The idea is to differentiate between addition and multiplication. In each case, you are in some sense doing one operation and then doing another. In addition, your step size is fixed, and you do one set of steps, then from that point you do another set of steps. In multiplication, you first do one set of steps with one step size, then you note where you are, reset yourself back to the beginning, then do a new set of steps with base step size equal to the step required to get from your origin point to the place you ended up after the first set. Perhaps a bit more complicated than is necessary, but I need to make sure what I’m saying is correct in both senses being discussed.
Yeah, but you are phrasing it much more complicatedly than necessary.
I mean, sure, if you think of (1-dimensional) vectors as operators that take points to points (on a one-dimensional line), and numbers as operators that take vectors to vectors, then addition of vectors is composition of operators on points, sure, and addition of numbers is pointwise addition of their outputs, which is to say, pointwise composition of their output operators on points. But all the same, multiplication of numbers is still just composition of operators on vectors, and that’s the simple point which can be illustrated simply. No need to drown it in all the talk about resetting base step sizes; the presence of additive structure doesn’t mean the explanation of multiplicative structure needs to become any more complicated.
For illustrating multiplication, just note that -4 * -5 = +20 because “Grow four times as large, and turn around to point the opposite direction” followed by “Grow five times as large, and turn around to point the opposite direction” equals “Grow 20 times as large, and stay pointing in the same direction”. That’s simple and accurate and the complete story, and presumably quite graspable. Talking about resetting base steps is just an unnecessarily more convoluted way of saying the same thing.
You also want to illustrate -4 + -5 = -9, and that’s fine. To illustrate that, take a displacement p, and make two copies: -4p, which is four times as big as p but pointed in the opposite direction, and -5p, which is five times as big as p but pointed in the opposite direction. Then add these copies together, which, as you say, is doing one after the another. So move out four times the distance of p in the opposite direction, and then move out five times the distance of p in the opposite direction. The cumulative result of course is the displacement -9p, nine times the distance of p in the opposite direction. So -4 + -5 = -9. I have no problem with the way you’ve illustrated this. But I think you’re making the illustration of multiplication more complicated than it has to be.
For completeness, proof that 0 * x = 0 for any x. I actually had to prove this once on a test in college.
By assumption
1 + (-1) = 0
multiply by x on the right on each side and distribute.
1x + (-1)x = 0x
It looks like I have to resort to another statement that I think I can prove, but this one might be an axiom: (-1)x = -(1x). I’ll get back to that.
1x + -(1x) = 0x
but the left side is the form “a + (-a)”, which equals 0 as they must be additive inverses.
0 = 0*x
You’ve never seen my bank balance, have you?
“Our balance is negative $50? So we have to raise $50 just to be broke?” - some sitcom
Thank you for leading me to this website! Duly bookmarked!
Addition and subtraction are easily seen as simple scalar demonstrations. I’m still struggling with the neg x neg illustrations, though. All the turning around has me turned around 
and the explanations make me wonder why the end result isn’t zero in the case of something like -4*-4. Intuitively, I know this is not the case and the answer is 16, but the examples are difficult for me to follow. Don’t get me started on division.
Certainly. If you want to assume that people know what vectors are, it gets easier. I’m trying to do it without referencing them.
ETA: Ah, the post right above makes this post obsolete. But why on earth try and do it without referencing 1d vectors? Introduce the idea of 1d vectors (as operators taking points to points, if you like; i.e., displacements), and then use them. Introduce whatever makes things simple, and then do things simply.
Plus, I think our OP has already essentially said they’re perfectly comfortable with 1d vectors, though perhaps I misremember.
For glowacks, not for people still struggling to understand neg * neg = pos:
Just to expand a little more on the equivalence between your (I feel) non-simple presentation of multiplication as composition and the (I feel) much simpler presentation:
These are all operators which take displacements to displacements. What you’re calling a “base step” is just the input to such an operator; what you’re calling “resetting the base step” is just a convoluted way of talking about composing such operators. I agree with your approach overall, but even as someone who knows what you’re talking about already, I find it very difficult to follow the phrasing in terms of “reset yourself back to the beginning” and so on.
I know you’re worried about distinguishing between the two senses of composition, one for addition and one for multiplication. But this worry needn’t lead you to abandon the simple description of composition as “Do this, then do that”. Distinguishing between the two senses is just distinguishing between the two different kinds of actions you might want to compose: actions which take points to points (displacements), and higher-order actions which take displacements to displacements (scalars/linear transformations/“numbers”).
