x -4 =20

This kind of thinking its fine at first. But if you apply it to real world problems, it just won't make sense.
How can I make sense of this. And also I can't find a book that tries to explain this.

 

[SteamKing]

http://en.wikipedia.org/wiki/Multiplication

First of all I must make a claim. Rules are shortcuts.

That's your claim. However, rules are rules. A rule is "one of a set of explicit or understood regulations or principles governing conduct within a particular activity or sphere." One should not confuse a rule with a shortcut; they are two different animals.

Shortcuts that people use to not bother in understanding the complete picture of something. And this includes the rules of calculation between negative and positive numbers. Furthermore, I believe that these rules are counterintuitive or perhaps not counterintuitive, but it fails to explain what it means.

You're all over the place here. Either something is counterintuitive or it is not. It cannot be both.
Sure, some people take shortcuts, but one should not confuse a shortcut with a rule.

Rule 1: When adding numbers with unlike signs, add the absolute value of the two numbers and copy the sign of the larger number.

http://en.wikipedia.org/wiki/Addition

Okay, but I just can't explain to myself how 5-20 is the same as 5+ -20= -15.

Are you familiar with the concept of the 'number line'? I believe that the number line can be used to explain 5 - 20 = -15 quite clearly.

Moreover, why is 5-(-4) equivalent to 5+4= 9

Number line again.

Furthermore, -5 x -4 =20

http://en.wikipedia.org/wiki/Multiplication

This kind of thinking its fine at first. But if you apply it to real world problems, it just won't make sense.
How can I make sense of this. And also I can't find a book that tries to explain this.

It seems to make sense to everyone who works in the sciences and engineering.

There are many books out there, both on arithmetic and algebra, which explain how addition, subtraction, and multiplication work with positive and negative numbers.

 

[symbolipoint]

These claims are just plain wrong. The shortcuts you describe are about how to use properties of numbers, and these properties become individually internalized upon careful study and practice. The descriptions about using those rules are given as help in applying them upon beginning to perform practice problems. The real focus is on learning to know and to use the properties of numbers.

You gave an example of 5+(-20) and 5-20. A picture can be made which allows us to more clearly understand addition or subtraction of signed numbers. We make a number line. We can also perform addition or subtraction on a number line. Adding positive numbers on a number line moves rightward. Adding any negative number on a number line moves leftward, which is essentially a subtraction.

To summarize in short, rules about numbers are generalizations which are true about numbers.

 

[berkeman]

First of all I must make a claim. Rules are shortcuts. Shortcuts that people use to not bother in understanding the complete picture of something. And this includes the rules of calculation between negative and positive numbers. Furthermore, I believe that these rules are counterintuitive or perhaps not counterintuitive, but it fails to explain what it means.



Rule 1: When adding numbers with unlike signs, add the absolute value of the two numbers and copy the sign of the larger number.

Okay, but I just can't explain to myself how 5-20 is the same as 5+ -20= -15.

Moreover, why is 5-(-4) equivalent to 5+4= 9

Furthermore, -5 x -4 =20

This kind of thinking its fine at first. But if you apply it to real world problems, it just won't make sense.
How can I make sense of this. And also I can't find a book that tries to explain this.

I learned the first two "rules" by being shown how a 1-dimensional "Number Line" works for explaining addition and subtraction. I think that was in first grade, IIRC., and it has made good sense to me since.

I'm not sure what the geometric motivation is for the 3rd multiplication rule, but I'm sure somebody will be able to explain it.

EDIT -- beat out by the speedy typists with lots more math knowledge than I have!

 

[fde645]

Guys. Addition is simply a combination of two parts to make up a whole. Now if one positive and negative numbers are to be combined why is it that the sum is a negative. And also, why is 5-20=5+(-20).

 

[fde645]

And when I claim rules to be shortcut I really meant it. Take for example, x+5=9, subtract both sides by -5 to transpose 5, is simply a rule, a shortcut "that people have invented even if you don't know what your trying to do." --Richard Feynman. There is no shortcut to geometry, as said by euclid, but this applies not only in geometry but the whole field of mathematics, then I propose that there should not be any generalization, shortcut, rule or whatever, in the explanation to some concept. When you add two numbers with unlike signs, and subtract their absolute value and copy the sign of the larger number, it does not explain why you should do it. And why addition, which is combination of two parts, numbers, whatever, when in terms of negative numbers suddenly became the opposite. Then because I cannot come up with an explanation and the number line is not my way in explanation basic arithmetic operations, please help me.
And the number line is not a very good model, in fact, in explaining anything. It only produces confusion and anxiety.

