How can you multiply two negative numbers?

[Dark Fire]

Ok, I know this seems very obvious to most people, but I have a problem understanding it.
First of all: I have not been taught in any way how something objective can have a negative amount, so I do not even understand results that ends up being negative (-1 apple, for example) - you can only consider it as potential negative; that it will in the future disappear: debt.
I can, however, accept it to be debt, and I understand the math/theory of it (who doesn't, you're taught this when you're 7 years old).

Ok, so now to the final: I get that you can multiply a negative with a positive.
You have 3$ in debt, and it occurs 3 times: -3$*3$=-9$
Now to what I don't understand, when you multiply TWO negative numbers: -3*-5=15
I know the math-part, which is that they clear each other out.
But in "practice", how the hell would it work out?
Not necessarily objective; our reality - universe, but some sort of a practical way to explain how two negative numbers can be multiplied, it doesn't make sense to me.
You have -3, -3 times, wtf!?
You can't have something -3 times! xD
The lowest value I can think of that you can multiply with is 0.

My question sort of is: how did mathematicians come with the conclusion of creating the rule that two negative numbers multiplied with each others is a positive number?
How? Why?
Was it just to create a way for us to keep on using this theoretical mathematics?

I'm sorry if this seem to be a foolish question, though I truly don't consider this foolish, regardless of how simple it might seem for the close-minded..
I find this of importance, to reveal and understand every inch of something, how can you otherwise be able to understand it, when you don't understand the source?

Thanks for potential replies and answers.

[Peeter]

Like you said, you can justify the rule for negative times positive easily enough with debt.

For negative times negative I like a geometrical comparision, where a negative one multiplication is a rotation of a vector in space by 180 degrees maintaining the length of the original vector (if you have just a number line, this becomes an inversion). With this comparision a positive times -1 becomes negative as with debt, and two -1 multiplications get you back to your original position in space, justifying the rule -1x-1 = 1.

ps. I don't think it's a foolish question. I had the same question back in grade school and didn't like the uninspired answer "it's just the rule" by my teacher either.

[Borek]

What if not? I mean - what (-1)*(-1) should be equal to if it will be not equal 1?

[Dark Fire]

For negative times negative I like a geometrical comparision, where a negative one multiplication is a rotation of a vector in space by 180 degrees maintaining the length of the original vector (if you have just a number line, this becomes an inversion).
Didn't get it.
I google'd vector and found en.wikipedia.org/wiki/Vector_graphics, where I read the first lines, and read the description beneath/above/beside the picture, and I didn't get it.
It just takes forever looking up English words from wikipedia, because the description of the word is again used with foreign words that I have to lookup..
Thanks anyways, for being polite, and what seem to be a good explanation.

What if not? I mean - what (-1)*(-1) should be equal to if it will be not equal 1?
As I wrote: "Was it just to create a way for us to keep on using this theoretical mathematics?" - as in, they had no greater answer, so instead of abandoning the idea of mathematics, they instead used their greatest answer.
I was sort of just kidding with it though; I'm quite sure there's a logical clearcut reason behind it.
I find this to be sort of alike to Imaginary numbers..
You can't squareroot a negative number, but in my opinion, can you neither square a negative number, even if squaring is defined as ^2 and based on general mathematics it would work out, but this is so theoretical for me, that I got problems accepting it and working with it, because how can you know, with advanced mathematics, what's right and not, when it's never clearcut, and seem slightly illogical, or even very illogical at times?

Btw, is it wrong to write -3*(-3)?

[Crosson]

You could think of it using division. If you have a debt of -10, and you divide it evenly between 5 people, then each of those people will have a debt of -2.

In symbols:

-10/5 = -2

Therefore you could ask the question: how many people does it take to split a -10 debt into a -2 debt for each person:

-10/-2 = 5

[Dark Fire]

Division works (= I get division), the issue is multiplication.
I guess multiplication isn't very clearcut after all..
I guess it is, as I suspected, very theoretical.

I just feel that I'm like.. Playing with fire, like working with what I don't understand - and I just don't like that..
Knowing what I'm working with, has always been my triumph card :P

[Crosson]

How about this (fictional) example:

Every time I go to the nightclub, I spend a hundred dollars. For me this is represented as c = -$100 (let c stand for "change in my money").

This means that if I go to the nightclub 5 times, I will spend 500 dollars, which is represented as c = 5*(-$100) = -$500.

What do we mean if I say I went to the nightclub -5 times? It means that I decided not to go out 5 of the times that I usually would have i.e. my friends call, but I tell them I won't be going out. Then the change in my money is c = -5*(-$100) = $500 which means that at the end of the month I will have 500 more dollars than usual.

[Borek]

Division and multiplication are in a way the same operation. If division works, calculate 1/(-2) first - it gives -0.5. Now you multiply -0.5*(-10) and result must be the same 5, so 5 = -10*(-0.5) = -10/-2

Borek
--
http://www.chembuddy.com
http://www.ph-meter.info

[Hurkyl]

ps. I don't think it's a foolish question. I had the same question back in grade school and didn't like the uninspired answer "it's just the rule" by my teacher either.
The problem is that you didn't ask what you wanted to know; "Why can you multiply two negative numbers?" is a much different question than "How might I use the arithmetic of negative numbers in real world situations?"