Composing operators which act on points is adding displacements. Composing operators which act on displacements is multiplying “numbers”. You can also add numbers, but doing so isn’t composition at the level of operators which act on displacements; it’s pointwise addition of their outputs, which, yes, is composition at the lower level of operators acting on points. None of this should interfere with your ability to simply describe composition at any level as “Do this, do that” for actions at that level.
Well, why do you think -4 * -4 would be 0? If you tell us why, we can help pinpoint what’s being miscommunicated in our descriptions.
You wouldn’t say that if you saw my bank statement.
To say x + y = 0 (and, equivalently, y + x = 0) is to say that x and y are reverse displacements; doing one and then the other, in either order, cancels out into doing nothing.
Clearly, every displacement has a unique reverse, which it itself is the reverse of. In other words, the reverse of the reverse of x is x.
To say N + 1 = 0 is to say that Nx + x = 0 for every x; i.e., N is the operator which takes everything to its unique reverse. Accordingly, because the “reverse of the reverse of x” is the same as “x”, we have that NNx = x for every x, which is to say, NN = 1. The latter following from the former is precisely the fact that the reverse of the reverse of any displacement is that original displacement itself.
I don’t know if you did it on purpose but that sentence you wrote actually gets to the heart of the understanding. We stop thinking of numbers as “things” and instead think of them as “transformations.”
The switch in thinking about numbers isn’t explicitly pointed out in typical grade school education so for many folks, the ideas behind “negative” numbers remain a riddle of weird rules.
Understandably, we as children get our “flawed” but actually quite usable and practical concept of “number” from our physical mapping of the real world:
We see an apple next to another apple –> we “map” that to “2”. We think of “2” as a noun.
We add another apple to the group –> we then “map” that to “3”. We think of “3” as a noun.
We see one sheep standing in each corner of a rectangular fence and we call that quantity “4”.
The fact that we can remember that “4” comes after “3” which comes after “2” is what we call “counting.”
At this point, we think of numbers as nouns instead of verbs. We don’t yet consider numbers as a “transforming process.” Today’s mathematicians think this is very rudimentary concept but this is actually quite a sophisticated leap in re-conceptualizing numbers. The early Greeks didn’t have the concept of negative numbers and yet they built ships to sail the sea; they harvested crops; the predicted where some of the stars would be at a certain times of the year.
It seems that we are naturally wired to implicitly mapping of numbers to “things.” We treat different numbers as different labels for things. There’s a fascinating book called The Number Sense that discusses all sorts of psychological experiments done on infants, chimpanzees, dogs, etc that supports this idea.
The problem with this simple mapping of counting physical objects (that works extremely well at the start) is that it doesn’t elegantly take us to more sophisticated ideas with negative and imaginary numbers using a seamless set of consistent concepts. The consistent concept that ties together the old world of “counting” to the new world of negatives & imaginaries is the idea of “transformation.” (This leads to fancier terminology like “scalars” “vectors” “fields” etc.)
It’s actually possible to “learn” negative and imaginary numbers without the concept of “numbers are transformations” but the end result is that the head is filled with strange rules. “I memorized that negative times negative is positive; don’t ask me why it works that way; that’s just the way it is and I got A’s on my tests.”
Ok, so far we’ve been “counting” apples. We see an apple, we label that distinction as “1”. We add another one – we label it as “2.” What if we want to add some extra “bookkeeping” information to that quantity? For example, I have 3 apples is one concept – but I also think of owing (debt) 2 apples to my friend. Where could we stick this extra concept of owing a debt? Could we tack that concept onto the number itself? Maybe we could use a modified number such as an underscore of “debt_” like this: debt_2 or debt_3 or debt_398472834. Maybe we’d have another set of numbers that all begin with the prefix “debt_” . How about other extra bookkeeping concepts such as apples in the “past” or apples already “eaten”? Now we’d have “past_2” or “eaten_2” – you might start to see that it’s cumbersome to attach all sorts of ad hoc prefixes to each number. Mathematicians love to whittle down all these seemingly unrelated concepts of description down to its essence. The concept of “negative” can elegantly can stand for “debt”, “past”, “eaten”, and ultimately “direction.” The negative is “opposite direction” acts as the unifying concept making all the rules of manipulating negative numbers look “right” instead of “weird”.
