## Why is a ‘minus times a minus equal to a plus’?

Among my friends I am the resident mathematics nerd and so I get asked all sorts of weird and wonderful little questions ranging from the trivial to the impossible to answer. One such question that came up while on a night out was ‘why is a minus times a minus equal to a plus?’

I have to admit that I couldn’t come up with an answer that satisfied my non-mathematical friend which bugged me no end so I asked around….’How would you explain it?’. I asked on Twitter, I asked in my office, I asked some math teachers I know. I even came up with a ‘proof’ (I am an ex-physicist so I apologise if this is considered to be far from rigorous) as follows.

From the distributive law of multiplication we have

a*(b+c) = a*b + a*c

which holds for all real numbers. So it will hold if we put a=-1, b=1 and c=-1 right? let’s do it!

(-1)*(1+(-1)) = (-1)*1 + (-1)*(-1)

1+(-1) = 0 and -1*1 = -1 so we get

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

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

rearrange to get

1 = (-1)*(-1)

Simple enough but not exactly intuitive is it? By the way, would any proper mathematicians care to comment as to whether this constitutes a proof or not?

Ideally I wanted an explanation that was completely intuitive and so far the best I have come across is the explanation from my friend Paul over at Crossed Streams who’s fiancee asked us this very question while we were all on a night out. Other good explanations involved ideas such as vectors and direction but when we tried those out on our target we were met with suspicion – ‘I thought we were talking about **numbers** and you explain using **vectors**!’

What fascinated me was that my tweets asking for explanations were possibly the most popular I have ever made. I had people phone me at my office to ask ‘So did anyone come up with a good explanation then?’

A minus times a minus is a plus….everyone knows it….but how would you explain it such that even the least mathematical person would grok it?

[Cross-posted at Crossed Streams]

How’s this for rationale for the influence of negative factors on the sign of a product (for the integers at least)?

If -a = 0 – a, and

if a * b = 0 + a + a + a + … + a wherein a appears b times as an addend,

then -a * b = (0 – a) * b = 0 + (0 – a) + (0 – a) + (0 – a) + … (0 – a) wherein 0 – a appears b times as an addend.

Hence -a * b = 0 – a – a – a – … – a wherein a appears b times as a subtrahend,

and thus -a * b = -(ab).

However if a * -b = 0 – a – a – a – … – a wherein a appears b times as a subtrahend,

then -a * -b = 0 – (0 – a) – (0 – a) – (0 – a) – … (0 – a) wherein 0 – a appears b times as a subtrahend,

then -a * -b = 0 + a + a + a + … + a wherein a appears b times as an addend,

and thus -a * -b = ab.

I think the distributive example is a good one, because the real, deep reason that negatives multiply to positives is that our very nice rules about positive numbers wouldn’t work with negatives unless it is the case that (-3)x(-4)=12, say.

Here’s another way to argue it though:

(-3)x(4)

is equal to

0-4-4-4 = -12.

That is, if multiplying by a positive integer is repeated addition, then multiplying by a negative integer is repeated subtraction. So

(-3)x(-4) = 0 – (-4) – (-4) – (-4) = 0+4+4+4 = 12

I would illustrate the distributive law argument with different numbers though:

(-5)x(10-10) = -50 + 50 = 0

A modification of the vector explanation should probably hold…

All numbers can be represented using Euler notation. Let’s start with a number, a*e^(j*0), that sits on the right hand side of the x-axis in the complex plane. If this number is now multiplied by -1, i.e. e^(j*pi), we end up with a number,

b= a*e^(j*0)*e^(j*pi)=a*e^(j*pi)

‘b’ then sits on the left hand side of the x-axis, being, as it is, the result of rotating ‘a’ by 180 degress. Now consider a further multiplicationn by -1 resulting in,

c=b*e^(j*pi)=a*e^(j*2pi)

This further rotation by 180 degrees causes ‘c’ to coincide with the location of ‘a’, thus proving that two minuses make a plus.

I think like everything in math it is just a convention with no deeper meaning. I am (almost) sure that the whole edifice could also be build on e.g. (+)*(+)=(-) and (-)*(-)=(-)

It just comes natural to us because we are used to it – but that means nothing – like e.g. the decimal system ( digital system) or standard analysis ( non-standard analysis).