[Dark Fire]

This means that if I go to the nightclub 5 times, I will spend 500 dollars, which is represented as c = 5*(-$100) = -$500.

What do we mean if I say I went to the nightclub -5 times? It means that I decided not to go out 5 of the times that I usually would have i.e. my friends call, but I tell them I won't be going out. Then the change in my money is c = -5*(-$100) = $500 which means that at the end of the month I will have 500 more dollars than usual.


I disagree with that way of putting it or something..
I would rather put it like this: -100$*0(5)
Do you lose your money? No, so it doesn't happen, which means multiplied 0 times.
Now how many times does it (not) happen? 5.
The conclusion is that you don't earn nor spend anything, because we're focusing at what really is happening, and not taking what you could have done with your money in consideration.
I would set up how much you have earned/spent as: +-100$*5=+-500$, which means you earn/spend 100$ depending on how you put it/what counts.

I get your example, but unfortunately it doesn't fit with my logic, however I was thinking like this when I was younger (which is why I accepted it earlier).

[tony873004]

Consider North the positive direction, and an increase in temperature to be positive as well.

If every mile north I drive, it gets 1 degree colder and I drive 10 miles north then the temperature has changed
10 miles * -1 degree/mile = -10 degrees
10 * -1 = -10, so the temperature has dropped 10 degrees

If I drive south instead, then this would be represented by -10 miles since I consider north the positive direction.
-10 * -1 = 10, so the temperature is 10 degrees warmer

[LukeD]

Dark Fire: Many times mathematics is developed for one thing (such as counting things or keeping track of accounting), and you find that you logically develop some thing that isn't used in that setting (for instance, multiplying two negative numbers together). In this case, the idea that two negatives multiplied together gives a positive is the simplest way of defining multiplication for two negative numbers.

Often times though, there are other applications other than the ones that you originally thought of in which these developments are used. For instance, vectors have already been said as an application, so I'll try to explain what they are and why multiplying two negative numbers together works here.

A vector can be intuitively thought of as a direction along with a magnitude. For instance, think of a velocity (a speed and direction): You can be driving north at 30 miles per hour.

Let's say that v is your velocity (for instance, north at 30 miles per hour). You can think of multiplying by a number x as going x times faster. For instance, 1v = v since going 1 times faster is the same as not changing your speed or direction at all. 2v then would be going twice as fast, or north at 60 miles per hour.

You can also multiply this by a negative number. So you can go -1*v, which is north at -30 miles per hour, but notice that this just means that every hour you're going -30 miles in the north direction, which is to say, that you went 30 miles south. Therefore, multiplying by a negative number turns you around.

Then notice that multiplying by a negative number turns you around twice, so negative times negative is positive.

----

One thing to note though is that this is a very different example than accounting. In accounting, there is no reason (at least as far as I can think of) why you would multiply two negative numbers together. However, in this new example, you do have situations in which you'd multiply two negative numbers together.

However, in both of these situations, the numbers that you are used to are applicable.

Note that I am saying that the numbers are "applicable". You should try to convince yourself that numbers are not the same as money or directions or anything like that; numbers themselves are not real and they are never the situation that you are modeling; they are only applicable to situations. And sometimes, some numbers are more or less applicable. For instance, you said that you can't take the square root of a negative number, but this really isn't accurate; it's just that in many of the real world situations you've thought of where you apply numbers, there is no interpretation of complex numbers. However, there was a post earlier where someone asked for physical applications of complex numbers, and people were able to come up with several.

[CRGreathouse]

How about this (fictional) example:

Every time I go to the nightclub, I spend a hundred dollars. For me this is represented as c = -$100 (let c stand for "change in my money").

This means that if I go to the nightclub 5 times, I will spend 500 dollars, which is represented as c = 5*(-$100) = -$500.

What do we mean if I say I went to the nightclub -5 times? It means that I decided not to go out 5 of the times that I usually would have i.e. my friends call, but I tell them I won't be going out. Then the change in my money is c = -5*(-$100) = $500 which means that at the end of the month I will have 500 more dollars than usual.

I think that's a great example.

A geometrical method would be to graph y = x * -1; if the straight line continues as usual to -1, it will be at y = 1.

As pattern completion:

4 * 2 = 8
4 * 1 = 4
4 * 0 = 0
4 * -1 = -4
4 * -2 = -8
. . .

3 * -2 = -6
2 * -2 = -4
1 * -2 = -2
0 * -2 = 0
-1 * -2 = 2

[LukeD]

Ah, I missed this or I would have said that it was a fairly good example of the use of multiplying negative numbers together in accounting.

How about this (fictional) example:

Every time I go to the nightclub, I spend a hundred dollars. For me this is represented as c = -$100 (let c stand for "change in my money").

This means that if I go to the nightclub 5 times, I will spend 500 dollars, which is represented as c = 5*(-$100) = -$500.

What do we mean if I say I went to the nightclub -5 times? It means that I decided not to go out 5 of the times that I usually would have i.e. my friends call, but I tell them I won't be going out. Then the change in my money is c = -5*(-$100) = $500 which means that at the end of the month I will have 500 more dollars than usual.

This only works if you set your 0 at your usual amount of spending.

Though, this often is the way you'd think of your spendings if you've set a budget.

