# Unsensible Rules/Laws of negative number operations

1. May 16, 2014

### 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.

2. May 16, 2014

### SteamKing

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

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.

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.
Number line again.
http://en.wikipedia.org/wiki/Multiplication
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.

3. May 16, 2014

### 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.

4. May 16, 2014

### Staff: Mentor

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!

5. May 16, 2014

### 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).

6. May 16, 2014

### 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.

Last edited: May 16, 2014
7. May 16, 2014

### gopher_p

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.

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.

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.

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 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.

8. May 16, 2014

### AlephZero

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).

9. May 16, 2014