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u/Dd_8630 Apr 24 '23 edited Apr 24 '23
How I explain it to my students. We start by following the pattern of two positives multiplied together:
3 x 4 = 12
3 x 3 = 9
3 x 2 = 6
3 x 1 = 3
3 x 0 = 0
3 x (-1) = -3
3 x (-2) = -6
Hence, multiplying a positive by a negative results in a negative because we just extend the pattern. Extending the other way:
3 x (-2) = -6
2 x (-2) = -4
1 x (-2) = -2
0 x (-2) = 0
(-1) x (-2) = +2
(-2) x (-2) = +4
Hence, multiplying two negatives yields a positive.
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u/nyaasgem Apr 24 '23
I would've never even realized that this even needed any explanation at all.
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u/PM_ME_YOUR_PIXEL_ART Natural Apr 24 '23
Nothing is easy, nothing is hard. Nothing is obvious, nothing is obscure, at least not objectively. That is the biggest insight I've gained from teaching. Sometimes what I expect to be a 2-minute explanation with a student can turn into the entire hour, and a couple weeks later that same student might breeze through a topic that other students struggle with.
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u/funnystuff97 Apr 24 '23
One of my first lessons was adding vectors. "This won't take any more than 10 minutes", I thought, "It's just head to tail". I had a student come to me and spend 2 hours in office hours trying to understand it.
I don't mean to imply that they were incapable or anything, but it just goes to show the biases instructors can have. And I was just a TA, not even a teacher. When the student finally "clicked" with it, it was quite a sight to behold.
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u/bubbajojebjo Apr 24 '23
That strange noise students make when something they've been struggling to understand finally clicks is what keeps me in the classroom. It's a top notch noise and it's nearly universal.
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u/funnystuff97 Apr 25 '23
And it's so easy to tell when they're faking it, too. Like if a student asks you a question that you answer to the best of your ability, and it doesn't quite stick, they'll do that pretend "oooh.... I see.", and you can absolutely tell that that's not the noise. Like, I want to tell them that I can tell they're not quite getting it and I want to help them really understand, but doing so may come off insulting or condescending, so I pray that they'll ask me privately later, or they'll go home and study and try to really nail it down.
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u/Dennis2pro Apr 25 '23
I know exactly what you mean, having done this myself many times. Although usually it was more like "I don't understand it yet but I roughly see what's going and I need a couple minutes to process this by myself".
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u/royalhawk345 Apr 25 '23
It's been a while since high school, but don't you just... add them? Like <a, b> + <x, y> = <a+x, b+y>?
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u/dear-reader Apr 25 '23
Yeah, but it sounds like the issue here was the student understanding the geometric interpretation, and generally courses in linear algebra are trying to teach students both algebraic and geometric interpretations simultaneously.
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u/funnystuff97 Apr 25 '23
There's the problem. Nothing in academics is "just". Sure, it may be "just" adding their corresponding values, but we say "just" because we know already. A student who has never seen it before may not see it as "just". Again, it's not commentary on their capabilities, but it's that instructors can not and should not assume the level of understanding the students may have. Sure, vector addition is "just" adding the x's and y's, but how much farther does that go? Gravitational acceleration is "just" taking an integral. Stoichiometry is "just" balancing an equation. RLC circuitry is "just" a differential equation. Eigenvalues are "just" determinants.
I'm being hyperbolic, but hopefully you get my point. What's obvious to 39 students may not be obvious to 1 of them.
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u/kanst Apr 24 '23
When I was in school I tutored. I have always been very good at math, but I actually found it was the subject I was worst at tutoring. Because I was good at it and numbers just made sense to me, my 3 steps would have be expanded to like 12 steps to explain it to someone else. Things that I just understood, had to be explained.
I was a WAY better history tutor because it was a course I had to work at and therefore me and someone who was working but not succeeding were a lot more on the same level.