[awvvu]

How about this:

A particle is moving left at a speed of 5 on a number line (so its velocity is -5). It starts at the origin at t = 0 s.

Where is the particle at t = -2 s?

x = v * t = -5 * -2 = 10.

So, 2 seconds before it's at the origin at t = 0, it is 10 to the right at t = -2.

[Daniel Y.]

http://mathforum.org/dr.math/faq/faq.negxneg.html

[mathwonk]

huh?

[fedaykin]

I sort of think of it this way:

\ (-3)(-3)

(The opposite of) 3 copies of (the opposite of) 3.

If you think of it on a number line, (the opposite of) means go in the other direction away from zero.

Let's also look at it this way. (3 * -3)(-1) = 9


3 copies of -3 is negative nine, the opposite of -9 is 9.

I don't mean to imply that these number are somehow opposed or anything; they're just on opposite sides of the number line.

I also want to mention that sometimes negative numbers do have a physical representation. Take the charges of the electron and proton. Both are non-zero, but when you add them together you get zero, no net electric charge.

[Kurret]

Consider this example:
you own 3$, then you have +3$.
If you ow someone 3$, then you have -3$, or, lets say you have a bill on 3$, then you must sooner or later pay these 3$, so that "variable" is worth -3$.
But if you ow someone a bill of 3$, ie you must give someone else a bill on 3$. Then you have -(-bill)=-(-3$).But when you give this bill to the other guy, he needs to pay you 3$, so "owing a bill to someone else" will give you 3$. Thus, -(-3$)=3$
:smile: :smile: :smile:

[PowerIso]

I disagree with that way of putting it or something..
I would rather put it like this: -100$*0(5)
Do you lose your money? No, so it doesn't happen, which means multiplied 0 times.
Now how many times does it (not) happen? 5.
The conclusion is that you don't earn nor spend anything, because we're focusing at what really is happening, and not taking what you could have done with your money in consideration.
I would set up how much you have earned/spent as: +-100$*5=+-500$, which means you earn/spend 100$ depending on how you put it/what counts.

I get your example, but unfortunately it doesn't fit with my logic, however I was thinking like this when I was younger (which is why I accepted it earlier).

Oh it works. But as stated earlier you have to say I usually make a net profit of 0 dollars per week and in my budget I make room for my nightclub visits. If one week, you become to busy to go night clubbing and you review your budget, you are going to notice a 500 dollar increase. Thus not spending 500 dollars for not going out five days a week will produce 500 dollars more than you would have had otherwise.

[Dark Fire]

If I drive south instead, then this would be represented by -10 miles since I consider north the positive direction.
-10 * -1 = 10, so the temperature is 10 degrees warmer

This is the same as using a board with an X-axis and Y-axis, which is the only "practical" way of thinking about it, though I wasn't satisfied with it~


Ok, let's add something I worked with in my head for the last hour (forgot to bring paper and pencil).
Imaginary Number = I
I = -9^-1

Figure that out.

[arildno]

But in "practice", how the hell would it work out?
Not necessarily objective; our reality - universe, but some sort of a practical way to explain how two negative numbers can be multiplied, it doesn't make sense to me.
You have -3, -3 times, wtf!?
You can't have something -3 times! xD
The lowest value I can think of that you can multiply with is 0.

My question sort of is: how did mathematicians come with the conclusion of creating the rule that two negative numbers multiplied with each others is a positive number?

Your question' isn't dumb at all.
You are perfectly right that you cannot "have" something -3 times, (debtfan or not!)

But, and that's the rub:
Why do you think "multiplication" necessarily means to "have something a number of times"?

[Dark Fire]

Why do you think "multiplication" necessarily means to "have something a number of times"?

It's the only way I've been able to understand multiplication, we were also taught to think of it that way when we were childs, and I never had a flash/bright idea/philosophy popping up, telling me maybe multiplication isn't that way of thinking after all: until now.

Thank you all for the replies, feeling good for not being a fool after all :)

[Peeter]

Is there a program similar to the TeX

install cygwin, also selecting the tetex packages when you do so.

[LukeD]

TeX doesn't calculate anything; it's just a type setting program. All it does is let you type up nice papers (though it is very useful for writing up your math homework in college if you're taking some advanced courses).

There are several programs that will act like very good calculators though. Mathematica is one (though it costs a lot of money); MATLAB is another. There are a few free ones, but I've never used them. For instance, there is Octave, which is very similar to MATLAB. It can be found at www.octave.org

However, these programs will usually only give you the answer without showing you how to calculate things. I don't know of any program that will calculate things step by step.

If you are just looking for a typesetting program like TeX, first of all, get LaTeX instead of TeX. TeX is quite outdated; whenever someone says TeX nowdays, they almost always mean LaTeX (LaTeX is what is used on the forums here, not TeX). If you want to get LaTeX, then MikTex is probably your best option. (Installing LaTeX in Cygwin works too, but unless you want to get familiar with a command line, it's probably going to be more difficult to use for you than MikTex)

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

By the way, what did you mean by i = -9^{-1}? -9^{-1} = - \frac{1}{9} not i

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

Also, you're correct in thinking that x*y does not mean that you have x amount of y (i.e. that you have 3 6s or something). For some numbers, you can give it this interpretation, but certainly not for all of them. What would it mean then to do (6+i)(1-2i)? That you have (6+i) of the number (1-2i)? That's meaningless. It gets even worse when you start looking at other things you can multiply together like matrices.