You can recognize if somebody hasn’t made the mental leap to treat numbers as abstract “transforms” – he’ll asks questions like these:
“I can see 2 apples in front of me. I can see that no apples is 0 apples. How do ‘see’ negative-2 apples? Since I can’t ‘see’ minus-2 apples it makes no sense.”
"The square root of -1 is ‘i’. Can someone show me a physical manifestation of ‘i’ so that I can understand it? What physical object in the world would be an example of ‘i’ ? "
We can’t provide tangible objects to represent these. This also applies to concepts like “debt.” Leaving aside the jokes, what does “debt” look like? We’ve never actually seen it. “Debt” is just a mental construct in our brains. If aliens visited our planet and scanned the entire surface, they’d see oceans, mountains, oxygen, humans, apples but no “debt.” They can’t see debt just like they can’t see any negative or imaginary numbers.
It was a bit of a joke. I go four units, then I turn around and go four units, I jump down, turn around, pick a bale of cotton, and now I’m back at zero. I don’t actually think that -4*-4=0; I just got lost in the translation.
You know… there happens to be a minor philosophical debate about the teaching approach for multiplication:
“multiplication is not repeated addition!”
PhD Mathematicians arguing with other PhD mathematicians about consistency and rigor from the very start vs realistic pedagogy for young children. Interesting thoughts from both sides of the argument.
Anyways, maybe if you conceptualize multiplication as repeated addition (as I also first learned it) and cling to that mental model, it makes the negative * negative illustrations harder to accept.
They also can’t see 2, really. They can see pairs of apples, but pairs of apples aren’t the same thing as 2.
Or rather, the sense in which a pair of apples illustrates 2 is the sense in which two sticks pointing in opposite directions illustrates -1, or two sticks pointing at right angles illustrates i. If you’re willing to speak of “seeing” 2 in the sense that the abstract transformation it represents has physical manifestations, then you should be just as willing to speak of “seeing” -1 or i, since those transformations also have just as good physical manifestations. They just happen to be transformations which act on things other than cardinalities. But that’s ok; different transformations act on different things.
You can have 3 children, but you can’t have 3.47 children. 3 acts on cardinalities, but 3.47 doesn’t act on cardinalities. But we all know 3.47 can be made to act on, say, distances. You can’t have a distance of -3.47 meters, but that’s ok. -3.47 doesn’t act on distances; but it can be made to act on directed distances (spatial vectors; displacements). One marathon runner can’t be 3.4 + 0.7i meters ahead of another, but that’s ok. 3.4 + 0.7i doesn’t act on 1-dimensional displacements; it acts on 2-dimensional displacements. Different transformations act on different things, but if one is willing to describe appreciating the presence of some natural physical interpretation of one abstract transformation as “seeing” the abstract entity in tangible form, one ought be equally willing to say so for the rest.
You do, however, illustrate well why the canonical choice of “debt” is a really bad example to choose to make the concept of negative numbers seem more concrete. Debt, and money in general, is just a game whose rules we made up. Of course we can match any mathematical concept to some game whose rules we made up. Nothing is really made all that more concrete in hearing about the application of some arbitrary math-game to some arbitrary other game. Speaking about reversals of displacements or such things more generally is a much better way to make “concrete” the applicability of the notion of negation, if concreteness is indeed a pedagogical goal one is after.
(That having been said, there are natural reasons why we made the rules of money and debt and so on the way they are, and the reason negative arithmetic applies in that context is because these model the same natural reasons that lead us to make the system of negative arithmetic the way it is. But this important connection isn’t an illustration of concreteness; it’s an illustration of the power of abstraction!)
a) 34 Start at zero and add 3, 4 times
b) -34 Start at zero and add (-3), 4 times
c) 3*-4 Start at zero and add 3, negative 4 times (go in the opposite direction)
d) -3*-4 Start at zero and add (-3), negative 4 times
Hopefully (a) is straightforward.
With (b) you start at zero, go back to -3, then to -6, then to -9 and finally to -12. The negative 3 tells you to go backwards, the 4 tells you how many times.
With (c) it’s the same thing except the 3 tells you how far to move, the -4 tells you how many times to do it, but it’s negative so you go backwards. You can see that it doesn’t matter [here] which part has the negative sign, either way you go backwards
Now, with (d), you are told to go backwards by three and do that backwards 4 times.
If it’s easier think of it this way, the first positive or negative sign tells you which way to move each time and the second positive or negative sign tells you to do that in the normal or opposite direction? More confusing, huh?