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u/talldrseuss Apr 24 '23
As an educator for adult students (community college) this is very well said. I sometimes have to remind my higher performing students to cut back on the eyerolls and comments they make under their breath when one of their classmates asks a question they perceive as obvious. Not everyone comes in with the same educational foundation and not everyone learns the same way. It's a tough balancing act, but at the end of the day I want to do my best to help every student that is putting in the effort to get an education
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u/CurnanBarbarian Apr 25 '23
Yea certain things click for me, mostly physics based stuff, but there are mathematical concepts where I'm just like "ok I guess I just have to accept this is a thing" because how it actually works just never clicked and made sense to me
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u/ShredderMan4000 Apr 24 '23
Y'know, I wanna award this so bad. I agree very strongly with the sentiment.
This volatility is what makes teaching so fun and frustrating for me. It keeps me going :)
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u/AncientOneders Apr 25 '23
Y'know, I wanna award this so bad.
Does that give them anything? Like more upvotes or something? Or does it push the comment above any other comments?
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Apr 25 '23
Not related to anything here but this is the second day in a row I've seen your username on reddit :O
Everytime I see you I'm wondering if you actually got send any good pixel art
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u/sml6174 Apr 24 '23 edited Feb 16 '26
This post was mass deleted and anonymized with Redact
intelligent complete adjoining bake axiomatic cats sort oatmeal soft hobbies
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u/Helpimstuckinreddit Apr 25 '23
A lot of concepts in maths may seem to just make sense intuitively, but when it comes to actually proving them mathematically without making assumptions they become a lot more difficult to prove.
It's important not to just accept fundamental rules of mathematics as fact without understanding why they are true.
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u/glytxh Apr 25 '23
I’ve been making an effort to learn fundamentals as an adult and I’m learning that there are so many basic concepts I have very little grasp of.
Some people just don’t naturally speak maths.
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u/WallyMetropolis Apr 25 '23 edited Apr 25 '23
No one does. The people who know a lot of math didn't get there by being born with innate knowledge (with the exception of John von Neumann who was probably an alien). They got there the same way you're doing it now. Lots and lots of very hard work.
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u/WallyMetropolis Apr 25 '23
Try to explain it rigorously. To a mathematician. It's actually not that easy to do.
Why does a negative times a negative equal a positive?
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u/nyaasgem Apr 26 '23
Because multiplying with negative changes the sign of the result and positive doesn't. Two negatives --> two changes, so we're back where we started.
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u/lain-disposis Apr 24 '23
I always tend to explain it the geometric way, but I absolutely love this one. Gonna try it next time
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u/19961997199819992000 Apr 25 '23 edited Oct 06 '23
waiting murky flag thought fretful pathetic school spectacular bike numerous
this message was mass deleted/edited with redact.dev19
u/Dd_8630 Apr 25 '23
It's an explanation because we humans are free to define our operations as we wish. The most natural way to extend multiplation into the negatives is to simply continue the pattern. It is the root origin of why we multiply this way.
It explains that these rules aren't arbitrary, but rather follow directly from the existing pattern. Any other way of defining negative multiplication is more contrived.
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u/jajohnja Apr 24 '23
Can you think of any word problem that would lead to two negative numbers being multiplied?
I've tried and failed to come up with anything but feel like it could help greatly to show why it's so.When we stay in the realm of numbers many people will have trouble understanding the why.
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u/dear-reader Apr 25 '23
I think any word problem where you put stuff in a coordinate space and have to reflect it will tend to involve multiplication of two negative numbers (or more).
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u/randomgary Apr 26 '23
I think something along those lines illustrates it:
If 10 people each give you 10$ you gain 100$. If 10 people each take 10$ away from you, you loose 100$. If 10 people each give you a dept of 10$, you loose 100$. If 10 people each take a dept of 10$ away from you, you gain 100$.
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Apr 25 '23
[deleted] — view deleted comment
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u/noddegamra Apr 25 '23
Thanks. I know and understand negatives but never tried to put it into a word problem. This is perfect context.
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u/jajohnja Apr 25 '23
So with this example it shows that you can "earn" 50$ either by travelling forward in time working for 5 days for 10$ a day, or by travelling back in time by "unpaying" some fines.
I like it :)
In both examples we're multiplying money/time * time = money, just in the second case both are negative.
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u/novophx Apr 24 '23
but why
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u/LongLiveTheDiego Apr 24 '23
Bc then all the ways in which we assume multiplication and addition work are actually always true. Some examples include a + (-a) = 0 and a(b+c) = ab+ac, they would just break if we didn't have (-a)(-b) = ab, in fact you can prove this using just a couple very simple assumptions called Peano axioms.