The only thing you have to explain is “-1 times -1 = 1” and the general rule follows easily also for a non-mathematician.

Explain that multiplication of a number by -1 is defined as reflection around zero (in the real number line if you don’t want to mention complex numbers). It is then obvious that if you do this twice to the number 1 (i.e., if you compute -1 times -1 times 1) you get back to where you started, i.e., to 1.

The fact that this definition makes the distributive rules work can then be taken as a lucky coincidence by non-mathematicians.

Thanks for all the comments people…

@valter – I like that way of looking at it and will try it out on our target when I next see her.

My high school algebra teacher had a pretty intuitive answer for this. According to him, multiplication was adding the number a to itself b times. Multiplying a negative a by a positive b was like adding the negative number a to itself b times. Alternatively, it was like adding b to itself a times, and then putting a negative out front. So, from the alternate view, adding another negative sign simply canceled out the original, resulting in a positive.

The distributive law is of course _why_ we require a negative times a negative to be positive (and it’s what I use to explain it, but I’m not at all convinced that it’s an accessible explanation).

I’ve heard the “debt” explanation used. Say I am given 2 tenners. My net worth changes by 2*10 (=20). Say I have 2 tenners and I burn them. My net worth changes by -2*10 (=-20). Say I take on two debts of 10 pounds each (example, someone buys me a pint in london, twice, and I owe them back for it): my net worth changes by 2*-10 (=-20). A debt is like burning money. Say I cancel my two debts. My net worth changes by -2*-10 (=20): I get richer!

I have to say I’m not a great fan of this explanation either, but I’ve seen it used.

If your audience accepts that multiplication is modeled by rectangles, the side lengths being the factors and the area being the product, then it is fairly easy to draw a picture. Set the side lengths to something like (a-b) and (c-d). You will end up drawing several rectangles (e.g. ac, a(c-d), etc). The area of the (-b)(-d) rectangle will be forced to be positive for the resulting equation relating the area of (a-b)(c-d) and ac to make sense. This is another visual demonstration of the distributive law, but might appeal better to intuition.

When you tire of this on, you’ll want to tackle the “why is dividing by a fraction the same as multiplying by its reciprocal?” question…

On the way to the chippy I was thinking: to work out “(minus something) times (something)” do the “times” without the minus, then minus the result, so it’s the same as “minus (something times something)”. We use that rule _again_ when both somethings have a minus. So “(minus something) times (minus something)” is “minus (something times (minus something))” which is (same rule again) “minus minus (something times something)”. “minus minus thing” is the same as “thing”. Why? (actually I think most people capable of posing the original question are probably happy to accept that minus minus thing is thing) Because of the rule we learnt when doing subtraction of negative numbers: “minus minus thing” is “0 – minus thing” which is “0 + thing” which is “thing”.

Algebraically: (-a)*(-b) == -(a*(-b)) == -((-b)*a) == -(-(b*a)) = b*a == a*b

Basically, it is an arbitrary convention. I read a whole book on this subject a few years ago. The author demonstrated that you could construct a coherent mathematics without this convention. He sketched out the beginnings of a system in which minus times minus is still minus.

If I can recall the name of the book, I’ll post it. But I read it a few years ago and didn’t keep it.

Thanks to the rather long memory of amazon dot com, I was able to find the book I was thinking of:

“Negative Math” by Alberto A. Martinez

http://www.goodreads.com/book/show/547416

Math teacher here.

There’s no such thing as minus. Only “opposites”

Say these out loud:

I have 8 dollars.

I have the opposite of 8 dollars. (what do you have?)

I have the opposite of the opposite of 8 dollars. (what do you have?)

1) 8

2) – 8

3) – – 8, which we know as “8”

The negative numbers are the concept of an opposite to the positives. Their very origin is in the question: “what number can I add to 8 to get zero?”

Your initial question is doomed by vocabulary to begin with: take “Minus times a minus is a plus”.

do the following replacements:

minus -> opposite

times -> of

plus -> normal

Surely we can agree that the opposite of an opposite is back to normal.