Instead, you should think of numbers as just being formal rules: abstract theoretical things. However, you can apply them to things that do have physical interpretations, and often times your intuition about these physical things carries over to abstract theoretical numbers as well.

[Dark Fire]

TeX doesn't calculate anything; it's just a type setting program. All it does is let you type up nice papers (though it is very useful for writing up your math homework in college if you're taking some advanced courses).
Ahh, never mind, it's a bug in the editing...
I first wrote my whole math, but then it looked all crazy, and it looked like it were calculating every part I wrote, so I edited it to only "-9^-1" and it looked like it were calculating the whole thing, until I refreshed the website...

And btw, I was taught that ^-1 means squareroot.
81^-2 = 3 because, 3^2 = 81

Here's my math:
I=-9^-1
I^2=(-9^-1)^2=81^-2=3

And if ^-1 doesn't mean sqrt then:
I=sqrt-9
I^2=sqrt(-9)^2=cuberoot(81)=3

[LukeD]

No, x^(-y) = 1/(x^y)

So (-9)^(-1) = 1/((-9)^1) = 1/(-9) = -1/9

However, x^(1/2) is the square root of x

You might want to look up how exponents work

(also 3^2 = 9, not 81)

[Trekky0623]

So let's assume we have a positive whole number, n.

-n = -1 * n

So let's just say n is 3.

You're adding -1 three times

(-1) + (-1) + (-1) = -3

When multiply by a negative, instead of addition, we use subtraction, so:

(-1) * (-3) = -(-1) - (-1) - (-1)

So (-1) * (-3) = 3

[Dark Fire]

(also 3^2 = 9, not 81)

Read again, it says cuberoot.

[d_leet]

Read again, it says cuberoot.

The cube root of 81 is also not equal to 3. 33=27.

[LukeD]

The cube root of 81 is also not equal to 3. 33=27.

Hey, it's your 2^{10}th post!

DarkFire: (81)^(1/4) = 3 and 3^4 = 81. Like I said, you may want to look up how exponents work. I think you're a little confused about them.

Also, one thing that you should know is that for non integer values of y, x^y is not a function. There are for instance two different values for x^{1/2}. If we let z denote one of the numbers, then -z is the other (it's just then when x is real and positive, we usually take it to be the positive number and when x is negative, we take it to be the "positive" imaginary number). In particular, if n is an integer, then x^{1/n} could be 1 of n different numbers. When y is irrational, there is an infinite number of things that x^y can be.

So this means that canceling exponentials by taking them to the 1/n power is not always valid. You need to be careful with this or you'll end up with nonsense like 1 = -1

[maze]

@ the original poster

You can also rephrase the question from the point of view of the distributive law,
1 = 1*1 = (-1+2)*(-1+2) = -1*-1 + -1*2 + -1*2 + 4 = -1*-1

Then you only need to justify the distributive law for negative numbers, which may be easier to think up real-world examples for (?)

[Peeter]

I blundered on the website of the url below, and searched on Euler just for fun and happened to find the following:

"An introduction to the elements of algebra, designed for the use of those who are acquainted only with the first principles of arithmetic. Selected from the algebra of Euler (1818)"

http://www.archive.org/details/introductiontoel00eulerich

His justification for this rule (paraphrasing) is that -a * -b, for positive a & b should have a different sign than a * -b which is negative, thus positive (exact text available above).

[maze]

Euler's argument is not particularly convincing IMO. The counterargument being, that for a negative times a negative, we might want to define it to be a whole other sort of number altogether (eg: -a*-b = ja*b, where j is -1*-1, analogous to i in complex numbers).

[Gib Z]

It may be simplest if we explain it as if numbers were vectors along the real number line.

For all numbers, we can split it into two things: Its magnitude, and its sign. We can interpret the sign as to be the direction it in going in, and magnitude is how far it goes. A positive sign just means go towards the right, and negative signs just mean switch directions. So having multiplying two negative signs can be seen as changing your direction but then changing back!.

[Peeter]

Euler's argument is not particularly convincing IMO. The counterargument being, that for a negative times a negative, we might want to define it to be a whole other sort of number altogether (eg: -a*-b = ja*b, where j is -1*-1, analogous to i in complex numbers).

His argument works nicely for the i of complex numbers too. Either way it flips the sign, and application twice gets you right back to where you were. This is why I like the geometrical view of negation ... it's a 180 degree rotation, or inversion, depending on the dimension of the space.

[Borek]

In terms of quaternions Eulers argument seems weak, I am with maze on that one.

[Peeter]

In terms of quaternions Eulers argument seems weak, I am with maze on that one.

I don't follow what you are getting at? You've got:

i(-j) = -k

so, in my mind his argument would say that one should have:

(-i)(-j) = k

[arildno]

It's the only way I've been able to understand multiplication, we were also taught to think of it that way when we were childs, and I never had a flash/bright idea/philosophy popping up, telling me maybe multiplication isn't that way of thinking after all: until now.

Thank you all for the replies, feeling good for not being a fool after all :)

It's better to think of multiplication as one particular map (amongst numerous others) between two numbers yielding a third number.