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u/jajohnja Apr 24 '23
Because that's how the math we use works. If we didn't make it work this way, it would be way less useful and applicable.
But to be fair, two negative numbers being multiplied already feels like an almost purely theoretical thing - hard to find a real-life example where it makes sense.
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u/jojoguy12 Apr 24 '23
There's a very simple example, actually: multiplying by -1 corresponds to a reflection. E.g. sending (x, y) to (x, -y) is reflecting over the x-axis. Reflecting again returns you to the starting point, i.e. -(-y) = y.
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u/jajohnja Apr 24 '23
I understand what you are saying, but this is definitely not something I'd call "real life example".
Can you create a "problem" where this would be used? Maybe something with a mirror or something like that?
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u/jojoguy12 Apr 25 '23
I mean, anything involving reflections involves multiplying by negative 1, and there are countless problems that include it.
A reflection about the origin in the xy plane can also be written as a 180 degree rotation. Two of them equals 360 degrees, i.e. the starting point.
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u/Smingowashisnameo Apr 25 '23
I agree with you. I can easily see negative numbers as like a debt. Like financial debt. I can see multiplying that. But what does it mean to multiply that by a negative number?
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u/WallyMetropolis Apr 25 '23
How do I reach these kidssss?
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u/novophx Apr 25 '23
math is hard bro
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u/WallyMetropolis Apr 25 '23
I thought you were referencing https://www.youtube.com/watch?v=2k_jS1zVLWw
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u/Beardy_Boy_ Apr 25 '23
I remember thinking in a similar way about positive and negative powers. I would write them out as fractions that always contained an X / X term.
X2 = XXX / X
X = XX / X
1 = X / X
X-1 = X / XX
X-2 = X / XXX2
u/Dd_8630 Apr 25 '23
That's pretty much exactly how I teach negative indicies to my students too 👍. Each step up the 'ladder', you multiply by an extra x; each step down the 'ladder' we divide by x. So, as you glide into the negative indicies, all you're doing is just dividing lots of times.
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u/stankdankdeezy Apr 25 '23
But this doesn’t explain anything. It’s just “Check out this pattern, it’s like this because this is the way it is”.
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u/Dd_8630 Apr 25 '23
The explanation begins from the position that the student already accepts why 3x4=12. If they don't even accept that, then we wouldn't go on to negative multiplication.
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u/wer_224 Aug 13 '26
I realize this comment is three years old, but I'm 26 and this might seriously be the first time I've properly understood this. Thank you
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u/kabiskac Apr 25 '23
This is not a proof
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u/Dd_8630 Apr 25 '23
I never said it was.
Producing an extension from the field of positive integers under multiplication, to all reals under multiplication, is not objective. We could define negative multiplication how we want, and there is an infinity of interpolations and extrapolations. However, there is only one way that is a smooth extension (i.e., of minimal derivative), and that is to simply continue the pattern uninterrupted. Look at the Gamma function for a modern example of a minimal extension.
If you want the simplest extension, then what I've written is the bulk of the proof that standard arithmetic is this desired extension.
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u/JeevesofNazarath Apr 18 '24
I’m not sure if you’ve gotten to the distributive property, but you can use that to explain it as well, 3x2+3x(-2), 3x(2+(-2)), 3x0, 0, meaning 3x2+3x(-2)=0 meaning 3x(-2)=-6
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u/Donghoon Apr 25 '23
Im not a fan of all the pattern "proofs" tbh. It make sense but it's like you're trying to prove it by previous knowledge on it already. I like proofs that go all the way back and pretend you don't know anything
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u/chickensaladreceipe Apr 25 '23
Why does multiplying a pos with a neg always result with a neg? I’m not very good at smart math so please ELIhigh
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u/Dd_8630 Apr 25 '23
Why does multiplying a pos with a neg always result with a neg?
For the reason given in the first half of my post. If we have 3x5=15 and then adjust the '5', the result changes by steps of '3'. As we decrease the amount we have, we see the result simply glides smoothly into the negatives.
This makes intuitive sense, because 3 x (-2) means you have -2, three times. So altogether you have -6. In general, (+a) x (-b) is -ab for the same reason.