(and technically the opposite is called the “additive inverse” since there are other kinds of opposites)

Other kinds like — multiplicative inverses!

Divide 100 tiles into two piles. (100 / 2)

Didn’t you just take “half of 100?” (1/2 * 100) remember, of = multiply.

Divide 100 tiles into four piles. (quarter of 100?)

Divide 100 tiles into 1 pile.

Divide 100 into … 1/2 pile. (how many more tiles do you need?)

Each time we are taking our 100 tiles and making them into a newly sized pile. We are given the number of piles to make, and we make them. When we need to make half a pile, what would the other half “be” ?

A last one to think about: divide 100 into 2/3 of a pile.

I think you could just see a – (minus) as a sign-change, and for multiplication we need to add the number of sign changes: 2 sign-changes off 1*1 makes 1*1….

Hey, i always approach this from the angle of direction, and specifically with videos of people walking forwards and backwards. I’ve uploaded the videos i use to my blog, john cleese and michael jackson. The kids in my class really seem to grasp the concept pretty much intuitively from spending fifteen minutes or so mucking around with these clips (we tend to finish on powerpoint being presented with them all simultaneously).

Hope this helps?

Ryan

There are several good explanations here: http://mathforum.org/dr.math/faq/faq.negxneg.html Ask Dr. Math is a great resource that I often refer to if I need to explain something mathematical to someone else.

I wouldn’t say that it’s a proof. It’s just verifying that the axioms hold.

Ray

I think that, despite our nice rules for the simple arithmetic operations, there are more to their personalities than we are willing to admit. In particular, although multiplication is commutative — and there’s no doubt of that — what’s being meant there is the binary or two-argument form of multiplication.

If, instead, a*b is seen through the eyes of Frege and Curry (logicians, but no matter), it might be parsed as [*(a)](b). Interpreted, the “*(a)” generates a function given the value of “a” which multiplies its argument by the value of “a”. Fed the value of “b” it produces the value of “a*b”. But “*(a)” might be considered the class of functions which “times by the value of ‘a'”.

Now, consider “*(-1)”. What does it do? Well, one thing it does is to map the positive real line into its negative, and the negative into the positive, leaving the exact zero where it belongs. If the value of “a” is positive, then “*(-a)” reflects and scales positive numbers into negatives, and negatives into positives.

So, one of my answers would be the reason why if the values of “a” and “b” are both positive, then “(-a)*(-b)” is positive is because of this latent characteristic of the binary operation of multiplication as a signed, scaling transformer.

There’s not really anything to explain. It’s a convention, just as it’s a convention that we accept Euclid’s fifth postulate. It’s a necessary convention so that the accepted rules of the numbering system follow what we would take as physical intuition (see many of the above arguments).

In the words of Abstract Algebra, multiplication by a negative reciprocal gives unity (the existence of an inverse). This is just a fancy way of stating that we want the real numbers, along with the operation of multiplication gives a group.

But no, if you’re a formalist, then there’s no real reason why -1*-1 = 1. It’s a matter of convention and a matter of making everything symmetric and pretty.

I think the following is a rigour explanation using only the axioms of a field:

Every element, a, in a field F has a additive inverse, denoted (-a), that is

a + (-a) = 0.

Now the additive inverse of (-a) is (-(-a)), so

(-a) + (-(-a)) = 0.

By adding a on both sides we get

a + (-a) + (-(-a)) = a

0 + (-(-a)) = a

(-(-a)) = a

Hence by the commutative and associative axioms we get

(-a)(-b) = (-1*a)(-1*b) = (-(-(a*b)))= a*b.

Best regards

Jonathan

Just realized that this of course doesn’t work for integers, since they do not form a field. An easy solution is of course to consider integers as rational numbers with 1 in the denominator. If we don’t want to consider rational numbers, you must notice that integers under addition is a abelian group and under multiplication a commutative monoid.

