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parallaxicality
2007-Jul-10, 02:40 PM
I'm sure that there's a perfectly rational and logical explanation for this, and I'm sure that, given that I only barely passed basic math in high school, I won't understand a word of it.

Anyway.

Here's what I don't get about negative numbers.

5 + -4 is the same as 5 - 4. This makes sense. -4 is the expression of the subtraction of four.

If I have five of anything, and I subtract a subtraction of four from it, am I not left with five?

5 x -4 is -20

Here I'm really lost.

If we multiply five and four, what we are in effect doing is adding five to itself four times. 5 x 4 is the same as 5+5+5+5

I assume the logic should hold for negatives. Since adding a negative is the same as subtracting a positive, then 5 x -4 should be the same as subtracting five from itself four times. Therefore 5 x -4 should be the same as 5-5-5-5. The problem is that 5-5-5-5 is -10, not -20.

-5 x -4 is 20

OK. Even accepting the idea that subtracting a negative is the same as adding a positive, this would be the equivalent of subtracting -5, that is, adding 5, to -5 4 times. In other words -5+5+5+5, which leaves us with 10.

I know I must be missing something rather basic, but I have no idea what it is.

NEOWatcher
2007-Jul-10, 02:54 PM
I'm sure that there's a perfectly rational and logical explanation for this, and I'm sure that, given that I only barely passed basic math in high school, I won't understand a word of it.
Well; here's some quickie answers (I have a math degree but sometimes just shoot of the hip and don't remember much). And; obviously, a more in-depth lesson will be required.

5 + -4 is the same as 5 - 4. This makes sense. -4 is the expression of the subtraction of four.
Good... we have a basis for discussion.

There are just some things in math that work out, but can't quite be related to real life, but this one is similar to a double-negative in language. (in-fact it's the same definition)

If I have five of anything, and I subtract a subtraction of four from it, am I not left with five?
You are reversing the removal of the 5 therefore putting 5 back in. [EDIT: sorry, that should be 4]

5 x -4 is -20
...If we multiply five and four, what we are in effect doing is adding five to itself four times. 5 x 4 is the same as 5+5+5+5
Just reverse it and say you are removing 4 and doing it 5 times which results in removing 20

I assume the logic should hold for negatives. Since adding a negative is the same as subtracting a positive, then 5 x -4 should be the same as subtracting five from itself four times. Therefore 5 x -4 should be the same as 5-5-5-5. The problem is that 5-5-5-5 is -10, not -20. No, the starting point is not the first 5. The starting point is zero, then you apply the operation the specified number of times.

-5 x -4 is 20
OK. Even accepting the idea that subtracting a negative is the same as adding a positive, this would be the equivalent of subtracting -5, that is, adding 5, to -5 4 times. In other words -5+5+5+5, which leaves us with 10.
Same double negative, you are reversing the removal of 4, 5 times, therefore adding 20.

I know I must be missing something rather basic, but I have no idea what it is.
I'm not sure where (in particular) to point you, but I do suggest going to a bookstore or library and do some more in-depth perusing of the subject.

Ban Me
2007-Jul-10, 02:59 PM
I'm sure that there's a perfectly rational and logical explanation for this, and I'm sure that, given that I only barely passed basic math in high school, I won't understand a word of it.

Anyway.

Here's what I don't get about negative numbers.

5 + -4 is the same as 5 - 4. This makes sense. -4 is the expression of the subtraction of four.

So if you begin with 1, and add 4 to it, you have 1+4=5. Then if you add -4, this undoes your earlier operation, and gets you back to one: 5-4=1.

If I have five of anything, and I subtract a subtraction of four from it, am I not left with five?

It is the same as before. If you begin with 9, and add -4 to it, you end up with 5, 9-4=5. Now if you subtract -4, you undo your earlier operation, and get back to nine: 5--4=9.

5 x -4 is -20

Here I'm really lost.

If we multiply five and four, what we are in effect doing is adding five to itself four times. 5 x 4 is the same as 5+5+5+5

But you did not add five to itself four times. You added 4 instances of 5 together.