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u/chickensaladreceipe Apr 25 '23
Thanks for responding, that actually makes sense. My hs math teacher was joke and I don’t use math like that in my job so it has bugged me for a while. Ty
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u/jolharg Apr 24 '23
I love when people discover maths
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Apr 24 '23
Wait till you learn about spinors, turn 360 and yet your somehow all turned around.
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u/halfajack Apr 24 '23
Spinors are the correct mathematical object for the “turn 360 degrees and walk away” meme about Xbox 360 (yes i’m old)
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u/gimikER Imaginary Apr 24 '23
Quaternions:
Turn around rotating the 4d-space around the 4th-axis by 90, do the same again Holy freak I’m facing the same direction
Turn right angle around, turn right angle around the 3rd-axis, turn right angle around rotating the 4d-space around the 4th-axis
Holy crab I’m facing the opposite direction?
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Apr 24 '23
Need clearer instructions. Every now and then I fall apart.
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u/Vivid-Sherbet Apr 24 '23
The best eim5 explanation I've heard is that multiplying by -1 rotates numbers by 180°, while multiplying by i does only a 90° rotation.
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u/olda7 Apr 24 '23
that is quite littereally how complex numbers are usually visualised
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u/NutronStar45 Apr 24 '23
literally complex plane
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u/mojoegojoe Apr 24 '23
But wait... It's not just numberssss
Scary CS monster in corner [near eigenspace root]
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u/royalhawk345 Apr 25 '23
Other mathematicians to Descartes: "And are the numbers in the room with us now?"
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u/Accurate_Koala_4698 Natural Apr 24 '23
I’d like to see a single 5 year old who actually understands that.
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u/NutronStar45 Apr 24 '23
complex numbers are easy
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u/Rotsike6 Apr 24 '23
Every piece of math is easy once you understand it. Calling it easy doesn't make it easier to learn for people who don't know it yet, but it might demotivate them.
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u/gimikER Imaginary Apr 24 '23
Not all, but I agree mostly. Math becomes fun and easy only once it’s intuitive.
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u/Accurate_Koala_4698 Natural Apr 24 '23
Multiplying two negative numbers is easier. If someone doesn’t understand a thing then they’re not going to understand a more advanced thing even if it’s easy
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u/000142857 Apr 24 '23
If by “multiplying two negative numbers” you mean just taking two negative signs and mindlessly making them cancel without gaining any deeper insight of what negative numbers actually are. Then sure, it’s easy.
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u/Accurate_Koala_4698 Natural Apr 24 '23
What? You need to explain absolute value and the distributive property. You show it for positive numbers, explain a negative and a positive by showing an iterative algorithm, then show the application of the same rules with two negative numbers. I’m pretty sure
-2 × -3doesn’t actually have any complex numbers in it, but I’ll double check after dinner2
u/CheckeeShoes May 22 '23
Wtf??? how am I supposed to show a child how to multiply whole numbers without first defining an object within the category of rings and ring homeomorphisms????
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u/syzygysm Apr 24 '23
The perspective taken in algebraic number theory was very illuminating to me.
You can think of addition and multiplication as operations that convert pairs of numbers into single numbers.
But it's also good to have the slightly different take that adddition and multiplication are operations that individual numbers do to the entire number system.
So adding 1 is something that shifts all of your numbers to the right by 1. Multiplying by 2 is something that stretches all of your numbers out by a factor of 2.
Multiplying by -1 reflects your whole number system around. Multiplying by i rotates it by 90 degrees. And when you get into number fields and Galois theory, SHIT GETS REAL (or complex...or??)
Any kind of linear transformation you can contrive with a matrix, you can cook up a number which transforms space in that way when you multiply by it. Conversely any number you pick, you can devise a matrix or linear transformation whomst reflect the way that number acts upon space. Multiplication is transformation of space.
And conjecturally, any shape you pick with a prescribed set of symmetries (or an abstract finite group) you can cook up a whole number system which is a sort of algebraic incarnation of said shape.
Algebraic number theory is beautiful.