1 * a = a, – – ”here ‘a’ is any number”

hence, 1 * (-1) = (-1)

also -1 * a = -a, – -”here ‘a’ is any number”

hence, -1 * (-1) = – (-1)

Negative of +1 = -1

negative of (-1) = 1, because a number plus the negative of the

number adds up to zero. i.e. -1 + x = 0, that means x=1

From above, (-1) * (-1) = – (-1) = negative of (-1) = 1

That shows (minus * minus ) yields plus

Another Soln

This is not a proof, however a demonstration for a curious kid why

negative times negative is positive. It may be helpful to think that

when dealing with the numbers ‘-’ or ‘+’ signs are identical to ‘-1

times’ or ‘+1 times’. Here is an example:

-5 = ‘-1 times’ 5

A number plus a negative number can result zero. Lets take the

number ‘1’ that is ‘+1’, then take ‘-1’. It could have been any other

number. The MAIN assumption is:

1 -1 = 0

Without losing the effect we can times the equation by ‘-’ or ‘-1’ then:

-1 -(-1) = 0

Since both equations are the same, that means:

-1 -(-1) = 1 -1

that means:

-(-1) = 1

it shows:

‘-’ times ‘-’ is plus

—————————————————-

This approach is also helpful to show why ‘+’ times ‘-’ is ‘-’

Let us safely assume ‘1’, which is ‘+1’, times a number results in the

number, i.e. ‘1 times’ 10 is 10. Then ‘1 times’ ‘-1’ is ‘-1’ . That is:

‘+1 times’ -1 = -1

That shows: ‘-’ times ‘+’ results ‘-’

It’s definitely not arbitrary and not just a convention. That is to say that changing it would be like stating 3 = 4.

To demonstrate it numerically rather than conceptually (ie through some real world application) I would first talk about why a negative divided by a negative equals a positive. I think most people more easily understand that any non zero value divided by itself equals 1. That is, -x / -x = 1 where x > 0 and thus a negative divided by a negative is a positive. If from that they can’t see that a negative times a negative is also a positive then an example may be in order.

I would offer the following from the above:

1 = (-1 / -1)

Knowing that a positive divided or multiplied by a negative is a negative also establishes that:

-1 = (1 / -1)

-1 = (1 * -1)

With these it is easy to show that a negative times a negative must be positive:

-1 = -1

-1 = (1 / -1)

-1 * (-1) = -1 * (1 / -1)

-1 * -1 = ((-1 * 1) / -1)

-1 * -1 = (-1 / -1)

-1 * -1 = 1

This isn’t a proof of course, but it should be enough to illustrate why our mathematical system requires that a negative times a negative equals a positive. The point here being that this isn’t dependent on any particular real world application, but is an inherent truth in mathematics.

Alternately, it might be better to demonstrate why it can’t be otherwise. For example:

-5 + 5 = 0

-1 * (-5 + 5) = -1 * (0)

(-1 * -5) + (-1 * 5) = 0

Now, if a negative times a negative is a negative then (-1 * -5) would equal -5 so:

(-1 * -5) + (-1 * 5) = 0

(-5) + (-5) = 0

-10 = 0

-10 is not 0 obviously so that isn’t possible mathematically.

However, if a negative times a negative is a positive then (-1 * -5) equals 5 so:

(-1 * -5) + (-1 * 5) = 0

(5) + (-5) = 0

5 – 5 = 0

0 = 0

Consequently, the fact that a negative times a negative is a positive isn’t just some convention for the sake of convenience, it’s absolutely necessary and true.

I also wanted to point out that the original example given doesn’t constitute a formal proof. It states that for the equation, a*(b+c) = a*b + a*c, the values a = 1, b = -1, c = 1 satisfy it such that 1 = -1 * -1. Just plugging in a specific set of values to an equation won’t prove anything except the behavior of the equation with those specific values.

I’ll demonstrate with an example. Suppose I want to prove that the sum of any number with itself is equal to the product of that number by itself. That is to say:

x + x = x * x

Using the above technique I could say let x = 2 so:

(2) + (2) = (2) * (2)

4 = 4

So showing that the equation is true for x = 2 is misleading. It doesn’t indicate anything about other values of x so it does not prove that the equation is true for all values of x.

Now back to the original problem. Stating that a negative times a negative equals a positive could be expressed as:

-x * -y = z where x, y and z are real numbers greater than 0

It’s not enough to provide a single set of values that satisfy the equation. A formal proof has to prove that for all positive values of x and y, there exists a positive value of z for which the equation is true. I suspect someone has developed a formal proof for this, but I don’t have an example to show and it’s certainly beyond my skills.

