My 6th- and 7th-grade students are pretty effective at calculating with negative numbers. They all know, for example, that 5 – (-2) = 7. Ask them why, and you’ll hear this:

“Because two negatives make a positive!”

Then, if you listen carefully, you will hear something else: the low rumble of my teeth grinding together with tectonic force.

“Two negatives make a positive” is one of those math slogans that drives me crazy, because it is so pithy, so memorable, so easy to apply… while also being so vague and non-mathematical that I’m amazed students find it useful at all.

We can all think of many, many cases where two negatives don’t make a positive. Rain on your wedding day plus grand larceny on your wedding day does not make for a winning combination, despite what “two negatives make a positive” would suggest.

It’s not even true with negative numbers, where -10 + -30 does NOT equal +40 (although I have seen students claim that it does, citing “two negatives make a positive” as their justification).

In fact, that’s one of my major complaints with “two negatives make a positive”: it is such a swift, over-arching generalization that students wind up applying it in places where it doesn’t make much sense.

In fact, “two negatives make a positive” doesn’t really make much sense anywhere.

What does make sense is a slight variant, less catchy but far more true: “The opposite of the opposite is just the thing itself.”

What’s the opposite of “the opposite of happy”?

Well, “the opposite of happy” is sad.

So the opposite of that is “happy” again.

For adding and subtracting with negatives, I tend to favor a debt model.

For multiplying and dividing with negatives, I think a slightly more abstract approach is necessary – it’s all about the properties of multiplication.

Good mental models are more effective than mantras like “two negatives make a positive,” I believe. But even if they weren’t – even if the use of mantras led to error-free computation with negatives – I’d still favor the “mental model” approach. Learning new models engenders the kind of rich thinking that math class is supposed to be about; learning new mantras engenders the uncritical thinking of the cult-follower.

52 thoughts on “The Catchy Nonsense of “Two Negatives Make a Positive””

Haha! Fantastic. I never liked that phrase either, for much the same reasons. I do like the visual of the number line and that the second negative simply changes the direction of the outcome. Minus a positive ends you toward the negative direction, but minus a negative switches the direction — toward the positive end. Even my autistic child gets this.

That multiplying by negative one changes the direction you’re pointing ( i.e. rotates you by 180), can be used beautifully to introduce the complex numbers, as multiplying by iota changes your direction by half! (i.e. 90 degrees)
You’re neither pointing forward nor back, you’re pointing somewhere completely different-the imaginary axis!

I first came across this in W.W.Sawyer’s wonderful book Mathematician’s Delight.

One Day I wondered to myself how does -1 * -1 = 1? in fact how do you multiply -x by anything. I understand that 2 * -3 = -6 because -3+-3=-6 however how could you even do the opposite to begin with? negative numbers don’t exist in physically. negative numbers only exist within concept and measurement(you can have negative velocity and money)(like the i axis but more relatable). However according to the repeating rule how could you even repeat a nonexistent number by anything? I have, since then, been trying to have someone prove that you can even multiply a negative by a positive. There have been many attempts such as:
my enemies enemy is my friend(1. not true in a 3 team fight 2. it is based off of word logic)
-3*2=2*-3 (this only works a discovered relationship[3+3=2+2+2] overrides the operator and even if it does then (-3)*2=2*(-3) would then force the negative to be applied first before the multiplication(like -1^2=-2 but (-1)^2=2) even if you were to flip the operators you must put a zero at the beginning in order for it to work[0-(2)-(2)-(2)] )
(3+-3)*2=0 thus 3*2+(-3*2)=0 (again this is based off of another discovered relationship)

so far the only answer that shows this as incorrect is that the -x*-x=+x is a rule, not a relationship, and that the idea that multiplication is just the repeated number of other numbers is only partially true.

Actually I agree. I’m a hardcore questioner of conventional wisdom.

I didn’t see multiplication (or division (even though I know that in some sense they are the same things)) by two negatives (even worse, division by a negative) as a legitimate idea because I never thought of the negative as a symbol meaning opposite.

I always read the symbol as literally negative quantities, such as removing objects from existence. For instance, if I had two apples I saw negatives as literally making those apples disappear. I never thought that the opposite of an apple was a missing apple (is it?). I do not think the opposite of apples are missing apples.

Though, perhaps the opposite of food is no food.

But take marble for instance. Is the opposite of a marble a missing marble? Does that idea make sense?

I would think using this idea would be context dependent and would limit where the idea of negatives would be allowed to be used.

For instance, if I said I wanted two groups of the opposite of two apples what would that mean? Maybe somebody would grab me two oranges, or two glass apples, rather than make my two apples disappear.

This thinking works well for up and down, and all directions because the opposite of up is in fact down. So perhaps the mathematical model that is defined by arbitrarily desired properties is not universally meaningful. In that, not all of mathematics has meaning and that any specific use case meaning is intentionally context dependent and should be stated before demonstration.

“Rain on your wedding day plus grand larceny on your wedding day ..” I’m not sure rain + grand larceny is a good thing on any day, but I applaud your specificity. Is there a really interesting story behind that, or do you just have a really vivid imagination?

If we want a phrase in English that implies the correct rule without abandoning “negative,” we can say “the negative OF a negative is positive” where the word “of” in many contexts conveys multiplication, not addition.

Of course, any attempt at a mnemonic or the like is not guaranteed to give deep understanding, but just a few pegs upon which to hang something that is fairly arbitrary. All sorts of mnemonic devices are great for recalling historical dates, lists (e.g., the presidents of the United States by number, the periodic table of elements), phone numbers, etc., where there is no completely logical reason that things should be as they are. For mathematics, that would be things that are conventions (e.g., order of operations) rather than fundamental to mathematics from a logical perspective. We could have had a different order of operations or a different base or a different way of interpreting composition of functions, etc., and things would not fall apart. But having the product of two negative numbers be positive has underlying logic that we can get at in various conceptual ways. as does having subtraction of a negative number being the same as adding the inverse of that number “make sense.” There are underlying reasons to have x^0 equal 1 for all non-zero x that can be explained. And whenever there is more of an explanation than “that’s the convention,” we should help students grasp the underlying logic rather than simply dismiss their need to know by saying, “That’s the rule. Remember it!”