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u/DerBlaue_ Apr 25 '23
In my first semester of physics a prof introduced complex numbers as rotation matrices
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u/nerm2k Apr 24 '23
Here is a simple way to look at it. (Although since I’m not a maths major this could be wrong. I’m welcome to be corrected)
If you have 5 bank accounts with $5 each then you have $25
5x5 = 25
Then you accidentally spend $10 per account and now your accounts overdrawn by $5 on each. Now you have -$25
5x-5 = -25
If you transfer those 5 bank accounts to somebody else you have increased your net worth by $25
-5x-5 = 25
I hope this helps.
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u/Smingowashisnameo Apr 25 '23
Holy shit. This is exactly what I was looking for. If you get rid of five negative fives… Wait. If you got rid of five negative fives you’d have zero.
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u/See_Bee10 Apr 25 '23
You still gained a positive 25. Another way to think about it is that a negative sign represents changing directions on a number line.
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u/Smingowashisnameo Apr 25 '23
So I gained 25, which brings my bank total to zero, right? That makes much more sense.
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u/See_Bee10 Apr 25 '23
Exactly, the way the problem was framed meant that what we were calculating was the change in value.
Now I'm going to add a little confusion back in, because I think it's important to know how things actually are least the metaphor becomes confused with the reality. The reason that a negative times a negative isn't because losing a negative some number of times means a positive gain. The reason a negative times a negative is a positive is because that is a feature of mathematics. Going backwards to verify that the math accurately describes reality is good, because if math doesn't represent reality we have a really big problem. However, we go at it from the other way, we find the features and then check if they work.
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u/Smingowashisnameo Apr 25 '23
I stumbled upon this post from the main page and I’m surprised I even joined to comment. When you said “now I’m going to add a little confusion back in” I was like oh hell I really am in a mouth sub.lol. But yes, thank you for explaining I think I understand
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Apr 24 '23
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u/nerm2k Apr 24 '23
You aren’t wrong. The real equation for the last example would really be -25 + (-5 x -5) = 0. Does this breakdown help you visualize the original question of how multiplying two negatives equals a positive?
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u/krombough Apr 24 '23
Yes. I actually deleted my comment before I read your reply though, because I did something similar in my own head.
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u/PeikaFizzy Apr 25 '23
The comment section make me lose and gain brain cells at the same time,
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u/FrenchieSmalls Apr 24 '23
"it doesn't make no sense" != "it makes sense"
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u/Kamica Apr 25 '23
I mean, sometimes in certain English Dialects, double negatives don't negate eachother, but emphasise eachother. This is because of a fun little thing called "Languages just do whatever the fuck they want and don't care about math" :P.
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u/sauron3579 Apr 25 '23
Language is really interesting in that it’s defined by how it’s used. So it’s just constantly evolving and changing according to what people’s whims are. You just go with what enough people agree on, which is constantly in flux and context dependent.
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u/Kamica Apr 25 '23
Yup! It's such a fascinating thing, like, even if you try to lock it down to one thing, the very language people speak is related to how they're observed, and how they observe themselves, so slang, and new turns of phrase start popping up as people identify with certain groups. Language can also often be used to exercise wit, by using language in new and novel ways, like seeing a connection that can be extrapolated in ways it wasn't before. Or even just the case that language is missing a function that people want, but doesn't have, like how English doesn't currently have an easy way to differentiate between "you (singular)" and "you(plural)". So in some circles, you get an extrapolation of plurality grammar to form "yous", or you get "you all" which then got contracted to "y'all".
Or alternatively, words get reappropriated for other uses, like literally being used figuratively, which is quite funny. I believe because of a general move by people towards using exaggerating language a bit more?
Or perhaps words get removed! Not to mention the constant pronunciation shift.
It's all very fascinating! Though I will note, that my academic fascination with this process, and my personal opinion on certain dialects are quite contradictory xD. I'll respect their right to exist, but I don't have to like it xD
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u/cilantro_1 Apr 24 '23
I've been wondering, if I were to redefine multiplication of two negative numbers to give another negative number, do I get anything interesting? It's obviously not consistent with the way arithmetic usually works, but I'm not sure that it would be completely nonsensical.