 

[gopher_p]

Guys. Addition is simply a combination of two parts to make up a whole.

As a definition of addition, this statement is fairly meaningless from a mathematical standpoint. It may be true that addition of positive integers somehow corresponds well with the real-world activity of combining two things into a larger thing, and that's basically how it works for positive integers. But when you start talking about negative numbers and fractions and what-not, that analogy starts to break down to the point of being worthless.

Now if one positive and negative numbers are to be combined why is it that the sum is a negative. And also, why is 5-20=5+(-20).

This is basically true by definition. Essentially you define the collection of symbols "-20" to denote the unique number such that when added to the number denoted by "20" gives a sum of "0", where "0" denotes the unique number such that when added to any number "x" results in a sum of "x". Then you define subtraction by x-y=x+(-y). There's no magic or proof or "natural" reason; it's that way because we say so. It turns out that this mathematical definition is particularly useful in making real-world problems easier to manage, and so we go with it.

And when I claim rules to be shortcut I really meant it.

If you really mean it, then you should be able to tell us exactly what you mean by "rule" and "shortcut" and justify why we should believe your assertion that rules are shortcuts. Furthermore you should be able to explain why your assertion has anything to do with mathematics. I would strongly encourage you to consider this your primary task before all others. You're not likely to find anyone here willing to consider your opinions even remotely seriously (aside from soundly refuting them) until you do so.

Take for example, x+5=9, subtract both sides by -5 to transpose 5, is simply a rule, a shortcut "that people have invented even if you don't know what your trying to do." --Richard Feynman.

It is a "rule" that whenever x+y=z you also have x=z+(-y). This is related to the "rules" that (i) whenever a=b, you also have a+c=b+c, (ii) that (a+b)+c=a+(b+c), that (iii) a+(-a)=0, and (iv) a+0=a. While it's a matter of philosophy whether or not these rules were invented or discovered (and therefore outside of the realm of discussion on this forum), I can assure you that they are well understood by any mathematician and understandable by anyone who bothers to take the time to understand them.

There is no shortcut to geometry, as said by euclid, but this applies not only in geometry but the whole field of mathematics, then I propose that there should not be any generalization, shortcut, rule or whatever, in the explanation to some concept. When you add two numbers with unlike signs, and subtract their absolute value and copy the sign of the larger number, it does not explain why you should do it. And why addition, which is combination of two parts, numbers, whatever, when in terms of negative numbers suddenly became the opposite. Then because I cannot come up with an explanation and the number line is not my way in explanation basic arithmetic operations, please help me.

There are explanations, but the rabbit hole is far too deep to cover it all. I didn't see any proper explanations for a lot of simple algebra until I was a grad student. At some point, you're going to need to take something as given, without any further explanation. The basic fact is that math does work, there are good reasons for why it works, and it does a darn good job of making peoples' lives easier than they would otherwise be without math.

 

[AlephZero]

There is no shortcut to geometry, as said by euclid, but this applies not only in geometry but the whole field of mathematics, then I propose that there should not be any generalization, shortcut, rule or whatever, in the explanation to some concept.

In "real life" that idea isn't practical, unless you plan to teach 5-year-old kids axiomatic set theory before they learn how to count apples.

And if you are having problems understanding the number line, you would probably have worse peoblems understanding http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html and what that has to do with why the product of two negative numbers is positive.

The best way to learn math, at any level, is start somewhere "in the middle" of a topic and either work "forwards" to more complicated applications, or "backwards" to the foundations, which make a lot more sense when you can see what they are going to lead to.

The reason for the "rules" about negative numbers is because we want arithmetic to have some general properties for all numbers, both positive and negative, such as
a + (-a) = 0
0a = 0
1a = a
a + b = b + a
ab = ba
a(b+c) = ab + ac
etc.

So for example
a(b + (-b)) = a0 = 0
and
0 = a(b + (-b)) = ab + a(-b)
so a(-b) must be equal to =-(ab).