Perhaps the worst instance of counter-productive use of mnemonics is the teaching of the execrable FOIL, a device that supplants teaching the mathematics (learning how to apply the distributive property) and hamstrings students from actually understanding the general case that they will need for multiplying any polynomials that aren’t both binomials. What a foolish and harmful misuse of mnemonics.

The situation of “two negatives makes a positive” has the underlying danger of not only the specific error (do we mean addition or multiplication, or doesn’t make a difference?) but spreading the pernicious notion that math is memorization, not understanding.

Well said!
Even I was going to write something like yours similar, I have now written a note to remove confusions related to what exactly ‘is’ a negative number.Many people think they know what is a negative number,which actually is incorrect.

Before everyone had digital cameras, people used film, and there was a negative of the film. If one took a negative of the film negative, one would obtain something that looked like the original photo. This is an example of “A negative of the negative is (just like) the original.”

Subtraction is just the addition of negative numbers.

I say we introduce negative numbers in the second grade. Then we can get rid of this subtraction nonsense, and reduce the number of basic operations that need to be taught.

During a lecture the Oxford linguistic philosopher J. L. Austin made the claim that although a double negative in English implies a positive meaning and in French a negative one, there is no language in which a double positive implies a negative. To which Morgenbesser responded in a dismissive tone, “Yeah, yeah.”

Actually,Humans have adopted the convention,that whenever they encounter with a pair of elements,having opposite nature(behaviour), it would be nice to name one of them as positive and the other as negative.The words ‘Positive’ and ‘Negative’ are used in mathematics JUST for the sake of showing oppositeness of the two quantities,ideas or elements.

Consider in Physics.Why have we named electronic charge as negative and protonic charge as positive? Have scientists figured out some ‘tattoo’ on electron showing its negative charge(and same for proton?)? No.
We could have worked with Electricity by naming the electron charge as positive and the proton charge as negative! Nothing will change! Except the terminology.

So,in Mathematics too, we find a natural number quantity and a negative integer quantity as opposite.Think that you just now have 100 pennies.So you have 100,with no complex confusion.
You can have whatever you can by your owned 100 pennies.

Again,consider, a different situation.This time you ‘owe’ 100 pennies to one of your friends.In this situation,it is far away of thinking what to buy,instead you are to think how to pay him back 100 pennies.

Quite different situations, more precisely, the situations are ‘opposite’ to each other.
So,here, we have a pair of opposites,both regarding 100 pennies.So,if in the first case,you have 100 pennies,then how much do you have in the second case? Yes,the opposite of what you had in the first situation.So what is the opposite of the number 100?

We can nicely smoothen this problem by using the conventional use of ‘SYMBOLS FOR OPPOSITES’,that is, the words ‘positive’ and ‘negative’. (And in the written sense, by using + and -)

So let the amount you had in the first case be positive(which is more likely to name ‘positive’),so, the amount you had in the second case is Negative 100!

Strictly speaking,the pair of Positive and Negative of any elements or ideas are ONLY used to denote a sign of ‘oppositeness’.

Also one more thing I would like to include here,that the sign minus( – ), is a symbol, that we PUT between 2 numbers.The subtraction operation is supposed to be performed BETWEEN 2 numbers.
Then what does it mean by, for instance, -5?
From what is 5 subtracted from? Does that literally make sense?
Here’s the answer:
-5, is just a short form of the result of the operation 0 – 5,no matter what the result is!
Obviously, if you have something nothing(that is 0), and still need to give someone 5 things of what you have,then obviously you are forced to do the operation 0 – 5 ! It would be difficult for us to employ special symbols for the ‘opposite’ numbers (for the natural ones) ,so we used the idea of using the symbols of positives to show magnitude of the negative ,and then simple putting a minus sign ahead.
In short, without loss of generality, -5 is just a compact form for writing 0 – 5.

John Allen Paulos makes the case in his classic book Innumeracy for using a debt model to understand not only addition with negatives but also multiplication with negatives, too.

Great point – when learning about fractions, I remember being told that ‘whatever you do to the top, you do to the bottom’, which is clearly only true for multiplication and division, but not addition and subtraction! Maths teachers need to make sure what they’re teaching is true, before worrying about catchy statements.

We start out learning about numbers, whole numbers, by adding and subtracting them. These are “positive” numbers. They’re in our real world. We can picture and hold representations of them. Let’s say they’re apples. Then we learn to multiply these positive numbers. Multiplication is a short-hand way of adding numbers. 2 x 3 is 2, essentially lined up 3 times. 2 + 2 + 2. So that’s 6.

So if we can go on to this magical realm of negative numbers, if we use the “number line” concept, with a line drawn with positive numbers ranged to the right side of the zero, and negative numbers ranged to the left, we would have -2 x 3 = -2 + -2 + -2 = -6. So far so good. We’ve got -2 x 3 = -6. You could also picture this, as recommended above, to think of this as a matter of directions. That’s okay too. But it always goes off the rails for me, and has done so, for over 50 years, at the notion of -2 x -3 = 6. I cannot find a way to express this as an addition question. Multiplication IS addition. Just as division IS subtraction.

One teacher tried to explain it thus: two bad people leave town three times. That a net benefit to the town, thus a positive outcome of +6. However the people leaving town is a positive experience, so thus I understand why it’s a positive outcome. If someone would just admit that this whole concept is basically not understandable, that we’re in some magical realm of the other language of mathematics, of theories, I’d be happy. I’ve had doctors of mathematics struggle to explain this.