I assume the logic should hold for negatives. Since adding a negative is the same as subtracting a positive, then 5 x -4 should be the same as subtracting five from itself four times. Therefore 5 x -4 should be the same as 5-5-5-5. The problem is that 5-5-5-5 is -10, not -20.

5 x -4 is the same as -5-5-5-5=-20.

This one might be easier if you think of 5x4 as adding 4 together 5 times.

5x4=4+4+4+4+4=20

Then

5x(-4)=(-4)+(-4)+(-4)+(-4)+(-4)=-20

-5 x -4 is 20

OK. Even accepting the idea that subtracting a negative is the same as adding a positive, this would be the equivalent of subtracting -5, that is, adding 5, to -5 4 times. In other words -5+5+5+5, which leaves us with 10.

Try 5+5+5+5 which is twenty. In both of the multiplication problems, you didn't negate one of the numbers being added together. Your last sum is:

-5+5+5+5=-5+(5+5+5)=5*(-1)+5*(3)=5*(-1+3)=5*(2)=10

But this is -5*(-2). If you want -5*(-4), you need to begin with 5+5+5+5.

For the multiplication problems, it might be easier to begin with a simpler one. Instead of 5 x -4, think about 5 x -1. Do you feel that should be 5 or -5?

I would recommend that you avoid complex numbers for a while.

Ken G
2007-Jul-10, 03:42 PM
If I have five of anything, and I subtract a subtraction of four from it, am I not left with five?
First of all, you're not crazy or stupid, these are profound issues that most of us take for granted out of familiarity moreso than understanding. I think what might help you is to recognize that all arithmetic conventions are arbitrary, we are free to choose the results to be anything we want-- and we do so to achieve some useful value in practice. So the question here is, what is the situation we are using arithmetic to model, and how can we choose conventions to get the "right" answer?

As for addition and subtraction, one model for which we'd like to use them has to do with keeping track of credits and debts, and you can think of a negative number as a kind of debt. Let's imagine that instead of paper money, we just use a paper IOU (I-owe-you). So any time we perform a service, someone else keeps an IOU on our behalf, which is like negative money to them and positive money to us. And of course, if someone else does something for us, we keep an IOU on their behalf, which is our negative amount of money. In that system, addition is the equivalent of merging IOUs into a single IOU-- if I have an IOU for \$4 and \$5 to you, I can replace them with a single IOU to you for \$9, that's addition. Addition of a positive and a negative number is like if you have an IOU to me for \$4, and I have an IOU to you for \$5, we just merge those into a single IOU that I have to you for \$1 (i.e., from your perspective, that's 5-4 = 1).

Now how about subtraction? In this way of thinking of numbers, subtraction is like splitting an IOU up into two pieces, and throwing one away. So if I hold an IOU to you for \$5, that's a +5 from your point of view, but if I do you a service, instead of writing up an IOU to me for \$4, and then adding it to mine to get 5 + -4 = 1 from your point of view, you could simply tell me to take my \$5 IOU to you, break it down into a \$4 and a \$1 IOU, and rip up the \$4 IOU, giving 5 - 4 = 1.

How about subtracting a negative? If I have an IOU to you for \$5, that's like you having +5. If you then do another service for me, you could either say, "write me up another \$4 IOU and merge (add) the two", or, equivalently, you could ask me to split my \$5 IOU into a \$9 IOU from me to you and a \$4 IOU from you to me, so that's like a +9 and a -4 from your point of view. Then the subtraction of the -4 is tantamount to your saying, "I just did you a service, so shall we just rip up my \$4 IOU to you?". That way, it's not 5 + 4 = 9, it's 9 + (-4) - (-4) = 5 - (-4) = 9. I think you were leaving out the "break down the IOU" step when you just got \$5.

I assume the logic should hold for negatives. Since adding a negative is the same as subtracting a positive, then 5 x -4 should be the same as subtracting five from itself four times.It turns out that we want multiplication to be commutative, so you can also think of this as adding -4 to itself 5 times, which is more clear to give you -20. But that does beg your question, as we don't necessarily need to make multiplication commutative-- we should be able to concoct some meaning to a negative number in the second position.

