Illustrating Why a Negative Times a Negative is a Positive Using the Number Line!

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The number line is incredibly flexible in the different mathematical operations it can illustrate. Of course, students become familiar almost immediately with how it can be used for addition and subtract, and the way it represents those as left and right shifts. Scaling helps us represent multiplication and division, and understand the connection between the two.
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Inevitably, though, students also want to understand how to represent multiplication by negative numbers. Although there are other models that can help us understand this (repeated subtraction, for example, rather than repeated addition), my personal favorite is to take advantage of scaling on the number. When we let our scale get closer and closer to zero, we can see the number line contracting, until at a scalar of zero, we see the entire number line collapse to a single point. Pushing the scale past zero reveals that multiplication by negatives is actually flipping the entire number line around. This means that the items above 0 now flip below zero (so, a positive times a negative is negative), but even better, also shows that the items below 0 now map above it (so, the famous maxim "a negative times a negative makes a positive").
Not only can this reveal to students deeper structure than we might think the number line capable of, it also primes them for a geometric understanding later on of complex multiplication, and the role of the imaginary unit.
#negativetimesanegative #multiplication #numberlinemath #numberline #scaling
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[@hanchen267]

Turn around Turn around again Guess what you’re facing the same direction

[@ssjsurfndude4]

This is an awesome way of visualizing multiplication. Is the program you are using in this video available online? Would love to share it with my students

[@akashkrish1681]

Never saw someone explaining this topic soooooo perfectly as you...

[@polymathematic]

thank you!

[@donepudiravi5718]

​@@polymathematic❤❤❤❤😊😊😊😊😊

[@lunthuak2114]

Wow he successfully managed to make me more confused than i was originally 👌

[@polymathematic]

oh no! what was clear to you before that isn't now?

[@bransonh.4096]

​@polymathematic logically your 1 min video makes no sense. "Negative times negative = positive because graph" that's pretty much your explanation

[@polymathematic]

@@bransonh.4096 lots of things are true because graph. what's the problem?

[@brunodonascimento4387]

Idk I didnt get it if you use fewer number it might make more sense. U didn’t explained the graph u just showed it too. But I agree we dumb lol😂

[@Bloody_biggerSister]

​@bransonh.4096 if ur so smart how about you make a video that teaches people math huh 😊

[@TheXeneco]

6 of something negative suggests a negative total. That makes sense. This illustration perfectly shows how a negative of something negative points to a positive. Never seen number lines used like this - and I’m glad to have seen it. So far, only seen them when I was first at school as a counting aid

[@redheadredemption3]

Thank you for giving a visual I can use in class to show students.

[@G_4J]

It all makes sense now

[@pixelapse9613]

Are these is the same thing as General relativity about spacetime contraction 🤔

[@polymathematic]

for sure

[@studipchowdhury9386]

I'm 6th grade student and this is the first time i understand the explanation.

[@star_ms]

That's actually very nice.

[@King_Konglish]

No idea why I got recommended this, but that’s neat haha

[@romanemul1]

Very nice intuitive "kick"

[@polymathematic]

thank you!

[@FleshEatingBeagle]

Stay on target. Stay on Target!

[@muhammadilhamrobani7042]

Wow this now makes sense

[@Xnoob545]

The better way is use the complex plane -1 is a 180° rotation away from 1 Multiply by -1 (rotate by another 180°) and you get 360° A full circle Back to 1

[@polymathematic]

i certainly agree that the fuller explanation brings in the complex plane, but i don't think i would say that's the "better" way if you're working with, say, 6th or 7th graders.

[@lucam8758]

Is there a mathematical proof or is it considered an axiom?

[@mayconbruno2676]

There is proof

[@randyzeitman1354]

I'm not getting it. Is the reason simply convention?

[@polymathematic]

i wouldn't necessarily say *just* convention. it's more that once you allow multiplication by negatives at all, the only way to make sense of multiplying two negatives is for the result to be positive.

[@johnchisholm2896]

​@@polymathematicjust make a video saying you don't understand why 😆😂🤣

[@polymathematic]

@@johnchisholm2896 just make a video saying you don't understand why 😆😂🤣

[@johnchisholm2896]

@@polymathematic kzbin.info/www/bejne/qHyXpIujlLKDjdE

[@chatbear69]

Brilliant way to visualize.

[@scottcaine4514]

So the reason why a negative x a negative is a positive is because the number line shows it is. That doesn't seem to quite explain it.

[@jorgejorge8878]

Are you dumb? The point is not that that's the REASON. This is just a visual intuitive way of seeing what's happening.

[@scottcaine4514]

@@jorgejorge8878 guy in the video: visually shows things happening on a number line. Also guy in the video: "and that's why a -ve times a -ve is a +ve." No I'm not dumb.

[@jorgejorge8878]

@@scottcaine4514 gotcha, sorry for the mean comment. Not passing through my best moment when I wrote it.

[@nonamem2932]

This video is not the perfect example of why negative times negative is positive. You can search for Khan Academy video if you are interested

[@kazedcat]

It can be proven rigorously that (-a)×(-b)=a×b

[@tunistick8044]

omg that really makes sense now

[@markstephensajul8244]

Let's make a more simple example "Turn around" "turn around again" Oh I'm facing the same direction

[@MaiONerds]

Sir can you explain why (not how) 500(or any no.)×(-1) = -500. I am a visual learner and i just can't explain to myself why it is so

[@007siddharthkar7]

Which graph calculator you use to show this..