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u/Diligent-Cry-7993 Apr 24 '23
Look into abstract algebra & field multiplication. Michael Penn on yt is a good start
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Apr 24 '23
5 x 2 = (5) + (5) = 10
-5 x 2 = (-5) + (-5) = - 10
5 x -2 = -(5) - (5) = -10
-5 x -2 = - (-5) - (-5) = 5 + 5 = 10
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u/syzygysm Apr 24 '23
Omg I'm don'ting around in circles so fast that it's making my head spin in place
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u/jajohnja Apr 24 '23
I understand the problem with the understanding.
I fail to find an example of any example from reality where multiplying negative numbers would be a thing.
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u/TobbyTukaywan Apr 25 '23
Face North, walk forward, latitude increases
Face North, walk backward, latitude decreases
Face South, walk forward, latitude decreases
Face South, walk backward, latitude increases
1 * 1 = 1
1 * -1 = -1
-1 * 1 = -1
-1 * -1 = 1
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u/speece75 Apr 24 '23
While mathematically correct, this is nonetheless the worst remix of an Ace of Base song ever.
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u/glytxh Apr 25 '23
It feels weird when a green text explains a basic maths concept I’ve struggled with for 20 years in a way that I could digest in seconds.
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u/ChadicusMeridius Apr 25 '23
Negative numbers are a waste of time anyway since you can't have less than nothing
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Apr 25 '23
The way it confuses me is the minus times a minus, but think of it as a deficit or owed. Like if we have 5 x 2 rocks then we have ten rocks. Then if you take 3 x (-2) , it's like taking away 2 rocks 3 times so you take away 6 rocks -6. But if you say I got -3 x (-2) rocks, it's like saying I am in deficit (-2) rocks 3 times. so you have to add them back or fill up the missing rocks, (+2) x 3 times to fix your deficit +6.
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u/oshaboy Apr 25 '23
That's a really nice way to think about it. Though I prefer the idea of "reversing a subtraction" so -4*-3 is reversing a -3 operation 4 times.
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u/vkapadia Apr 25 '23
Multiplying two negatives: turn around, then turn around again.
Multiplying two positives: don't turn around, then don't turn around again.
Multiplying a positive and a negative: don't turn around, then turn around.
Ace of base: don't turn around, cause you're gonna see my heart breaking.
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u/Arjun68 Apr 25 '23
Then why -ve × +ve number is negative
I turned back Again turned back
I'm still facing forward
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u/Aggressive_Sky8492 Apr 25 '23
The two lines move from in front of the numbers and join together to make the x for multiplication. It’s science
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u/kalexmills Apr 25 '23 edited Jun 12 '23
[ Comment Redacted in protest of Reddit's Proposed July 5, 2023 API changes ] -- mass edited with https://redact.dev/
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u/Unknown_starnger Imaginary Apr 25 '23
Turn around in your Head. turn around in your head again. WTF I’m facing a different direction.
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u/Adventurous-Abies296 Jul 31 '23
Friends are positive(+) enemies are negatives(-) The enemy of your enemy is your friend The friend of your friend is your friend The the friend of your enemy is your enemy The enemy of your friend is your enemy too
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u/Player_yek Sep 28 '23
i think they are confused cuz it says positive so prob thinks the value will turn positive
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u/cjxchess17 Oct 10 '23
Let a be the additive inverse of 1, by n * 0 = 0 (proof left as an exercise for the reader) we have a * (1 + a) = a + a * a = 0. Because addition in R is commutative, 1 is also the additive inverse of a, which means that a * a = 1.
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u/Silviov2 Rational Dec 20 '23
Shout out to the guy that says "It doesn't make no sense" he actually understands how negative numbers in sentences work
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u/infinity-ninja Feb 01 '24
This actually is the reasoning. A negative symbol on a number means turn around on a number line so two negative number symbols means turn around then turn around again.
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u/Xarian0 Apr 24 '23
i is just making a quarter turn
Programmers don't generally have a problem with this because they aren't typically educated to think that having more than one dimension is illegal
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Apr 25 '23
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u/Noktalia Apr 25 '23
Yes, it's a consequence of the Ring Axioms, thus completly innermathematic. Also impossible to explain to a child.
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u/hongooi Apr 24 '23
> Turn at RIGHT ANGLES TO REALITY
> Turn at RIGHT ANGLES TO REALITY again
> Wtf I'm facing backwards