I think the whole concept of multiplication as "adding itself a certain number of times" is actually too limited to answer your question-- it really only applies when you have a positive number (preferably an integer) in the second position. A more generally useful way to think of multiplication, which can handle non-integers in either position and is automatically commutative, is to think of it as a way to get the area of a rectangular tile that has one corner anchored to a point we'll call "the origin". The wrinkle is that the area on the top of the tile is positive, by convention, and the area on the bottom is negative-- that's how we can specify if it's the top or bottom of the tile we are looking at. Then we say that lengths that go from left to right (along the "x axis"), starting at the "origin", are positive lengths, and lengths going from right to left are negative. Similarly, lengths that go from the origin (along the "y axis") to above the origin are positive, but lengths that go from the origin to below the origin are negative. So we put one corner of the tile at the origin and find its area by multiplying its lengths along the x and y axes, and check our result by inking the tile with a reference unit area, over and over.

Note that multiplying two positive numbers has the origin at the lower left corner of the tile, and then we say we are looking at the "top" of the tile and get a positive area. But a positive times a negative, keeping the same corner at the origin, requires flipping the tile upside down (either right to left or up to down, depending on which number is negative)-- so gives a negative area. Multiplying two negatives requires that we flip the tile twice (since we are always keeping the same corner at the origin)-- and so we're back to the top of the tile and get a positive area.

Multiplication and addition come up a lot in physics, obviously, and these are the "right" conventions to use, because they spawn a useful arithmetic. Why-- nobody knows.

hhEb09'1
2007-Jul-10, 03:48 PM

If I have five of anything, and I subtract a subtraction of four from it, am I not left with five?So, you know the right answer, but you disagree with it? Shouldn't this be in ATM? :)

5 x -4 is -20

Here I'm really lost.

If we multiply five and four, what we are in effect doing is adding five to itself four times.No, to itself three times.

I assume the logic should hold for negatives.Yes, it does, of course. You've just extracted the wrong principal.

-5 x 4 = -5 plus -5 plus -5 plus -5

negative five four times

WaxRubiks
2007-Jul-10, 03:48 PM
taking away a negative is like taking away some debt from someone, it is the equivalent of giving them money.

-5*-4 is similar it is the repeated addition of -4 except that it is the opposite operation ie taking away -4 so, starting with zero as your base you have 0--4--4--4--4--4, which is 20. I dunno, it is difficult to understand.

John Mendenhall
2007-Jul-10, 04:23 PM
I'm sure that there's a perfectly rational and logical explanation for this, and I'm sure that, given that I only barely passed basic math in high school, I won't understand a word of it. [snip]

I know I must be missing something rather basic, but I have no idea what it is.

Hey, at least you know to ask. The posters before me have done a fine job of explaining. One thing that wasn't mentioned is that the operations of (and this discussion has been limited to integers) addition and subtraction, including negative numbers, must give only negative and positive numbers, and the results must be consistent and reversable. As I recall, the mathematical description is that the set of integers is closed with respect to addition and subtraction. It is possible to construct systems in which this is not the case, but first learn the common real world operations.

One thing that causes problems in our system is that we use + and - to indicate positive and negative numbers, and also to indicate the operations of addition and subtraction. The sign of a number and the operation intended, if any, are really separate things. If you can find an older HP calculator which uses Reverse Polish Notation (RPN), the difference quickly becomes clear. It's also fun to see if you can figure out how to use the calculator without reading about RPN first.

pzkpfw
2007-Jul-10, 08:54 PM
Of course, you all know negative numbers are evil, don't you?

Godlike Productions -- --- Negative Numbers and Other Frauds --- (http://godlikeproductions.com/bbs/message.php?page=1&messageid=364286&showdate=7/10/07&mpage=1)

:-)

Jerry
2007-Jul-10, 09:28 PM
Here I'm really lost.