That’s not to say that the example in the original post isn’t good for explanations, but it’s just not a proof.

In mathematical terms, having -1*-1 = +1 is a requirement for the sets of integers, real and complex numbers to constitute what is called a vector space.

All you guys are proving here start from one or more of the axioms of a vector space (look them up on Wikipedia), such as distributivity, commutativity and associativity. If -1*-1 = -1 these sets would no longer constitue vector spaces and everything would go very, very wrong as you have all shown! :-)

lets think in practical terms:

if a gambler describes exactly mathematically what he does when he places a bet for example.

he has $100 then he places $20 for betting ,if he looses ,he takes away $20 from his $100 now having only $80,the house adds $20.Now he wins he adds $20 ,his total is $120 and the house looses taking away $20.

Lets describes it mathematically

money in the beginning minus ( -) money bet ,if he looses it the house wins(+) and he subtracts (-) the money bet

house wins 20

100 – (+ 20) = 80 the minus with the plus is minus

money in the beginning minus ( -) money bet ,if he wins it the house looses(-) and he adds (+) the money bet

house looses 20

100 -(- 20)=120 the minus with minus is plus

Another way is to see math as a language for example,if the language is logical,math should be as well.

We can’t use double negative because it does not mean what we want to say for example:

I don’t have money => correct, if we ask , what you don’t have ? the answer is money, correct.

but if I say,I don’t have no money,we ask ,What you don’t have ? the answer is no money,so you have some money at least.

Lets think of minus and the word no as a switch.The word yes or the plus sign doe not change the meaning in the sentence,they exist by default.But the word no ,contradicts the object.

for example,in the ballot

vote yes for tax or vote no for no tax is the same thing.You’re voting for tax.

@Scott

Very helpful reply. First thing I have read that made sense of negative numbers. Thank you.

@vJD

Nothing in math is a convention with no deeper meaning. Please take into account that everything is false until proven otherwise.

Looking at it more concretely, here is my 2 cents worth.

Let us bank. If we deposit money into our account, our balance would increase in a positive direction. If we withdraw money, our balance would decrease (move in the opposite i.e. negative direction).

If I had no money in the account and I decide to deposit 2 cents per day for three days, after three days my balance would be 6 cents.

i.e (2)(3) = 6 cents (positive time positive = positive).

If I then decide to withdraw 2 cents per day for three days, after three days my balance would be 6 cents smaller (i.e 0 logically).

i.e (-2)(3) = -6 cents (negative times positive = negative).

Now if I saved/deposited 2 cents everyday for three days, that means that three days ago my balance would have been 6 cents smaller than it is today.

i.e (2)(-3) = -6 cents (positive times negative = negative).

Lastly, if I withdraw 2 cents everyday, three days ago my balance would have been 6 cents more than it is today.

i.e. (-2)(-3) = 6 (negative times negative = positive).

PS it is not money, but the lack of money that is the root to all evil!

think of it as moving in a certain direction when given an instruction.

+1 = move to right (pos.)

-1 = move to left (neg.)

* +1 means follow the directions exactly.

* -1 means invert the direction given.

so, 1) +1cm. * +1 means go right 1cm.

2) -1cm. * +1 means go left 1cm.

3) -1cm. * -1 means go left but invert the direction given and go right.

or a driving analogy,

+1*+1 = indicate right and turn right.

-1*+1 = indicate left and turn left.

-1*-1 = indicate left and turn right.

bad drivers will understand. :-)

What we need is the following powerful learning system to make it intuitive as well as mathematically pure.

This is how it is taught in the new Maths Makes Sense approach published by Oxford University Press.

…………………………………………………………….

We know that if we add any number to its negative we get zero. The most intuitive way I know for demonstrating this idea is:

* flat ground being zero

* a heap of soil being 1 and

* a hole being -1.

So a filled up hole has the same value as flat ground – zero.