(That's only a very quick summary - get a textbook on abstract algebra if you want more).

 

[jbunniii]

And the number line is not a very good model, in fact, in explaining anything. It only produces confusion and anxiety.

How about accounting, then? Such as a simple cash register transaction: I owe you $5, I pay you $20, now you owe me $15. This is succinctly expressed as 5 - 20 = -15.

 

[fde645]

As a definition of addition, this statement is fairly meaningless from a mathematical standpoint. It may be true that addition of positive integers somehow corresponds well with the real-world activity of combining two things into a larger thing, and that's basically how it works for positive integers. But when you start talking about negative numbers and fractions and what-not, that analogy starts to break down to the point of being worthless.

Okay I accept the rebuke. However, how do you propose for me to understand basic arithmetic operations, including negative number operations? I am sorry but I can't wait till my college professor explains it to me. And I am not implying that mathematics is not a working concept, I know it works that is precisely why I am studying it. I have trouble accepting something just because someone said so. I don't observe blind faith. I want to accept something because I understand it and because it is something I can explain on my own.

 

[fde645]

To me basic operations in arithmetic is this,
In addition, you are given two parts, which you combine to get a whole.
In subtraction, you are given the whole and another part, and you are trying to get the other part.
In multiplication, you are given an individual factor and the multiplier, and you multiply or clone the individual factor, then you combine those clones, to get a product.
In division, you are given the product or dividend and the multiplier or perhaps the individual factor, but you are trying to get either the individual factor or multiplier depending on the divisor.

Now this insights only work for positive integers. And I am completely lost now that you guys mentioned the importance of number line. That seems to me really confusing. I simply can't understand how fractions are multiplied in the number line but I have also my own definition or interpretation of multiplication of fractions that does not involve the number line.

I must know what to do.

 

[fde645]

Maybe I should work on the number line and understand it. Anyone can recommend any sites or books for this matter? I simple can't push the insights that I precedingly describe, not with negative numbers and fractions. Maybe with positive numbers those works.

 

[Matterwave]

You are free to make whatever definitions of addition, subtraction, multiplication, and division as you like, as long as the definitions are logically consistent (you are of course also free to make logically inconsistent definitions, but then what is its purpose?). In your scheme, you can restrict yourself to the natural numbers if you like. But understand then that your scheme will have very limited practical applicability.

But people, over the last several thousand years, have already come up with a nice standard set of definitions. You are free to not use these definitions. The price you pay is that before you talk to anyone else regarding these matters, you have to first convince that person to use your definitions. This would be akin to me going around and telling everyone I met "hey, let's not use English, let's use a language that I just came up with and nobody but me knows". Probably you will get few people to come over to your side, so to speak.

If at the end of your journey, you want to use the standard set of definitions, then you can start by learning the basic definitions and rules set aside in basically any math book on arithmetic.

 

[fde645]

And that is exactly what I learned from the book on arithmetic. And all my definitions are consistent on the standard definitions of arithmetic. May I remind that arithmetic only deals with natural numbers. But Algebra expands arithmetic into negative numbers and the like. Thus when I learned algebra it contradicts my definition, subsequently making the confusion.

 

[symbolipoint]

And that is exactly what I learned from the book on arithmetic. And all my definitions are consistent on the standard definitions of arithmetic. May I remind that arithmetic only deals with natural numbers. But Algebra expands arithmetic into negative numbers and the like. Thus when I learned algebra it contradicts my definition, subsequently making the confusion.

That is another claim or statement which is absolutely false. Further, if what you learned of algebra contradicts any of your definitions, then either your definitions are wrong, or you did not learn algebra, or neither of the two.

 

[gopher_p]

Okay I accept the rebuke. However, how do you propose for me to understand basic arithmetic operations, including negative number operations? I am sorry but I can't wait till my college professor explains it to me. And I am not implying that mathematics is not a working concept, I know it works that is precisely why I am studying it. I have trouble accepting something just because someone said so. I don't observe blind faith. I want to accept something because I understand it and because it is something I can explain on my own.

Unless you're a (pure) math major taking a course in foundations of math or abstract algebra, your college professor is not going to explain basic arithmetic operations. I think you are underestimating the level of difficulty and overestimating the value in truly understanding what is going on. Furthermore, I will promise you that acquiring the knowledge necessary to truly understand basic arithmetic will only open new areas in which you lack understanding; the process will raise more questions than it answers, I guarantee.