I’m a little late to this particular party, but I’m hoping someone might take me on, and explain this. Where am I going wrong with my “multiplication is addition” thinking and how do we apply that to the multiplication (addition) of negative numbers?

This “negative times negative makes a positive” idea is a hard one! Lots of mathematicians throughout history quibbled with it or resisted it.

One way to approach it is with this pattern:

-3 x 3 = -9
-3 x 2 = -6
-3 x 1 = -3
-3 x 0 = 0
-3 x -1 = __

For the sake of the pattern’s consistency, it would be nice if the final result were “3.” This is a peculiar kind of thinking (“I need the rules to be consistent, and the only consistent rule is ___”), native to math but found a few other places (law comes to mind). I can’t think of anywhere else it comes up in the K-6 math curriculum.

In any case, I see what you mean when you call it “not understandable.” It’s not true for any physical reason. It’s more that we want the rules of mathematics to be consistent, and it would be inconsistent if -3 x -2 gave us -6 (or any other result).

A little late, but not fifty years late…thanks, Ben. My mind is slightly less boggled. I’m groping my way to acceptance of this,(trying to build a bridge, not a wall) if only because, as you say, it’s a matter of pattern. That makes some sense, if we accept those rules.

My inner 15 year-old is still balking somewhat. How does something from the negative world suddenly become something real and tangible, in the real world,
-3 x -1 = 3. That seems like magic. Positive three, I can hold that in my hand.

Is there any other world, other than directions (vectors?), or negative/positive number lines, where negative numbers exist? Maybe that would help.

I was reading some history of math yesterday and came across this quote, perhaps relevant:

“in common life, most quantities lose their names when they case to be affirmative, and acquire new ones so soon as they begin to be negative: thus we call negative goods, debts; negative gain, loss; negative heat, cold; negative descent, ascent; &c: and in this sense indeed, it may not be so easy to conceive, how a quantity can be less than nothing”

In other words, there are lots of cases where negative numbers exist, but we tend to give them other names (like “debt”).

I tend to encourage my students to think about negatives in terms of debt:

Positive number = asset
Negative number = debt
Add = add
Subtract = take away

Thus, “5 – -3” is to start with $5, and then to have $3 of debt removed from your ledger. Hence the result of $8.

(Not sure if that’s helpful; just riffing!)

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I’ve tried the money analogy with removing debt gaining money and gotten some mileage out of it with a subset of students, but it’s not one of my go-to models. It definitely resonates for some people though and is good to have in the repertoire.

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“in common life, most quantities lose their names when they case to be affirmative, and acquire new ones so soon as they begin to be negative: thus we call negative goods, debts; negative gain, loss; negative heat, cold; negative descent, ascent; &c: and in this sense indeed, it may not be so easy to conceive, how a quantity can be less than nothing”

Thanks Ben, that is an interesting quote! I’m making some progress here with that one. Great examples of real world negative values.

I’m still picturing a “negative” debt times a negative “what?” that would equal a positive something. If I’m overdrawn in my chequing account, and I pay off my $25 debt by $5, 6 times (Surely, the “6” is a positive value?) , I’m $5 to the good. Can we express that as:
-25 + (-5 x 6) = ? or
-25 + (30) = 5

But using the rules of mathematics, instead, we get:
-25 + (-5 x 6) = ?
-25 + (-30) = -55
But I know that’s not what happens in the real world of my chequeing account.

I know you mathematicians must be shaking your heads slowly, sadly. I’ve made some basic mistake in reasoning, and can’t think what it is. Maybe time to go back to the Khan Academy and take *all* of my maths over again, and try to understand this better.

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Seems like you’re playing a bit fast and loose (unintentionally) with when to use negative signs.

In your example, your debt of 25 units is -25. But paying it off in increments is +5, not -5. So your calculations should be -25 + (5 * 6) which actually means you overpaid +5 units, right?

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Michael Paul, thanks for your comment:

“Seems like you’re playing a bit fast and loose (unintentionally) with when to use negative signs.

In your example, your debt of 25 units is -25. But paying it off in increments is +5, not -5. So your calculations should be -25 + (5 * 6) which actually means you overpaid +5 units, right?”

I’m playing fast and loose with *everything*, unintentionally. 🙂

It seems to me that the entire debt is a negative, so why would I suddenly classify any part of it as a positive? It seems that the I’m paying off -5 (the debt, which is a negative amount) six times, ( in this example, just to bring me into the positive realm.)

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I understand how you might think that, but let’s take it from another perspective.

Imagine that red chips each represent -1 and black chips each represent +1.

Also, we agree that a pair of chips of opposite colors (one red, one black) represents 0.

Make a large circle on a piece of paper. When the circle is completely empty, it represents 0. Otherwise, we need to do some sort of adding and/or removing of chips to represent other numbers. (Keep in mind that there are multiple ways to represent positive and negative integers with this model. For example, 3 black chips = +3, but so does 5 black and 2 red chips).

How would you use chips to represent a debt of 25 units?

The simplest way would be to place 25 red chips in the circle, representing -25.

Now, how do you show a payment of 5 chips?

Either you remove (subtract) 5 red chips, or you add 5 black chips, pair them with 5 red chips making five “zero pairs” which can be removed without changing the value of what’s in the circle (which will now be -20, no matter which of those two things you do).

We might think of the first action, removing negatives, as someone ‘forgiving’ part of a debt.

We might think of the second action as someone (you) paying off part of a debt.

But the net effect of these is identical. You’ve gone from -25 to -20, and IMPROVEMENT in your situation of +5.

This suggests that subtracting a negative (removing red chips) is equivalent to adding a positive (adding black chips).

Now, doing the second action six times represents adding +30 to -25 leaving a surplus of +5.

You can think of this as resulting from paying (adding) +30 (+5 * +6).