If we multiply five and four, what we are in effect doing is adding five to itself four times. 5 x 4 is the same as 5+5+5+5

I assume the logic should hold for negatives. Since adding a negative is the same as subtracting a positive, then 5 x -4 should be the same as subtracting five from itself four times. Therefore 5 x -4 should be the same as 5-5-5-5. The problem is that 5-5-5-5 is -10, not -20.

I'm not against redefining the conventions, but using this set of definitions, I come up with -15, not -10. Edited to add: Opps! added wrong, you are corredt.

Don't feel alone. The standard definitions are rather arbitrary, and don't work at all well when trying describing the physical universe. This is why imaginary numbers were invented: to describe reality!

hhEb09'1
2007-Jul-10, 09:49 PM
This is why imaginary numbers were invented: to describe reality!I don't think so :)

Ken G
2007-Jul-10, 11:17 PM
The arithmetic of imaginary numbers also uses the same basic rules of subtraction and multiplication of negative numbers that we've all described above, embedded in the more "complex" rules you need for imaginary numbers.

hhEb09'1
2007-Jul-10, 11:31 PM
The arithmetic of imaginary numbers also uses the same basic rules of subtraction and multiplication of negative numbers that we've all described above, embedded in the more "complex" rules you need for imaginary numbers.The only other rule is that i2=-1. Doesn't sound very complex to me :)

01101001
2007-Jul-10, 11:44 PM
The only other rule is that i2=-1. Doesn't sound very complex to me :)

You're imagining things.

pzkpfw
2007-Jul-11, 12:27 AM
The discussion seems to have moved on, but it's now lunch-time for me (and I'm bored because I'm at a client site where I'm carefull where I surf to) so I'll make the post I wanted to make this morning. Here's a "real world" example that may or may not help.

I have an Apple tree. It has 20 apples on it today, and I know that 2 apples fall off it every day.

So every day I can subtract two apples from the tree, or add negative 2 apples to the tree.

How many apples will I have on the tree tommorrow?
= 20 + -2
= 18

[There's an implied "x 1" there, for the one day of losing 2 apples.]

How many apples will I have on the tree in three days time?
= 20 + (3 x -2)
= 20 + -6
= 20 - 6
= 14

Here's the kick: how many apples might I guess were on the tree [i]three days ago?
= 20 + (-3 x -2)
= 20 + 6
= 26

gain apples.]

Hopefully, that's a simple "real world" example of how multiplying two negatives and getting a positive, makes some kind of sense.

Ken G
2007-Jul-11, 05:57 AM
The only other rule is that i2=-1. Doesn't sound very complex to me :)
'Twas merely a pun. But if you think about it, that "other rule" is quite strange indeed. Yet the arithmetic it spawns is remarkably powerful.

Ken G
2007-Jul-11, 06:02 AM
Hopefully, that's a simple "real world" example of how multiplying two negatives and getting a positive, makes some kind of sense.

An excellent example indeed. I actually like that better than considering the area of a tile to be positive on one side and negative on the other, because the arrow of time is of such commonplace importance.

Maksutov
2007-Jul-11, 06:06 AM
I may be old-fashioned, but it seems that using a number line (http://en.wikipedia.org/wiki/Number_line) would help the OP with a lot of his questions and concerns.

Plus an investigation of absolute value might help too.

NEOWatcher
2007-Jul-11, 12:07 PM
I may be old-fashioned, but it seems that using a number line (http://en.wikipedia.org/wiki/Number_line) would help the OP with a lot of his questions and concerns.
Yes; that works too, and I was just thinking that... a negative number is the same as a positive number, it just goes the other way or makes things go the other way.

Sometimes, not having the step by step knowledge leading up to a concept is very detremental to not understanding the concept.

I remember in school, never remembering the quadratic equation, but any time there was a use for it on the test, I would always derive it from what I knew about how it was built.

I also did some tutoring in college. Mostly nursing students who had to take the 1 term "intuitive" calculus as opposed to the 3 term "analytical". (I always hated that name, because intuition had nothing to do with it) They were just taught formulas, and I would help them understand where the formula came from. It seemed to help.