I reckon you are perfectly content writing papers without worrying about why, in English, we typically start a sentence with a noun/object followed by a verb. You probably never gave a second though as to why an adjective precedes the noun it modifies in English whereas it follows in other languages. It's likely that you don't know the language of origin of most of the words that you use, nor do you likely care. And none of that really matters in the grand scheme of things. You are perfectly capable of writing those papers without that understanding.

That's not to say that you aren't capable of understanding these things. It's really just a matter of acquiring knowledge. And it's commendable if you are truly interested. It's that, for the vast majority of people, it's completely unnecessary (and often detrimental) to have that knowledge.

As for not observing blind faith ... You're going to find, as you get older, that you need to take a lot on faith. It is impossible in this day and age to truly know much of anything about more than a couple of (likely related) subjects. You're going to need to trust that those who do know what they're talking about won't give you bad information and learn to recognize the signs that someone maybe doesn't really know what they're talking about.

 

[fde645]

And why is it absolutely false? The definitions I have is based on arithmetic and not on algebra. Algebra deals with positive and negative integers, Arithmetic only deals with positive integers. As I have said my definitions are based on the arithmetical definitions, it is simply my interpretation but it is the same. My definitions are based on positive integers and thus deal with arithmetic and not on algebra. I simply used the freedom to use my own creativity in understanding the known rules of arithmetic.
As Lipang Ma say,"it is not enough to know how, we must know why". And this is precisely the reason why many hate mathematics because we are inducing them to just accept why something is that way rather than just simply how to do it.

And thus when I try to combine my definition in arithmetic to algebra together they seem to break apart. Arithmetic only gives you the liberty to subtract smaller from the larger and is not given the ability to do the opposite. Because in Algebra you are given the negative numbers and not on Arithmetic. You seem to claim something without providing the evidence.

 

[Mark44]

And that is exactly what I learned from the book on arithmetic.

What book was that?

And all my definitions are consistent on the standard definitions of arithmetic. May I remind that arithmetic only deals with natural numbers.

My first algebra class was when I was in 9th grade (in the US). Before that time my math classes were devoted to arithmetic with fractions (rational numbers) and decimal numbers (real numbers). I think you might not be using the standard meaning of "natural numbers," which are 0, 1, 2, 3, and so on (although some don't include 0 in this set).

But Algebra expands arithmetic into negative numbers and the like. Thus when I learned algebra it contradicts my definition, subsequently making the confusion.

Bringing in negative numbers merely extends the rules; the rules don't change to give you a different answer for the old problems that don't include negative numbers.

One question you asked was why 5 - (-4) is equal to 9. You can always rewrite a subtraction problem to an addition problem that has the same result.

5 - (-4) = 5 + -(-4) = 9
The middle expression is 5 + the opposite of -4. By "opposite" I really mean the additive inverse of -4, the number that is on the other side of 0 on the number line, and the same distance away from 0. A number and its additive inverse always add to zero.

 

[fde645]

I reckon you are perfectly content writing papers without worrying about why, in English, we typically start a sentence with a noun/object followed by a verb. You probably never gave a second though as to why an adjective precedes the noun it modifies in English whereas it follows in other languages. It's likely that you don't know the language of origin of most of the words that you use, nor do you likely care. And none of that really matters in the grand scheme of things. You are perfectly capable of writing those papers without that understanding.

That's not to say that you aren't capable of understanding these things. It's really just a matter of acquiring knowledge. And it's commendable if you are truly interested. It's that, for the vast majority of people, it's completely unnecessary (and often detrimental) to have that knowledge.

As for not observing blind faith ... You're going to find, as you get older, that you need to take a lot on faith. It is impossible in this day and age to truly know much of anything about more than a couple of (likely related) subjects. You're going to need to trust that those who do know what they're talking about won't give you bad information and learn to recognize the signs that someone maybe doesn't really know what they're talking about.

Well I thought I already implied that I am interested in knowing what it really means. That is precisely the reason of this thread. And regarding faith, do you really want to teach other people on faith? Rather than by reasoning? That is precisely why religion is failing to compete with science because one presumes it to be absolutely true before knowing what it really means. And by understanding it truly by yourself you become confident in teaching it to others. I must say that your post is misleading.