That mapping, called multiplication has numerous "properties" set down as so-called axioms.

To give you an idea about what I mean with "map", we will look at the "addition map A", associating to two numbers a,b a third number c called their "sum".

For example A(1,1)=2 (i.e we associate a=1 and b=1 with the number c=2, calling c the sum of a and b)

Similarly, we will have, for example, A(5,1)=6 A(7,13)=20 and so on.

Such a map has a very intuitive representation as a table.


Your multiplication table, with row numbers and column numbers is precisely such a map, or rather, a PART of such a map.

Thus, rather than thinkinking of multiplying two numbers as meaning "having so and so much" of a quantity, you should rather think of multiplication as a particular set of properties with the multiplication table, those properties enabling you to calculate the product, either by looking up in a pre-written table, or perform the multiplication by "multiplying", i.e, assigning the right c-number to the chosen a,b-numbers.

[maze]

But that is dodging the question. We can all make up number systems with all sorts of properties (or maps, or sets with operations, or however you want to describe it). In fact, people already have done so, and the results are often interesting.

However, what takes math beyond being a random game played with symbols is the strange coincidence that numbers, sets, and so forth actually have a connection to reality (or at least connections to things with a connection to reality). That is what the origional poster is getting at - what is the connection between the formalism and real world things (like paying for a $10 pizza split 5 ways).

[Borek]

I don't follow what you are getting at? You've got:

i(-j) = -k

so, in my mind his argument would say that one should have:

(-i)(-j) = k

What I meant was that we can extend the idea of sign. We can assume (-1)(-1) = 1, we can assume i^2 = -1, or (-i)(-j) = k - whatever. We can also assume that each combination like -+-++ is a completely new entity. Eulers argument will be wrong then, as it works only when the final result has to have one of two possible signs.

In a way Eulers argument is what I hinted at at the very beginning of this thread.

[ehj]

Well.. let's say that two negatives multiplied gave another negative:

(-1)(-1) = -1 and at the same time 1*(-1) = -1
That would imply that (-1)(-1) = 1*(-1) <=> 1 = -1 which is obviously not true.
It probably is a convention, although a convention that makes sense as we've just seen!

[Crosson]

But that is dodging the question. We can all make up number systems with all sorts of properties (or maps, or sets with operations, or however you want to describe it). In fact, people already have done so, and the results are often interesting.

However, what takes math beyond being a random game played with symbols is the strange coincidence that numbers, sets, and so forth actually have a connection to reality (or at least connections to things with a connection to reality). That is what the origional poster is getting at - what is the connection between the formalism and real world things (like paying for a $10 pizza split 5 ways).

I think you can't see the purpose of real numbers in an isolated problem, for example about mortgage debt, since even semi-intelligent human calculators can semi-consciously carry their negative signs through a word problem --- e.g. "I never add a positive to a negative, but rather I treat this as a subtraction of two positives in the order of larger - smaller and assign the the parity of the sum to be that of the larger number." or "obviously not paying a $500 expected mortgage for 5 months is not a savings of (-$500)(-5) but rather to ($500)(5); although these are the same, certainly only a fool with the mind of a machine would actually need to think in terms of the former."


In other words, negative signs only make sense inside of an established system. Consider the formula for average velocity of an object along a line. If we want there to be such a universal equation, we need a convention: positions along the line are represented by real numbers with the particle considered to be at position x = 0 at time t = 0 and the positive axis in the direction along the line most angularly near to north.

This means that the average velocity during the a trip of duration t to position x is given by the equation:

v = x/t

Where negative velocities correspond to negative net displacements. The bottom line is that instead of doing the book keeping in our heads, we can establish the conventions and express ourselves with more brevity. This is not apparent in the system I used as an example, where the weight of the conventions exceeded the value of the "universal" formula I gave, but that is because I wanted it to make the point while still being accessible to everyone.

Furthermore, your critique so far has only been half an argument, since you have made the unconcious assumption that numbers represent either magnitudes, cardinals, or ordinals. In fact, neither the negative reals nor the negative integers are any of these, but rather they are inverses of addition. If that is not real to you, than you are drawing your own arbitrary line in the sand without explaining and defending what you think it means to "connect to real world things."

[arildno]

coincidence that numbers, sets, and so forth actually have a connection to reality (or at least connections to things with a connection to reality).

Nope. That is an attribute of the real world, not of maths.

Maths IS, essentially, games of our own making.

[CRGreathouse]

Complex numbers allow any polynomial in real (or complex) numbers to be solved. In the same way, (possibly negative) integers allow any monic linear equation to be solved, and rational numbers allow any linear equation to be solved.

Without negatives it's hard to factor polynomials.

[Borek]

Nope. That is an attribute of the real world, not of maths.

Maths IS, essentially, games of our own making.

My take is that world have no choice, but to follow math. And while we can treat math as a "game of our own making" math in fact does exist on its own. Just because we are sometimes surprised by the fact that some esoteric math theory describes the way umpth dimension curls in the presence of blablah field proves math existence. After all, world worked the way it does long before we learnt how to count to three.

[LukeD]

Just because we are sometimes surprised by the fact that some esoteric math theory describes the way umpth dimension curls in the presence of blablah field proves math existence.