The only way things “grow” with addition/subtraction is by adding positives or subtracting negatives.

Adding negatives or subtracting positives makes the total DECREASE.

Not sure that’s helpful, hope it is.

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after reading all of this i find that my bank has been screwing me for years.

Michael Paul, good heavens, you’ve done it. You’ve cracked it!

I’ve been sitting here this morning with a piece of paper with a circle drawn on it, and some squares and triangles—-I wasn’t up for colouring :).

It works.

AND, because multiplication is basically shorthand for addition, i.e.
-5 (debt repayment) x -6 (removing 6 times from circle) = +30.

Late yesterday, I found a Reddit discussion on this topic. There was a link to a module on negatives at purple math. I can’t provide the link It seems I can’t post a link in this comment.

They had an interesting suggestion for understanding this, which got me a little further down the road:

“Imagine that you’re cooking some kind of stew in a big pot, but you’re not cooking on a stove. Instead, you control the temperature of the stew with magic cubes. These cubes come in two types: hot cubes and cold cubes.

If you add a hot cube (add a positive number) to the pot, the temperature of the stew goes up. If you add a cold cube (add a negative number), the temperature goes down. If you remove a hot cube (subtract a positive number), the temperature goes down. And if you remove a cold cube (subtract a negative number), the temperature goes UP! That is, subtracting a negative is the same as adding a positive.

Now suppose you have some double cubes and some triple cubes. If you add three double-hot cubes (add three-times-positive-two), the temperature goes up by six. And if you remove two triple-cold cubes (subtract two-times-negative-three), you get the same result. That is, –2(–3) = + 6.”

Thanks again to you, and to Ben, for your generous patience in working with me. Never too late to learn, good golly!

You’re welcome. Congrats on making sense of a difficult, abstract idea. Hot cubes/cold cubes is another good model. There are many. Whatever works.

I never stop learning new math, a subject I slept through in high school, avoided in college, then embraced personally and professionally in my thirties. Perseverance, patience, searching for good models online, and belief in your ability to learn and grow at any age are powerful, helpful attitudes and practices. I am 67 and always pushing my mathematical envelope.

Think about this, Michael Paul. I’ll always remember you as the person who helped me with this *thing* I’ve been struggling with for years. It was nice of you to do it. I’m excited to try to move beyond this, and review my high-school maths on the Khan Academy website.

I still don’t understand it, the majority of math just seems like garbage. Rules just for the sake of rules, no one ever mentions how the majority of people, will never need to use 80% of math in real life. To the one person who is saying “even my autistic son understands”, that’s very condescending…probably should have thought about posting that.

Funny, since there was just a comment here that brought me back to the topic, I’m surprised that I didn’t mention in any of my previous remarks that I’ve come to feel in the last couple of years of teaching algebra to adults at a technical school that one of the problems with “Why does multiplying two negatives give a positive?” or worse, “Why does two negatives make a positive?” which as has been discussed does not specify or imply an operation and hence is a meaningless claim, is that it misses one of the more helpful linguistic interpretations that is, I think, perfectly in tune with the underlying mathematics.

What I’m talking about is the fact that the negative sign (often carelessly and misleadingly called “the minus sign,” which implies subtraction for many people) is “really” an “opposite” sign (or additive inverse sign). So when we write, 3 – (-3) = 6, I think a very useful way to read that is “3 plus the opposite of negative 3” which is equivalent to “3 plus 3” which is unequivocally equal to six. What I’m suggesting is to always read “subtraction” as addition, which is how it’s actually defined: e.g., 3 – 3 = 3 + (-3) or in words, “three plus the opposite of three” which better equal 0, and happily, it does.” And then we should be comfortable reading 3 – (-3) as I suggested: “3 plus the opposite of negative 3” which is 3 + 3 = 6.

The problem stems from overloading the “-” symbol to mean both negative for the sign of a number and the sign for the operation subtraction. And so consecutive “negative signs” is a real bear to think through. As we’ve discussed in this thread, you can just make it a “rule,” but most of us would agree that that is an unsatisfying, mechanical approach.

I suggest that it’s better to read a single negative sign between two numbers as “adding the opposite of the second argument to the first argument. And if another negative sign follows the first one immediately, it can’t be anything but “the opposite or additive inverse of the number attached to me.”

This becomes even more useful when dealing with variables. Do we tell students to read -(-x) as “negative negative x”? I did, for years. But now I read that as “the opposite of the opposite x.” Most of my students tell me that this has made an enormous difference for them with correctly interpreting algebraic expressions with negative signs. Keep in mind that students tend to assume that “x” alone is positive and that “-x” is negative when as most of us know from experience, no such assumptions can be made. After all, “x” might equal 0 and then x = -x. And if x is < 0 (negative), then x < -x, something that befuddles many students on standardized tests.

Now, I will leave it as an exercise for any actual readers to see if it's sensible to carry the above forward into multiplication/division. I believe that it is.

Michael Paul, the idea of opposites is interesting. Maybe the number-line is not our friend. This is really working for multiplication for me, without having to turn my mind into a pretzel thinking about hot and cold cubes. But it falls apart with division for me. I must be missing something.

So thinking of opposites:

a) (-3) x (-4) = 12 because the opposite of (-3) is 3 and the opposite of (-4) is 4, and 3 x 4 = 12

b) (-12) / (-4) = 3 because the opposite of (-12) is 12, and the opposite of (-4) is 4, and 12 / 4 =3

This is true:
c) 12 / 4 = 3

But is this also true?

d) 12 / (-4) = 3 if 12 is divided by the opposite of (-4) which is 4 this becomes 12 / 4 = 3

Can both c) and d) both be true? Bear with me. I’m >63 🙂

My 7th grader was super frustrated and could not just accept “two negatives make a positive” and I couldn’t explain it for the life of me. This was the lightbulb for her, the opposite of the opposite is the the thing itself. Thank you for the super helpful article!!!