 

[Mark44]

And why is it absolutely false? The definitions I have is based on arithmetic and not on algebra. Algebra deals with positive and negative integers, Arithmetic only deals with positive integers.

Ordinary arithmetic is not limited to just positive integers. When you add, subtract, multiply, or divide fractions, you are not dealing with integers - these are rational numbers. Also, algebra is not limited to integers.

As I have said my definitions are based on the arithmetical definitions, it is simply my interpretation but it is the same. My definitions are based on positive integers and thus deal with arithmetic and not on algebra. I simply used the freedom to use my own creativity in understanding the known rules of arithmetic.
As Lipang Ma say,"it is not enough to know how, we must know why". And this is precisely the reason why many hate mathematics because we are inducing them to just accept why something is that way rather than just simply how to do it.

And thus when I try to combine my definition in arithmetic to algebra together they seem to break apart.

Then the definitions you are using are wrong.

Arithmetic only gives you the liberty to subtract smaller from the larger and is not given the ability to do the opposite.

This is silly. If you have $25 in your bank account, and write a check for $30, do you suppose that the people at the bank have to dig out an algebra book to figure that you are overdrawn by $5?

Because in Algebra you are given the negative numbers and not on Arithmetic. You seem to claim something without providing the evidence.

 

[fde645]

What book was that?

Arithmetic for the Practical Man. Well it does not exactly say my definition I created my own interpretation based on the rules of arithmetic don't you guys make your own understanding of it and not just accept what is there being laid in front of you?

My first algebra class was when I was in 9th grade (in the US). Before that time my math classes were devoted to arithmetic with fractions (rational numbers) and decimal numbers (real numbers). I think you might not be using the standard meaning of "natural numbers," which are 0, 1, 2, 3, and so on (although some don't include 0 in this set).

Nope. That is my natural numbers. I don't know what you mean.

5 - (-4) = 5 + -(-4) = 9
The middle expression is 5 + the opposite of -4. By "opposite" I really mean the additive inverse of -4, the number that is on the other side of 0 on the number line, and the same distance away from 0. A number and its additive inverse always add to zero.

you mean "the middle expression is 5 + the opposite of 4?

 

[fde645]

This is silly. If you have $25 in your bank account, and write a check for $30, do you suppose that the people at the bank have to dig out an algebra book to figure that you are overdrawn by $5?

In arithmetic you don't know anything about negative integers. In arithmetic you only subtract smaller from the larger, and not the opposite.How are you suppose to subtract 30 from 25 and not know the negative integers. How can you represent the debt, below sea level depth without knowledge of algebra? But if you know algebra you know that $25-$30= -$5 dollars overdrawn.
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?

 

[Mark44]

What book was that?

Arithmetic for the Practical Man. Well it does not exactly say my definition I created my own interpretation based on the rules of arithmetic don't you guys make your own understanding of it and not just accept what is there being laid in front of you?

It sounds like you created a faulty interpretation based on your misconceptions about the "rules of arithmetic."

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.

5 - (-4) = 5 + -(-4) = 9
The middle expression is 5 + the opposite of -4. By "opposite" I really mean the additive inverse of -4, the number that is on the other side of 0 on the number line, and the same distance away from 0. A number and its additive inverse always add to zero.

you mean "the middle expression is 5 + the opposite of 4?

No, I mean what I wrote, "5 + opposite of -4".

 

[pwsnafu]

And that is exactly what I learned from the book on arithmetic. And all my definitions are consistent on the standard definitions of arithmetic.

The axioms of arithmetic of natural numbers are available here. You'll notice they look very different to what you proposed.

May I remind that arithmetic only deals with natural numbers.

False. (Elementary) arithmetic includes integers and rationals (remember, a fraction is a division of two integers).

And regarding faith, do you really want to teach other people on faith? Rather than by reasoning?

The problem here is that difficult results require prerequisites.
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.

That is precisely why religion is failing to compete with science

Religion does not compete with science.

because one presumes it to be absolutely true before knowing what it really means.

All mathematics does this to some extent. Mathematics has a set of axioms (statements which are presumed true) and then proves results (theorems) using logic from those axioms. Mathematics is the process to get to B from A, but we do not care if A is "actually" true.

