How can you multiply two negative numbers?

[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.