You're confusing theory with fact. No one knows that any of our physics describes things how they actually are. In fact, most people suspect that every piece of physics that we have to date is actually completely wrong (though correct to some approximation)

[Borek]

In fact, most people suspect that every piece of physics that we have to date is actually completely wrong (though correct to some approximation)

I am in minority (which is most likely the majority) then. Still, even if that's true there is no reason for the real physics to not follow math. Experience so far tells us that that's the way it is.

[LukeD]

I am in minority (which is most likely the majority) then. Still, even if that's true there is no reason for the real physics to not follow math. Experience so far tells us that that's the way it is.

Sure, it's possible that the "ultimate laws of physics" are describable by mathematics. But how does that make the math that describes them real? For any phenomenon, there is an infinite number of mathematical frameworks that describe it. If some framework that describes reality uses mathematical concept A, how does that make A "real"?

You're throwing around works like "real" and "mathematics" (not to mention "real physics"), but I have no idea what you even mean by those words, so I can't actually discuss them (and these are by no means words that have well accepted definitions).

Anyway, this conversation is off topic and should probably be moved elsewhere if you wish it to continue (there is a philosophy board which I suspect would be better suited for this)

[Diffy]

most people suspect that every piece of physics that we have to date is actually completely wrong (though correct to some approximation)

Is this true? Do you have a source or is this an opinion?

So if we take the mass of an object, and say said object is accelerating at some rate and we multiply the mass and the acceleration, we would not have the force? Is this due to an error in formula (perhaps we should subtract?), or numerical inaccuracy in measuring mass and acceleration?

[Borek]

http://www.physicsforums.com/showthread.php?t=237609

[LukeD]

Is this true? Do you have a source or is this an opinion?

So if we take the mass of an object, and say said object is accelerating at some rate and we multiply the mass and the acceleration, we would not have the force? Is this due to an error in formula (perhaps we should subtract?), or numerical inaccuracy in measuring mass and acceleration?

I may have been exaggerating when I said every piece of physics, but every physicist I've spoken to or heard speak has expressed that they don't believe that our best physics theories are exactly correct (as in down to every detail and accurate on all scales).

If some mathematical structure happens to describe what is going on with a physics theory, but the theory doesn't work everywhere, then the math only approximately describes the situation. Therefore, the math is an idealization, so it wouldn't be considered "real". That's why I'm not considering any physics theories that are only approximately correct. What I meant is that every physicist I've heard speak has expressed that every theory that we have is only approximately correct.

----

By the way, what you said is just the definition of a force. There's no physics in a definition; it's just naming something. I take it that what you meant then is conservation of momentum. Maybe conservation of momentum is correct; it is one thing that few people who think could be violated. It would certainly surprise most people. So that might be one thing that most people do think is correct.

[Dark Fire]

we can assume i^2 = -1

I'm not really into math atm, however, doesn't that also mean that I^3=1.
But cuberoot of 1 is only 1, not the sqrt of -1.

My take is that world have no choice, but to follow math. And while we can treat math as a "game of our own making" math in fact does exist on its own.

And either way is there no way to prove either, at least not closer than the distant future, so discussing either theory is, in my opinion, of the same level of discussing Christianity versa Buddhism; you're going nowhere.

[CRGreathouse]

I'm not really into math atm, however, doesn't that also mean that I^3=1.
But cuberoot of 1 is only 1, not the sqrt of -1.

No, that means i^3 = -i and i^4 = 1. The four fourth roots of 1 are 1, i, -1, and -i. (There are four roots thanks to the fundamental theorem of algebra.)

[Dark Fire]

No, that means i^3 = -i and i^4 = 1. The four fourth roots of 1 are 1, i, -1, and -i. (There are four roots thanks to the fundamental theorem of algebra.)

That told me nothing but to memorize that it's ^4 and not ^3: I did not become any wiser.

[CRGreathouse]

I did not become any wiser.

Is that because you already knew the fundamental theorem of algebra, or because you declined to learn it? I think it's a radically important part of mathematics.

[Dark Fire]

Is that because you already knew the fundamental theorem of algebra, or because you declined to learn it? I think it's a radically important part of mathematics.

It's because I decided not to google it-- rather pray for you to explain.

[LukeD]

Dark Fire: I see that you still haven't looked up how exponentiation works, but I'll be happy to refresh you.

If n > 0 is an integer, then x^n means x*x*x*...*x (with n terms).
Exponentiation is then extended continuously to most real numbers (except for 0^0 which is usually undefined and 0^(-a) where a is any positive real number) via the following properties which you can prove in the case where we had x^n and n > 0 was in integer

(x^a)*(x^b) = x^(a+b)
(x^y)^z = x^(y*z)
(x^z)*(y^z) = (x*y)^z
x^0 = 1 (as long as x is not 0)

We then just assume that as long as x is not 0, then we have at least one number defined by x^y

By the first property, if x is not 0, we have that (x^a)*(x^(-a)) = x^0 = 1, so this tells us that x^(-a) = 1/(x^a)

By the second property, we have that if x is not 0, then (x^(1/a))^a = x^(a/a) = x^1=x