 

[fde645]

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.

 

[pwsnafu]

In arithmetic you don't know anything about negative integers.

No, the arithmetic of the natural numbers doesn't say anything about negative integers.
The arithmetic of the integers does indeed say things about negative integers.

In arithmetic you only subtract smaller from the larger, and not the opposite.

On the natural numbers you can only subtract a smaller natural number from a larger natural number.

You seem to have this strange idea that "arithmetic = natural numbers" and "algebra = everything else". We are telling you that is not true.

How are you suppose to subtract 30 from 25 and not know the negative integers. How can you represent the debt, below sea level depth without knowledge of algebra? But if you know algebra you know that $25-$30= -$5 dollars overdrawn.

Again this is arithmetic not algebra.

Algebra has two meanings in mathematics: elementary algebra is the study of the properties of polynomials, abstract algebra is the study of sets with n-ary operations. Neither is being talked here.

 

[pwsnafu]

Axioms, correct me If I am wrong is not absolutely true but highly likely that is true. Well mostly because it is self evident.

An axiom is a statement is assumed to be true for the purposes of logic. Why the researcher assumes it is true is irrelevant. It could be...

1. the researcher considers it self evident,
2. the researcher is arguing that it corresponds to observations of the real world,
3. the researcher like that set of axioms,
4. the researcher is not aware of other sets of axioms,
5. the researcher is considering a counterfactual,
6. the researcher is in a happy mood,
7. the researcher is taking an established set of axioms, and is researching what happens when one of the axioms happens to be false,
8. the researcher doesn't care.,

or any combination of the above.

 

[micromass]

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
$$(-3)\cdot 5 = (-3) + (-3) + (-3) + (-3) + (-3) = -15$$
and we will also get
$$3\cdot (-5) = (-5) + (-5) + (-5) = -15$$
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:
$$\begin{array}{cccccc} 1 & 2 & 3 & 4 & 5 & ...\\ 3 & 6 & 9 & 12 & 15 & ...\end{array}$$

We already agreed that the use of repeated addition forces upon us the rule $3\cdot (-5) = -15$. So we can extend the previous:

$$\begin{array}{ccccccccccc} ... & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4 & 5 & ...\\ ... & -9 & -6 & -3 & 0 & 3 & 6 & 9 & 12 & 15 & ...\end{array}$$

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 $-3$:
$$\begin{array}{cccccc} 1 & 2 & 3 & 4 & 5 & ...\\ -3 & -6 & -9 & -12 & -15 & ...\end{array}$$

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:

$$\begin{array}{ccccccccccc} ... & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4 & 5 & ...\\ ... & 9 & 6 & 3 & 0 & -3 & -6 & -9 & -12 & -15 & ...\end{array}$$

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.

 

[fde645]

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?

 

[micromass]

Very insightful post Micromass. I. M. Gelfand and A. Shen, Algebra is this the book you are talking about?

Yes. He has some other books which are meant to be for high school students, such as books on trigonometry and coordinate geometry. Watch out though, he has several very advanced math books too so be careful what you purchase. Gelfand is really a top mathematician and it is very rare for top mathematicians to write high school books. Usually they are written by people who do not study science and math on a professional level. Other good books on high school level written by professional mathematicians are Lang's book on basic math and geometry, and Euler's books.

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.

There are actually two stages here. A first stage is selecting which axioms we accept as true. This is where "inductive reasoning" enters mathematics. We might look at nature to select our axioms, or we might be lead by math itself or simply by elegance. The axioms we select are quite arbitrary, we can study any axiomatic system we want. But not all axiomatic systems will be useful and thus not all axioms will be worth studying. Applicability of mathematics in science and engineering is a final judge of whether our axioms are good ones.

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.

What book can anyone suggest for abstract algebra?

Abstract algebra is something that is usually studied in advanced mathematics classes at university. If you are currently learning basic algebra such as solving equations, then you're probably not ready yet for abstract algebra. A good book that I recommend is Pinter's "book on abstract algebra". It is suitable for high school students who have completed things like Algebra I and Algebra II. Occasionally, it will make use of calculus and trig in the exercises, but that's not essential. Don't be depressed if the book is too difficult for you, this is normal. Complete the high school curriculum of mathematics first before you go to more advanced mathematics like abstract algebra.