The ‘two negatives make a positive’ is actually a grammatical issue in language. For certain languages, like standard English multiple negation can lead to a weakened affirmative, viz “I don’t know nothing”, meaning “you must know something”.

Haha! Fantastic. I never liked that phrase either, for much the same reasons. I do like the visual of the number line and that the second negative simply changes the direction of the outcome. Minus a positive ends you toward the negative direction, but minus a negative switches the direction — toward the positive end. Even my autistic child gets this.

Love your posts and cartoons. 😀

Me too!

That multiplying by negative one changes the direction you’re pointing ( i.e. rotates you by 180), can be used beautifully to introduce the complex numbers, as multiplying by iota changes your direction by half! (i.e. 90 degrees)

You’re neither pointing forward nor back, you’re pointing somewhere completely different-the imaginary axis!

I first came across this in W.W.Sawyer’s wonderful book Mathematician’s Delight.

One Day I wondered to myself how does -1 * -1 = 1? in fact how do you multiply -x by anything. I understand that 2 * -3 = -6 because -3+-3=-6 however how could you even do the opposite to begin with? negative numbers don’t exist in physically. negative numbers only exist within concept and measurement(you can have negative velocity and money)(like the i axis but more relatable). However according to the repeating rule how could you even repeat a nonexistent number by anything? I have, since then, been trying to have someone prove that you can even multiply a negative by a positive. There have been many attempts such as:

my enemies enemy is my friend(1. not true in a 3 team fight 2. it is based off of word logic)

-3*2=2*-3 (this only works a discovered relationship[3+3=2+2+2] overrides the operator and even if it does then (-3)*2=2*(-3) would then force the negative to be applied first before the multiplication(like -1^2=-2 but (-1)^2=2) even if you were to flip the operators you must put a zero at the beginning in order for it to work[0-(2)-(2)-(2)] )

(3+-3)*2=0 thus 3*2+(-3*2)=0 (again this is based off of another discovered relationship)

so far the only answer that shows this as incorrect is that the -x*-x=+x is a rule, not a relationship, and that the idea that multiplication is just the repeated number of other numbers is only partially true.

Love it

Actually I agree. I’m a hardcore questioner of conventional wisdom.

I didn’t see multiplication (or division (even though I know that in some sense they are the same things)) by two negatives (even worse, division by a negative) as a legitimate idea because I never thought of the negative as a symbol meaning opposite.

I always read the symbol as literally negative quantities, such as removing objects from existence. For instance, if I had two apples I saw negatives as literally making those apples disappear. I never thought that the opposite of an apple was a missing apple (is it?). I do not think the opposite of apples are missing apples.

Though, perhaps the opposite of food is no food.

But take marble for instance. Is the opposite of a marble a missing marble? Does that idea make sense?

I would think using this idea would be context dependent and would limit where the idea of negatives would be allowed to be used.

For instance, if I said I wanted two groups of the opposite of two apples what would that mean? Maybe somebody would grab me two oranges, or two glass apples, rather than make my two apples disappear.

This thinking works well for up and down, and all directions because the opposite of up is in fact down. So perhaps the mathematical model that is defined by arbitrarily desired properties is not universally meaningful. In that, not all of mathematics has meaning and that any specific use case meaning is intentionally context dependent and should be stated before demonstration.

“Rain on your wedding day plus grand larceny on your wedding day ..” I’m not sure rain + grand larceny is a good thing on any day, but I applaud your specificity. Is there a really interesting story behind that, or do you just have a really vivid imagination?

“Rain on your wedding day” from Alanis Morissette… and grand larceny because hey, why not?

(My own wedding day was mostly a matter of dust-storms, only some of them emanating from the dance floor.)

This is definitely a not bad post!

Love you comment!

OOPS – meant “your”

If we want a phrase in English that implies the correct rule without abandoning “negative,” we can say “the negative OF a negative is positive” where the word “of” in many contexts conveys multiplication, not addition.

Of course, any attempt at a mnemonic or the like is not guaranteed to give deep understanding, but just a few pegs upon which to hang something that is fairly arbitrary. All sorts of mnemonic devices are great for recalling historical dates, lists (e.g., the presidents of the United States by number, the periodic table of elements), phone numbers, etc., where there is no completely logical reason that things should be as they are. For mathematics, that would be things that are conventions (e.g., order of operations) rather than fundamental to mathematics from a logical perspective. We could have had a different order of operations or a different base or a different way of interpreting composition of functions, etc., and things would not fall apart. But having the product of two negative numbers be positive has underlying logic that we can get at in various conceptual ways. as does having subtraction of a negative number being the same as adding the inverse of that number “make sense.” There are underlying reasons to have x^0 equal 1 for all non-zero x that can be explained. And whenever there is more of an explanation than “that’s the convention,” we should help students grasp the underlying logic rather than simply dismiss their need to know by saying, “That’s the rule. Remember it!”

Perhaps the worst instance of counter-productive use of mnemonics is the teaching of the execrable FOIL, a device that supplants teaching the mathematics (learning how to apply the distributive property) and hamstrings students from actually understanding the general case that they will need for multiplying any polynomials that aren’t both binomials. What a foolish and harmful misuse of mnemonics.

The situation of “two negatives makes a positive” has the underlying danger of not only the specific error (do we mean addition or multiplication, or doesn’t make a difference?) but spreading the pernicious notion that math is memorization, not understanding.

I was going to post something very similar. Well said.

Well said!

Even I was going to write something like yours similar, I have now written a note to remove confusions related to what exactly ‘is’ a negative number.Many people think they know what is a negative number,which actually is incorrect.

Woohoo!! I read some of this out loud to my family and we all loved it. Keep it up!