1 (the heap) + -1 (the hole) = 0 (flat ground)

With this (heaps and holes) idea as a learning system the maths works beautifully and intuitively.

so -2 + 3 = 1 (imagining 3 heaps filling 2 holes resulting in one heap)

-3 x 2 = -6 (imagining 3 holes being dug 2 times resulting in 6 holes)

-6 / -2 = 3 (imagine 6 holes grouped into groups of two holes and counting the groups. How many groups? 3. This is the grouping type of division)

-6 / 2 = -3 (imagine 6 holes being shared between two. Each would get 3 holes. This is the sharing type of division)

TRICKIER

3 – -2 = 5 —— Try to picture the following in your head or draw it out ——

Imagine 3 heaps to begin with. Trying to take 2 holes away is now tricky as there aren’t any holes there to begin with. But there are lots of zeros!! And we know that 0 = 1 + -1. So if we convert two zeros we create another two holes and heaps. Now we have 5 heaps and 2 holes. So now we can take away -2 (tho holes we just created) leaving… 5 heaps.

The TRICKIEST of all is the negative x the negative.

But here it is as a proof

-3 x -2 =

We know that -2 = (0 – 2)

So -3 x -2 = -3 x (0 – 2)

(expanding) = -3 x 0 – -3 x 2

= 0 – -6

= (6 heaps and 6 holes) – (6 holes)

= 6 heaps

I attempted an answer to this question in my own way in my article :

https://ytelotus.wordpress.com/2013/06/24/arithmetic-operations/

Let me know if that makes sense.

Well, I am no mathematician but I cannot find errors in the original proof. In the past I heard it was a convention, but I was not sure since even with (-)(-)=- it would be needed the proof for (+)(+)=-. Also I saw errors in the other prooves like: -(-4)=4 we still do not know it. Most of the other are life problems constructions not math. Finally Tim: You add the unknow asumption that -x/-x=x since the division is difined by multiplication, and we can proof the follow up of the original to the rest of the negatives: Now we know (-1)(-1)=1 this implies (-x)(-y)=(-1)(x)(-1)(y)=1(xy).

Multiplication (when positive) is easily understood as repeated addition.

number * multiplier = result

number + number + number + …. + number = result

Conceptually you have a crutch that’s easily grasped: in a box of bricks, 1 brick multiplied by 4 is easily visualised as having 4 bricks in the box.

What about a positive number multiplied by a negative number? It’s much harder to visualize the outcome of 1 brick multiplied by -4. What does it even mean? What about -1 bricks? Can one have a negative brick?

We can visualize a negative brick as little hole (or gremlin) in our brick box that eats up a brick when added leaving no hole (or gremlin) and no brick. When one writes “1 brick * -4” this does not mean you start with 1 brick in the brick box and do something to it; rather it means make 4 brick-eating-holes the size of 1 brick. And that difference is very important to understand.

Let’s reverse the numbers: if we have a big brick the size of 4 bricks and we multiply by -1 we now take it to mean “make 1 brick-eating-hole the size of 4 bricks”. Even though the construction of the hole is different the effect on future bricks added to the box is the same. This lets us we say the result is equal (despite the different structure).

So what happens when we have a brick-eating-hole multiplied by -4 (or -1 * -4)? Well from the above we would say it means “make 4 brick-eating-holes of size -1 bricks”. Hmmm – “of size -1 bricks” – that makes little intuitive sense.

We need a different approach to try to understand this. We know from the above that multiplying by -1 makes a brick-eating-hole exactly the size of brick being passed to it. It’s exactly what’s needed so that when the brick and brick-eating-hole come together nothing remains (sometimes called zero!). In a sense multiplying by minus 1 creates the inverse of a brick: a brick-eating-hole. Similarly multiplying a brick-eating-hole by -1 gives us it’s inverse: a brick. Switching away from thinking of things as negatives but rather as inverses helps our intuition: especially when we remember that “* -1” creates the inverse of an object.

Hopefully most people can see how with ease how multiplying by -4 is the same as multiplying by 4 and then by -1. So we have

-1 * -4 =

-1 * 4 * -1 =

(-1 + -1 + -1 + -1) * -1 =

4

In English in our metaphor it means take a brick-eating-hole (-1) and make 4 of them, then take the inverse of that which is 4 bricks.

Or something like that! Actually the maths teacher response in the comments does a much better job of explaining things. Think opposites (or inverseses), not negatives.