In particular, this means that x^(1/2) is one of the square roots of x

-----

i^2 = -1 means that i*i=-1
Then multiplying by i again gives
i^3 = -1*i = -i
Then we can evaluate i^4 as
i^4 = i^(2+2)=(i^2)*(i^2) = (-1)*(-1)=1

---

By the way, I said before that we just assumed that we had some number x^y that satisfied the required properties, but at least in the case where x and y are real numbers and x is non zero, we can prove not only that at least one such number exists, but we can prove that x^y is continuous in both x and y

[Dark Fire]

i^2 = -1 means that i*i=-1
Then multiplying by i again gives
i^3 = -1*i = -i


I know the general of how exponents work, which is why I've never bothered to look it up, correct.
We never learned about exponents which is less than 1, including negative numbers, however I've seen exponents with negative numbers earlier (not @ school), but never {0, 1} or [0, 1] I forgot which one is which (never learned this @ school either, I haven't done high-school).
I just haven't done school/math in a couple of years, but yes, I realize that it was wrong twice, it should be ^4, actually without even reading the whole sentence, though my mistakes were irrelevant in both cases, so I honestly think you were going off-topic (you in plural).
i^4=1
but
1^0.25!=i
where ! means not equal to, and don't go off-topic again if 1^0.25 is wrong :P
If it's wrong, then please tell me if you want, but add a reply to what's on-topic:
I^4 is 1 but 1 4root isn't only i but 1, meaning i can also be 1 since 1*1*1*1=1 right?
2root = squareroot, 3root = cuberoot, 4root = unnamed.

[DavidWhitbeck]

Well.. let's say that two negatives multiplied gave another negative:

(-1)(-1) = -1 and at the same time 1*(-1) = -1
That would imply that (-1)(-1) = 1*(-1) <=> 1 = -1 which is obviously not true.
It probably is a convention, although a convention that makes sense as we've just seen!

You can prove that (-1)(-1) = 1 just by saying that R with the + and x operators form a field (as part of the definition of R).

(-1)(-1) - 1 = (-1)(-1) + (-1)(1) by the defn of 1
= (-1)(-1 + 1) by distributive law
= -1(0) since -1 is the additive inverse of 1
= 0 by the defn of 0
but
1 - 1 = 0 since -1 is the additive inverse of 1

but now we have shown that
(a) (-1)(-1) is the additive inverse of -1 and
(b) 1 is the additive inverse of -1.

But additive inverses are unique. Therefore (-1)(-1) = 1. Is there flaw in that argument? I'm weak with fields so if you poke holes in it I won't be sad.

[CRGreathouse]

It's because I decided not to google it-- rather pray for you to explain.

A polynomial of degree n with real (or complex) coefficients has exactly n roots, counting multiplicity. So there are six roots for x^6, x^6 + 3x^4 - 2x, and pi * x^6 - e * x^5 + 3.132432567. The fact that there are n nth roots of unity is just a special case of this. In the case of roots of unity, they are always distinct (unlike (x - 2)(x - 2) which has two roots, both 2).

[LukeD]

I know the general of how exponents work, which is why I've never bothered to look it up, correct.
We never learned about exponents which is less than 1, including negative numbers, however I've seen exponents with negative numbers earlier (not @ school), but never {0, 1} or [0, 1] I forgot which one is which (never learned this @ school either, I haven't done high-school).
I just haven't done school/math in a couple of years, but yes, I realize that it was wrong twice, it should be ^4, actually without even reading the whole sentence, though my mistakes were irrelevant in both cases, so I honestly think you were going off-topic (you in plural).
i^4=1
but
1^0.25!=i
where ! means not equal to, and don't go off-topic again if 1^0.25 is wrong :P
If it's wrong, then please tell me if you want, but add a reply to what's on-topic:
I^4 is 1 but 1 4root isn't only i but 1, meaning i can also be 1 since 1*1*1*1=1 right?
2root = squareroot, 3root = cuberoot, 4root = unnamed.

Like I said before, x^y isn't exactly a function, it can have multiple, or even infinite values. It's just that a certain specific one called the principle value is taken as the value that people usually mean when they write x^y.

This means that x^y is not invertible. So we might have that a^2 = b^2 but not that a=b.
For instance, 1^2 = (-1)^2, but 1 is not -1.

Also, as you pointed out i^4 = 1, but 1^(1/4) = 1 (again, this is just a convention, it could just as well be -1, i, or -i)

There is nothing wrong or contradictory about this, it is just that the exponent operator does not strictly have a single value.

[elarson89]

I haven't read all 5 pages but i understand the question. I wanna restate it because I am a bit confused on it too, but in more of a general sense.

It's more intuitive to use the natural numbers in math, but somewhere along the line multiplication was generalized to include decimal, irrational, imaginary, and negative numbers. How was this set generalized, specifically with negative numbers.

Another question I have, how was exponentials generalized to include the negative numbers. (its easy to see the imaginary generalization thanks to Euler) Why is it that a negative exponent is just the reciprocal with a positive exponent?