Before everyone had digital cameras, people used film, and there was a negative of the film. If one took a negative of the film negative, one would obtain something that looked like the original photo. This is an example of “A negative of the negative is (just like) the original.”

of course if you are talking about quarks where anti up isnt down and anti down isnt up!

Subtraction is just the addition of negative numbers.

I say we introduce negative numbers in the second grade. Then we can get rid of this subtraction nonsense, and reduce the number of basic operations that need to be taught.

Ben – you never fail to make me smile 🙂 Thanks for another great post.

This post is not not very good.

During a lecture the Oxford linguistic philosopher J. L. Austin made the claim that although a double negative in English implies a positive meaning and in French a negative one, there is no language in which a double positive implies a negative. To which Morgenbesser responded in a dismissive tone, “Yeah, yeah.”

Actually,Humans have adopted the convention,that whenever they encounter with a pair of elements,having opposite nature(behaviour), it would be nice to name one of them as positive and the other as negative.The words ‘Positive’ and ‘Negative’ are used in mathematics JUST for the sake of showing oppositeness of the two quantities,ideas or elements.

Consider in Physics.Why have we named electronic charge as negative and protonic charge as positive? Have scientists figured out some ‘tattoo’ on electron showing its negative charge(and same for proton?)? No.

We could have worked with Electricity by naming the electron charge as positive and the proton charge as negative! Nothing will change! Except the terminology.

So,in Mathematics too, we find a natural number quantity and a negative integer quantity as opposite.Think that you just now have 100 pennies.So you have 100,with no complex confusion.

You can have whatever you can by your owned 100 pennies.

Again,consider, a different situation.This time you ‘owe’ 100 pennies to one of your friends.In this situation,it is far away of thinking what to buy,instead you are to think how to pay him back 100 pennies.

Quite different situations, more precisely, the situations are ‘opposite’ to each other.

So,here, we have a pair of opposites,both regarding 100 pennies.So,if in the first case,you have 100 pennies,then how much do you have in the second case? Yes,the opposite of what you had in the first situation.So what is the opposite of the number 100?

We can nicely smoothen this problem by using the conventional use of ‘SYMBOLS FOR OPPOSITES’,that is, the words ‘positive’ and ‘negative’. (And in the written sense, by using + and -)

So let the amount you had in the first case be positive(which is more likely to name ‘positive’),so, the amount you had in the second case is Negative 100!

Strictly speaking,the pair of Positive and Negative of any elements or ideas are ONLY used to denote a sign of ‘oppositeness’.

Also one more thing I would like to include here,that the sign minus( – ), is a symbol, that we PUT between 2 numbers.The subtraction operation is supposed to be performed BETWEEN 2 numbers.

Then what does it mean by, for instance, -5?

From what is 5 subtracted from? Does that literally make sense?

Here’s the answer:

-5, is just a short form of the result of the operation 0 – 5,no matter what the result is!

Obviously, if you have something nothing(that is 0), and still need to give someone 5 things of what you have,then obviously you are forced to do the operation 0 – 5 ! It would be difficult for us to employ special symbols for the ‘opposite’ numbers (for the natural ones) ,so we used the idea of using the symbols of positives to show magnitude of the negative ,and then simple putting a minus sign ahead.

In short, without loss of generality, -5 is just a compact form for writing 0 – 5.

John Allen Paulos makes the case in his classic book Innumeracy for using a debt model to understand not only addition with negatives but also multiplication with negatives, too.

Debt is A model, but it’s not THE model. There isn’t a definitive model. And I don’t care if Paulos or Jesus H. Gauss says otherwise.

Great point – when learning about fractions, I remember being told that ‘whatever you do to the top, you do to the bottom’, which is clearly only true for multiplication and division, but not addition and subtraction! Maths teachers need to make sure what they’re teaching is true, before worrying about catchy statements.

Minus x minus = plus. Such heartwarming nostalgia. I actually want to go back to school now. Thank you for this post. 🙂

We start out learning about numbers, whole numbers, by adding and subtracting them. These are “positive” numbers. They’re in our real world. We can picture and hold representations of them. Let’s say they’re apples. Then we learn to multiply these positive numbers. Multiplication is a short-hand way of adding numbers. 2 x 3 is 2, essentially lined up 3 times. 2 + 2 + 2. So that’s 6.

So if we can go on to this magical realm of negative numbers, if we use the “number line” concept, with a line drawn with positive numbers ranged to the right side of the zero, and negative numbers ranged to the left, we would have -2 x 3 = -2 + -2 + -2 = -6. So far so good. We’ve got -2 x 3 = -6. You could also picture this, as recommended above, to think of this as a matter of directions. That’s okay too. But it always goes off the rails for me, and has done so, for over 50 years, at the notion of -2 x -3 = 6. I cannot find a way to express this as an addition question. Multiplication IS addition. Just as division IS subtraction.

One teacher tried to explain it thus: two bad people leave town three times. That a net benefit to the town, thus a positive outcome of +6. However the people leaving town is a positive experience, so thus I understand why it’s a positive outcome. If someone would just admit that this whole concept is basically not understandable, that we’re in some magical realm of the other language of mathematics, of theories, I’d be happy. I’ve had doctors of mathematics struggle to explain this.

I’m a little late to this particular party, but I’m hoping someone might take me on, and explain this. Where am I going wrong with my “multiplication is addition” thinking and how do we apply that to the multiplication (addition) of negative numbers?

This “negative times negative makes a positive” idea is a hard one! Lots of mathematicians throughout history quibbled with it or resisted it.

One way to approach it is with this pattern:

-3 x 3 = -9

-3 x 2 = -6

-3 x 1 = -3

-3 x 0 = 0

-3 x -1 = __

For the sake of the pattern’s consistency, it would be nice if the final result were “3.” This is a peculiar kind of thinking (“I need the rules to be consistent, and the only consistent rule is ___”), native to math but found a few other places (law comes to mind). I can’t think of anywhere else it comes up in the K-6 math curriculum.