Draw a number line from -20 to 20 with ticks every 1 unit. Put your finger at zero.
(+3)*(+4), move your finger 3 units (+)forward 4 times in the (+)same direction that you were already heading
(-3)*(+4), move your finger 3 units (-)backwards 4 times in the (+)same direction that you were already heading
(+)3*(-4), move your finger 3 units (+)forward 4 times in the (-)opposite direction that you were heading
(-3)*(-4), move your finger 3 units (-)backwards 4 times in the (-)opposite direction.
Now that I’ve written that out, I see it doesn’t make a whole lot of sense like that, but I’ll leave it in on the chance that it might help.
How about this.
“Don’t not eat an apple” means your should eat an apple
“I won’t not ride my bike” means you will ride your bike
Does that help? -3*-3 is just a double negative. If you’re told not to do something twice it means you should do it, if you are told to go backwards twice, it means go forward.
Wait, this might do it.
Think about it this way, You are standing on a giant number line, the first + or - sign tells you which way to face, the second tells you which way to walk.
So…
(+3)(+4), face the positive side and walk forward
(-3)(+4), face the negative side and walk forward
(+3)(-4), face the positive side and walk backwards (you can see the same result as the above example with this)
and finally
(-3)(-4), face the negative side and walk backwards…you are now heading in a positive direction.
Does that help?
Incidentally, if you wanted to cast this reasoning into algebraic form, it would become “NN + N + 1 = NN + 0 = NN while also NN + N + 1 = (N+1)N + 1 = 0N + 1 = 1, so NN = 1”.
One doesn’t even need commutativity of addition or distributivity of multiplication over addition on the right; if the displacement x has a left inverse y (some way to undo it afterwards) which in turn has a left inverse z, then z must equal x (by consideration of x followed by y followed by z), so that x is the unique double left inverse of itself, with N above playing the role of any function (not necessarily linear) choosing left inverses for every displacement.
I know the “English double-negative grammar” is a common analogy but I don’t think it’s a good idea to map English phrases to math concepts like this.
Some students would misapply it as “-3 + -3 = 6”
For multiplication, don’t think of -4 as describing a way of walking from one point to another.
The things which stretch from one point to another are called… pointed sticks. (Actually, they’re called vectors or displacements or translations or differences or what have you, but you can think of them as pointed sticks). They have some length, and they point in some direction (one end is marked differently from the other). Nothing else about them matters except how long they are and how they’re oriented; any two pointed sticks of the same length and orientation, no matter where you happen to place them, count as equal.
For multiplication, think of numbers as ways of turning sticks into other sticks. They’re instructions, they’re computer programs. They take a stick as input and produce another stick as output.
-4 is the program “Make it four times as large as it currently is. Now turn it around 180 degrees.” -4 * any stick is another stick. I could draw a stick from LA to NYC, about 3000 miles long and pointing northeastish. I could then multiply this stick by -4, which means making another stick, four times as large and pointing in the opposite direction. So -4 * (the stick from LA to NYC) = a stick about 12000 miles long and pointing southwestish.
Ok, so what? So I can multiply that by -4 again. It becomes four times as large as that, and changes direction again. So now -4 * -4 * (the stick from LA to NYC) = a stick about 48000 miles long and pointing northeastish.
This is 16 times as large as the original stick from LA to NYC, and points in the same direction as it. So -4 * -4 * (the stick from LA to NYC) = 16 * (the stick from LA to NYC).
Of course, there’s nothing special about the stick from LA to NYC. -4 * -4 * any stick = 16 * that same stick.
And that’s the sense in which -4 * -4 = 16. The instructions “Make it four times as large as it currently is. Then turn it around 180 degrees. Then make that four times as large as it currently is. Then turn that around 180 degrees” have the same cumulative effect as the simpler instructions “Make it 16 times as large as it currently is”.
[Pedantic disclaimer: Actually, the distance from LA to NYC is somewhat less than 3000 miles, and the straight-line direction cuts through the inside of the Earth. But, whatever.]
Well, my head hurts, but I’ve never been good with spatial concepts. If I can see it demonstrated, I can usually understand it. Sort of like my previous thread about Einstein and ESP: the answer is never resolved to my satisfaction. Why these things come to me early in the morning is, I think, a bigger issue. I suspect it has something to do with less blood flowing to my penis than to my brain nowadays.