How can you multiply two negative numbers?

[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 unconscious 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 learned 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]

https://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 want to 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 what's 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 what's 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 what's 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.