In any case, I see what you mean when you call it “not understandable.” It’s not true for any physical reason. It’s more that we want the rules of mathematics to be consistent, and it would be inconsistent if -3 x -2 gave us -6 (or any other result).

A little late, but not fifty years late…thanks, Ben. My mind is slightly less boggled. I’m groping my way to acceptance of this,(trying to build a bridge, not a wall) if only because, as you say, it’s a matter of pattern. That makes some sense, if we accept those rules.

My inner 15 year-old is still balking somewhat. How does something from the negative world suddenly become something real and tangible, in the real world,

-3 x -1 = 3. That seems like magic. Positive three, I can hold that in my hand.

Is there any other world, other than directions (vectors?), or negative/positive number lines, where negative numbers exist? Maybe that would help.

I was reading some history of math yesterday and came across this quote, perhaps relevant:

“in common life, most quantities lose their names when they case to be affirmative, and acquire new ones so soon as they begin to be negative: thus we call negative goods, debts; negative gain, loss; negative heat, cold; negative descent, ascent; &c: and in this sense indeed, it may not be so easy to conceive, how a quantity can be less than nothing”

In other words, there are lots of cases where negative numbers exist, but we tend to give them other names (like “debt”).

I tend to encourage my students to think about negatives in terms of debt:

Positive number = asset

Negative number = debt

Add = add

Subtract = take away

Thus, “5 – -3” is to start with $5, and then to have $3 of debt removed from your ledger. Hence the result of $8.

(Not sure if that’s helpful; just riffing!)

I’ve tried the money analogy with removing debt gaining money and gotten some mileage out of it with a subset of students, but it’s not one of my go-to models. It definitely resonates for some people though and is good to have in the repertoire.

“in common life, most quantities lose their names when they case to be affirmative, and acquire new ones so soon as they begin to be negative: thus we call negative goods, debts; negative gain, loss; negative heat, cold; negative descent, ascent; &c: and in this sense indeed, it may not be so easy to conceive, how a quantity can be less than nothing”

Thanks Ben, that is an interesting quote! I’m making some progress here with that one. Great examples of real world negative values.

I’m still picturing a “negative” debt times a negative “what?” that would equal a positive something. If I’m overdrawn in my chequing account, and I pay off my $25 debt by $5, 6 times (Surely, the “6” is a positive value?) , I’m $5 to the good. Can we express that as:

-25 + (-5 x 6) = ? or

-25 + (30) = 5

But using the rules of mathematics, instead, we get:

-25 + (-5 x 6) = ?

-25 + (-30) = -55

But I know that’s not what happens in the real world of my chequeing account.

I know you mathematicians must be shaking your heads slowly, sadly. I’ve made some basic mistake in reasoning, and can’t think what it is. Maybe time to go back to the Khan Academy and take *all* of my maths over again, and try to understand this better.

Seems like you’re playing a bit fast and loose (unintentionally) with when to use negative signs.

In your example, your debt of 25 units is -25. But paying it off in increments is +5, not -5. So your calculations should be -25 + (5 * 6) which actually means you overpaid +5 units, right?

Michael Paul, thanks for your comment:

“Seems like you’re playing a bit fast and loose (unintentionally) with when to use negative signs.

In your example, your debt of 25 units is -25. But paying it off in increments is +5, not -5. So your calculations should be -25 + (5 * 6) which actually means you overpaid +5 units, right?”

I’m playing fast and loose with *everything*, unintentionally. 🙂

It seems to me that the entire debt is a negative, so why would I suddenly classify any part of it as a positive? It seems that the I’m paying off -5 (the debt, which is a negative amount) six times, ( in this example, just to bring me into the positive realm.)

I understand how you might think that, but let’s take it from another perspective.

Imagine that red chips each represent -1 and black chips each represent +1.

Also, we agree that a pair of chips of opposite colors (one red, one black) represents 0.

Make a large circle on a piece of paper. When the circle is completely empty, it represents 0. Otherwise, we need to do some sort of adding and/or removing of chips to represent other numbers. (Keep in mind that there are multiple ways to represent positive and negative integers with this model. For example, 3 black chips = +3, but so does 5 black and 2 red chips).

How would you use chips to represent a debt of 25 units?

The simplest way would be to place 25 red chips in the circle, representing -25.

Now, how do you show a payment of 5 chips?

Either you remove (subtract) 5 red chips, or you add 5 black chips, pair them with 5 red chips making five “zero pairs” which can be removed without changing the value of what’s in the circle (which will now be -20, no matter which of those two things you do).

We might think of the first action, removing negatives, as someone ‘forgiving’ part of a debt.

We might think of the second action as someone (you) paying off part of a debt.

But the net effect of these is identical. You’ve gone from -25 to -20, and IMPROVEMENT in your situation of +5.

This suggests that subtracting a negative (removing red chips) is equivalent to adding a positive (adding black chips).

Now, doing the second action six times represents adding +30 to -25 leaving a surplus of +5.

You can think of this as resulting from paying (adding) +30 (+5 * +6).

The only way things “grow” with addition/subtraction is by adding positives or subtracting negatives.

Adding negatives or subtracting positives makes the total DECREASE.

Not sure that’s helpful, hope it is.

after reading all of this i find that my bank has been screwing me for years.

Michael Paul, good heavens, you’ve done it. You’ve cracked it!

I’ve been sitting here this morning with a piece of paper with a circle drawn on it, and some squares and triangles—-I wasn’t up for colouring :).

It works.

AND, because multiplication is basically shorthand for addition, i.e.

-5 (debt repayment) x -6 (removing 6 times from circle) = +30.

Late yesterday, I found a Reddit discussion on this topic. There was a link to a module on negatives at purple math. I can’t provide the link It seems I can’t post a link in this comment.