[Egor50]

To DarkFire Specifically:

We are of like mind on the conceptual swallowing of a negative multiplier. Like you I see how negative values can add, subtract, and divide. But in multiplying I am conceptually snagged to see past zero as a mutiplier no matter how I try to construct it. You just can't multiply negatively. All negative values really say to me is, "its always some positive amount of negative value! No matter how I think of it I can't see past zero as the multiplier dead end. And this follows all of the math examples that try to show me otherwise. Now that said...Can I ignore my own inner concept that you can't multiply anything in this whole universe less than 1 and follow the "rule as explained", yes. But I am not at ease with it. It seems to violate some basic thought rule in me. And its one of those conceptual things I coined the "perceptual snag". I think the world is full of them. And due to natures constraints in the way we think, we are blind to them. And these perceptual snags remain invisible corrupting lots of areas of recorded thought. I think the idea of a negative multiplier is strong enough to be a "LAW". "There can be no multiplier less than the smallest fraction toward infinity. anything beyond (less than) that is ZERO! And how can you go negative (if the concept of negative is - = < 1) times "something" counted? Furthermore if you look at whats there in all negative math references you will see that even though math calls it negative, the value can always be interpreted as some Positive amount to a negative value. this of course really only applies to the "real world" value of negatives. But I can argue this point in its simplest form against any negative value.

[Egor50]

To DarkFire Specifically:

We are of like mind on the conceptual swallowing of a negative multiplier. Like you I see how negative values can add, subtract, and divide. But in multiplying I am conceptually snagged to see past zero as a mutiplier no matter how I try to construct it. You just can't multiply negatively. All negative values really say to me is, "its always some positive amount of negative value! No matter how I think of it I can't see past zero as the multiplier dead end. And this follows all of the math examples that try to show me otherwise. Now that said...Can I ignore my own inner concept that you can't multiply anything in this whole universe less than 1 and follow the "rule as explained", yes. But I am not at ease with it. It seems to violate some basic thought rule in me. And its one of those conceptual things I coined the "perceptual snag". I think the world is full of them. And due to natures constraints in the way we think, we are blind to them. And these perceptual snags remain invisible corrupting lots of areas of recorded thought. I think the idea of a negative multiplier is strong enough to be a "LAW". "There can be no multiplier less than the smallest fraction toward infinity. anything beyond (less than) that is ZERO! And how can you go negative (if the concept of negative is - = < 1) times "something" counted? Furthermore if you look at whats there in all negative math references you will see that even though math calls it negative, the value can always be interpreted as some Positive amount to a negative value. this of course really only applies to the "real world" value of negatives. But I can argue this point in its simplest form against any negative value.

[epkid08]

To DarkFire Specifically:

We are of like mind on the conceptual swallowing of a negative multiplier. Like you I see how negative values can add, subtract, and divide. But in multiplying I am conceptually snagged to see past zero as a mutiplier no matter how I try to construct it. You just can't multiply negatively. All negative values really say to me is, "its always some positive amount of negative value! No matter how I think of it I can't see past zero as the multiplier dead end. And this follows all of the math examples that try to show me otherwise. Now that said...Can I ignore my own inner concept that you can't multiply anything in this whole universe less than 1 and follow the "rule as explained", yes. But I am not at ease with it. It seems to violate some basic thought rule in me. And its one of those conceptual things I coined the "perceptual snag". I think the world is full of them. And due to natures constraints in the way we think, we are blind to them. And these perceptual snags remain invisible corrupting lots of areas of recorded thought. I think the idea of a negative multiplier is strong enough to be a "LAW". "There can be no multiplier less than the smallest fraction toward infinity. anything beyond (less than) that is ZERO! And how can you go negative (if the concept of negative is - = < 1) times "something" counted? Furthermore if you look at whats there in all negative math references you will see that even though math calls it negative, the value can always be interpreted as some Positive amount to a negative value. this of course really only applies to the "real world" value of negatives. But I can argue this point in its simplest form against any negative value.

Who says you have to think of numbers like you do money? Why not think of it like vector space?

[Diffy]

If you can conceptually swallow how negative value's can divide, then think of multiplication as division.

Let a, and b, be negative numbers,

Then a x b = a / (1/b)

[Egor50]

No you are not feeling my snag. Its not that I can not conceptualize the math rules for negative numbers. Even in vector space, its only use is as a reference below some scalier reference. At best as a concept, I still see it as non multiplier in any sense unless there is a corresponding negative value. Do you get me? Its deeper to me. Just "CONCEPTUALLY" a stand alone concept. Negative values can not be used as the multiplier. Multiplicand? Yes!

[BSMSMSTMSPHD]

Proof that (-1)(-1) = 1

Let a be any real number. Then, a = (a + 1) - 1, and so, a2 = [(a + 1) - 1]2.

Now, just expand both sides and simplify:

a2 = (a + 1)2 - 2(a + 1) + (-1)(-1)

a2 = a2 + 2a + 1 - 2a - 2 + (-1)(-1)

0 = -1 + (-1)(-1)

1 = (-1)(-1).

[arildno]

No you are not feeling my snag. Its not that I can not conceptualize the math rules for negative numbers. Even in vector space, its only use is as a reference below some scalier reference. At best as a concept, I still see it as non multiplier in any sense unless there is a corresponding negative value. Do you get me? Its deeper to me. Just "CONCEPTUALLY" a stand alone concept. Negative values can not be used as the multiplier. Multiplicand? Yes!

Your "concept" about distinctive roles of multipliers and multiplicands is totally WORTHLESS, since multiplication is commutative.

Your "concept" is just another misconception.