# How can you multiply two negative numbers?

• Dark Fire
In summary, the conversation discusses the concept of negative numbers and how they can be multiplied. It is explained that this rule was likely created to make theoretical mathematics more practical and usable. The conversation also includes a comparison to geometrical rotations and the idea of dividing a negative number to get a positive result. The importance of understanding the source and reasoning behind mathematical rules is emphasized.
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.

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.

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

Peeter said:
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.

Borek said:
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)?

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

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

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. 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 Last edited by a moderator: Peeter said: 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?" Crosson said: 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). Last edited: 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 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. Crosson said: 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

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.

Crosson said:

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. 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. huh? 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. Last edited: Consider this example: you own 3$, then you have +3$. If you ow someone 3$, then you have -3$, or, let's 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$)=3$Last edited: Dark Fire said: 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.

I = R

tony873004 said:
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.

Last edited:
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"?

arildno said:
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 :)

Last edited:
Dark Fire said:
Is there a program similar to the TeX

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

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.

Last edited:
LukeD said:
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

Last edited:
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)

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

-n = -1 * n

So let's just say n is 3.

(-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

LukeD said:
(also 3^2 = 9, not 81)

Dark Fire said:

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

d_leet said:
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

Last edited:
@ 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 (?)

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).

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).

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!.

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