They had an interesting suggestion for understanding this, which got me a little further down the road:

“Imagine that you’re cooking some kind of stew in a big pot, but you’re not cooking on a stove. Instead, you control the temperature of the stew with magic cubes. These cubes come in two types: hot cubes and cold cubes.

If you add a hot cube (add a positive number) to the pot, the temperature of the stew goes up. If you add a cold cube (add a negative number), the temperature goes down. If you remove a hot cube (subtract a positive number), the temperature goes down. And if you remove a cold cube (subtract a negative number), the temperature goes UP! That is, subtracting a negative is the same as adding a positive.

Now suppose you have some double cubes and some triple cubes. If you add three double-hot cubes (add three-times-positive-two), the temperature goes up by six. And if you remove two triple-cold cubes (subtract two-times-negative-three), you get the same result. That is, –2(–3) = + 6.”

Thanks again to you, and to Ben, for your generous patience in working with me. Never too late to learn, good golly!

You’re welcome. Congrats on making sense of a difficult, abstract idea. Hot cubes/cold cubes is another good model. There are many. Whatever works.

I never stop learning new math, a subject I slept through in high school, avoided in college, then embraced personally and professionally in my thirties. Perseverance, patience, searching for good models online, and belief in your ability to learn and grow at any age are powerful, helpful attitudes and practices. I am 67 and always pushing my mathematical envelope.

Think about this, Michael Paul. I’ll always remember you as the person who helped me with this *thing* I’ve been struggling with for years. It was nice of you to do it. I’m excited to try to move beyond this, and review my high-school maths on the Khan Academy website.

A pleasure and privilege. We’re all in similar boats on the river. Feel free to ask for ideas anytime: mikegold@umich.edu

You’re absolutely right – 10 + – 30 does not equal + 40. It equals -40. + 10 – – 30 would equal + 40.

I still don’t understand it, the majority of math just seems like garbage. Rules just for the sake of rules, no one ever mentions how the majority of people, will never need to use 80% of math in real life. To the one person who is saying “even my autistic son understands”, that’s very condescending…probably should have thought about posting that.

Funny, since there was just a comment here that brought me back to the topic, I’m surprised that I didn’t mention in any of my previous remarks that I’ve come to feel in the last couple of years of teaching algebra to adults at a technical school that one of the problems with “Why does multiplying two negatives give a positive?” or worse, “Why does two negatives make a positive?” which as has been discussed does not specify or imply an operation and hence is a meaningless claim, is that it misses one of the more helpful linguistic interpretations that is, I think, perfectly in tune with the underlying mathematics.

What I’m talking about is the fact that the negative sign (often carelessly and misleadingly called “the minus sign,” which implies subtraction for many people) is “really” an “opposite” sign (or additive inverse sign). So when we write, 3 – (-3) = 6, I think a very useful way to read that is “3 plus the opposite of negative 3” which is equivalent to “3 plus 3” which is unequivocally equal to six. What I’m suggesting is to always read “subtraction” as addition, which is how it’s actually defined: e.g., 3 – 3 = 3 + (-3) or in words, “three plus the opposite of three” which better equal 0, and happily, it does.” And then we should be comfortable reading 3 – (-3) as I suggested: “3 plus the opposite of negative 3” which is 3 + 3 = 6.

The problem stems from overloading the “-” symbol to mean both negative for the sign of a number and the sign for the operation subtraction. And so consecutive “negative signs” is a real bear to think through. As we’ve discussed in this thread, you can just make it a “rule,” but most of us would agree that that is an unsatisfying, mechanical approach.

I suggest that it’s better to read a single negative sign between two numbers as “adding the opposite of the second argument to the first argument. And if another negative sign follows the first one immediately, it can’t be anything but “the opposite or additive inverse of the number attached to me.”

This becomes even more useful when dealing with variables. Do we tell students to read -(-x) as “negative negative x”? I did, for years. But now I read that as “the opposite of the opposite x.” Most of my students tell me that this has made an enormous difference for them with correctly interpreting algebraic expressions with negative signs. Keep in mind that students tend to assume that “x” alone is positive and that “-x” is negative when as most of us know from experience, no such assumptions can be made. After all, “x” might equal 0 and then x = -x. And if x is < 0 (negative), then x < -x, something that befuddles many students on standardized tests.

Now, I will leave it as an exercise for any actual readers to see if it's sensible to carry the above forward into multiplication/division. I believe that it is.

Michael Paul, the idea of opposites is interesting. Maybe the number-line is not our friend. This is really working for multiplication for me, without having to turn my mind into a pretzel thinking about hot and cold cubes. But it falls apart with division for me. I must be missing something.

So thinking of opposites:

a) (-3) x (-4) = 12 because the opposite of (-3) is 3 and the opposite of (-4) is 4, and 3 x 4 = 12

b) (-12) / (-4) = 3 because the opposite of (-12) is 12, and the opposite of (-4) is 4, and 12 / 4 =3

This is true:

c) 12 / 4 = 3

But is this also true?

d) 12 / (-4) = 3 if 12 is divided by the opposite of (-4) which is 4 this becomes 12 / 4 = 3

Can both c) and d) both be true? Bear with me. I’m >63 🙂

hi nice cartoons bye lol

My 7th grader was super frustrated and could not just accept “two negatives make a positive” and I couldn’t explain it for the life of me. This was the lightbulb for her, the opposite of the opposite is the the thing itself. Thank you for the super helpful article!!!

I’m so glad it was helpful – and that’s great that she pushes for an explanation until she’s satisfied! It’s a great mathematical habit.

The ‘two negatives make a positive’ is actually a grammatical issue in language. For certain languages, like standard English multiple negation can lead to a weakened affirmative, viz “I don’t know nothing”, meaning “you must know something”.

True! In contrast to, say, Spanish, where “No digo nada” still communicates a negative.