[@chrisalex001]

This whole thing is confusing to students because mathematical expressions are never truly thoroughly explained to students. Most people think that visually 2 x 2 is two 2s multiplying by each other. In mathematics that is actually wrong. What 2 x 2 actually indicates is that the number 2 has to be repeated two times. The same thing happens in 2 x 3 (the number 2 is to be repeated 3 times). If it were 3 x 2 (then it would be the number 3 being repeated 2 times). Visually most students get the wrong impression when they read math because they just visually see numbers on the board and don't realize what the mathematical expression is really saying. This becomes a problem when dealing with negative numbers. This is also why people have a hard time understanding why fundamentally any number times 0 is equal to zero. They visually see, let's say as an example, 5 x 0 = 0 and think, wait how is 5 boxes times 0 boxes equal to 0 boxes if I already have 5 boxes to begin with? Well, it's because mathematical expressions are extremely technical and exact. 5 x 0 is equal to 0 because 5 has been repeated 0 times (and that is the answer) nobody cares about the fact that 5 boxes are still present after we attempted to multiply them 0 times. The same is true of 5 x 1 = 5. There we see that 5 boxes repeated 1 time is equal to 5 boxes again. This redundancy in mathematics, though absolutely necessary to solve problems with precision, is not intuitive to most people, and it gets harder when they truly depend on these basics concepts in order to actually think in mathematical terms later on. it's like learning a language and not being clear on what a subject and verb is. You won't be able to write coherent sentences or extrapolate information accurately.

[@polymathematic]

No, see, that's a fundamental error a lot of people make. Multiplication isn't "really" repeated anything. When you multiply 3 × -2 you're not repeatedly adding threes. When you multiplying -3 × -2 you're not repeatedly adding either threes or twos. When you multiply π × e you're not repeatedly multiplying *anything*. This is why it's important to think of multiplication as its own fundamental operation entirely separate from the fact that it happens to also give the correct result to repeated addition problems. Thus the metaphor in this video of "scaling". Scaling more naturally gets at what multiplication really is, which is why it's more useful for justifying the idea behind "a negative times a negative is a positive."

[@danielharry6569]

Damn… I wish this was my math teacher

[@polymathematic]

appreciate it!

[@thesocialscientist1]

Why does the number line flip around?

[@Questiala124]

To prove it we can use the formula 1-x=-(x-1), applying this to 1/2 we get 1/2=-(-(1/2)) Scaling by 2x we get x=-(-x). Now we prove -1*x=-x -1*x=(0x)-x=x-x-x=-x. Now Applying this on -1 one we get -1*-1=-(-1)=1. multiplying by an we get (-a)(-b)=ab, thus our proof is complete.

[@AfroNerd-cv1dl]

Yo! I promise you Google is in my head. I was just thinking about this earlier today

[@kingduckdodo6672]

reversal red

[@menzisaclown]

What software you using? Its so nice

[@polymathematic]

i built the graph in Desmos. You can see it here: www.desmos.com/calculator/nt8eokihfv. Thanks for watching!

[@menzisaclown]

@@polymathematic Thank you for the reply. I enjoy the videos, please keep the good work up

[@chefboiarby304]

I like that this accidentally creates an optical illusion where it seems like the number lines bow inward or outward depending on the mapping lines.

[@eepiioh_k]

It's easier for me to visualize multiplying 2 negatives as if it were money, like I owed debt to a debtor/s who forgave my debt.

[@jmrs_]

When the number line is actually somewhat useful

[@Lialvas]

random comment: What about roots of negatives? "Pulls of argand and polar"

[@Blue-bb9ro]

What app/program is this?

[@polymathematic]

the graph is done in Desmos. you can play with it here: www.desmos.com/calculator/w8iouadxhe

[@sail2byzantium]

Still not quite getting it. It might be good if we could see a larger graph--and get yourself out of the picture so to see it more clearly.

[@voxman428]

If I have a debt of 50 and square the debt in a week I have a debt of 2500 so why _50×_50=2500

[@polymathematic]

well, one issue here would be what on earth does "squaring" the debt mean in the real world. if you can describe that, we can try to make sense of it in a real world context.

[@jorgejorge8878]

awesome

[@johnperez3454]

yeah but why tho it doesnt really explain it, just shows this number line

[@gojosatoru1442]

But than why don’t positive times positive equals negative?

[@carultch]

Because multiplying by positive doesn't reverse the direction of the original number. An odd number of negative numbers multiplied is negative. An even number of negative numbers multiplied is positive.

[@chrisalex001]

Why not just simply explain this by showing a horizontal number line in which you illustrate that when a negative number is multiplied by a positive, it is multiplying in the same negative side of the number line. In other words -2 x 2 is actually -2 in the negative side of the number line multiplying by two in that same negative direction. The opposite happens when you multiply -2 x -2 because the direction is reversed (by having a double negative). In other words, if you say lets multiply -2 by another (also) negative two, you are actually going toward the positive side of the number line because you are indicating that negative two has to multiply in a negative direction of its already negative position on the number line, and thus in a positive direction. This would be a much simpler way to illustrate this whole concept.

[@polymathematic]

I do. You can check out this video for a horizontal representation of the same concept. kzbin.info/www/bejne/qKGsaGulnZefn68

[@amosmoses5630]

I hate the scale of the graph lol

[@johnchisholm2896]

That's like say because it just does. I award you zero points

[@deepi1235]

If you still got confused after this clear explanation. Sry bro Maths is not for you . :)

[@tzbq]

💀