Why is the product of two negative numbers positive? Math Foundations

#learn #math #foundations
When multiplying numbers, many people know that multiplying two negative numbers makes a positive number, but in this video, we will explain why! We explore how negative numbers are related to simple multiplication.
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Chapters:
0:00 Positive times Positive
3:37 Positive times Negative
5:24 Negative times Positive
6:56 Negative times Negative
7:37 Sidenote | Multiplying Negatives
11:28 Negative times Negative Completed
13:02 Summary
13:50 End Screen

[@xniyana9956]

One of the best explanations I've seen of this topic. Well done!

[@jilgk2043]

I searched for so long to find someone who explains it with a visual representation and acknowledges that it creates a knot in the head. Many just present (-1)x(-1)=1 as a fact that needs to be accepted.

[@UnknownSENKu]

you help me a lot to understand this. Thank you... Wish you got millions subscriber in few days...

[@christianduval8374]

Even more important is to realize that the flipping is actually a rotation left into a transverse plane 1st (we call it "times i"). And then rotate again ( hence i^2) to reach the final flipped state caused by x -1.

[@JerichoDeGuzman-rm1kd]

This is the explanation that hammered down the nail for me. Thank you very much my great sir

[@jazmina8947]

omg most helpful god bless you thank you!!!

[@LuminescentPictures]

I got it, then I didn't get it... The equation and/or demonstration at 11:55 only makes sense if you start with the number 1... But that number came right out of nowhere, and is not in the written statement. How, in reading (-3)x(-4), am I supposed to know that I need to start with the number 1, then complete the equation, versus starting with the number zero, or even starting with the number 17 or 39? conversly (just to make my questioning point very clear) when I multiply 3 x 4 to equal 12 I am starting with 3 then multiplying it by 4 to get 12.... I feel like in that example, I am starting with zero, then doing the work. I wouldn't even know how to start at 1 when multiplying 3 x 4... I mean, that could be 13 as 12+1, or it could be 3+1x4... oh... is that also 12? if we do the multiplication first? or would it be assumed that starting at 1 for 3 would equal 4 then times 4 to get 16? I Don't know, but still, you can see what my original question is getting at right?

[@eduardobarreto6095]

Excellent explanation, thank you

[@ogeidnomar4601]

(-1)= flips number to negative (-1)= flips number to positive And the pattern repeats This is an easy way for people to understand negative integers in multiplication and division. Interestingly, integers with addition and subtraction is a bit more challenging.

[@andyrzimm]

I'd explain -3*-4 as taking away negative 4 three times. If we're starting at zero, taking away negative four three times leaves us at positive 12, because taking away a negative is the same as adding.

[@123chomps]

This is great. Thinking about negative numbers as "amounts being taken away" is so helpful. To build off this, let's say you borrowed (borrowing is a negative amount) 3 dollars from your friend 4 times (-3*4=-12). So, you owe (negative amount) your friend 12 dollars. Let's also say your friend relieves you of your debt. So, -3*-4 is removing (removing is a negative amount) the -4 times you were in debt (debt is a negative amount) for -3 dollars which will mean you keep (a positive amount) the 12 dollars your friend gave you.

[@sampatkalyan3103]

this explanation doesn't not fit with the definition of multiplication. we have to create a new definition for multiplication.

[@sampatkalyan3103]

still it is assuming that -*- is positive

[@sampatkalyan3103]

@@123chomps here what is happening in this example that you are doing 3*4 then you are making it negative. but if we go by the what he said by amounts being taken away for example in -3 * 4 if -3 is added 4 times because 4 is positive number so it is -12 no lets go opposite what if 4 is taken away negative three times 4-4-4 which is 4. so tell me what the definition of taking away a something if - is present. by itself or with what. same logic applies with -3*-4. it doesn't work

[@sampatkalyan3103]

@@123chomps before you counter my comment explain what is 3*4 is it adding 3 4 times. and if this is true then what is 3*-4.

[@sholuakimara7534]

Basically the first number is your trajectory and the second number dictates if it continues that direction or flips the other way. Negative flips while positive keeps it going. Interesting

[@Exegesis66]

This is like my 8th video on the topic, still...don't....get it! Your basically creating an arbitrary rule then applying that rule, "Multiplying by -1 creates a flip around 0." But you didn't arrive at that rule by the same rules that you used to describe the previous equation so how valid is your rule? i.e. So, 3 x 4 = 3 sets of 4 which = 12. And 3 x -4 is 3 sets of -4 which = -12. Great. Makes sense. Ok, well, if 1 x 1 is one set of 1 which = 1....THEN...-1 x -1 = one negative set of -1 which = -1. That's consistent logic across all three equations. But a negative x a negative is NOT trying to be consistent with it's counterparts (like we would expect and what is intuitive) instead YOU (meaning mathematicians) are FORCING it to be consistent with something else, namely, the distributive property. But WHY must it be consistent with the distributive property when it is inconsistent with the previous intuitive rules that were set up???? No one answers this question.

[@bjrnvindabildtrup9337]

I totally agree.

[@alanjamey2777]

No your wrong ,see carefully -1 x 1 means the multiplier positive one means that but -1 x -1 here it is multiplied by -1 times not 1 times ,so you cannot say that one times of -1 equals -1 ,there is no one times here.its times -1.So basically -1 x -1 means if you subtract the 1 which is in negative quantity you get positive.I can't clearly wrap my head around this way but ,the easy way is we defined a general set of rules that goes well within algebraic and real world applications.

[@roysmith3198]

@@alanjamey2777 Its about understanding multiplication can be repeated subtraction. If you subtract a positive number, have less. If you subtract negative, you now have more because you have less debt. Negative numbers were creates 1500 years ago. It way a way to keep track of people getting products at the store when they had no money. Credit was created with negative numbers. When you subtract debt you are going in the positive direction.

[@alanjamey2777]

@@roysmith3198 yeah but this way of understanding doesn't get you to understand it fully ,if you think you understand it fully with this depiction of why - x - =+ you haven't asked enough questions but i understand it now the real answer is we need negative number arithmetics to use in algebra and these negative arithmetics make sense so that's the actual answer.The way i see it simple -1-(-1) is logically 0 ,it's subtracting the values with the same value so it's logically 0 ,so what we are actually doing is adding it so - x - is + and there are so many ways to prove this ,again the real answer is this system is useful in mathematical applications and is relationally correct with other arithmetics

[@talastra]

Hey Corey. I agree with you. And I tried working something out. My response is above, but I'll re-paste it here (with edits) in case you see it. Rather than thinking of multiplication as repeated addition and subtraction in terms of a number line (as the example above crashes on), let x(y) [x*y] mean something like "put y blocks from your hand onto a table x times," then we see that the operation involves either moving blocks (in your hand) onto the table or (from the table) into your hand. Nothing to do with a number line. So (just to work through all the cases), if we first do 3(4), this moves 3 sets of 4 blocks (from my hand) to the table, giving 12 blocks on the table (as expected). Then given -3(4) or 3(-4), this will involve in some way moving 4 groups of blocks 3 times from the table to my hand, leaving zero blocks on the table (as expected, and thus convincing myself that I have somehow "reversed" what I did with 3(4) to begin with]. However, if there were not already blocks on the table, then after 3(-4) or -3(4), I will have borrowed or stolen or subtracted 12 blocks from the table to my hand, and there will be a "hole" or "deficit" of -12 blocks on the table (as expected). But wait. No one told us that we could "steal" from an empty table in this way, but apparently we can! Also, no one told us that we apparently have a stock of blocks in our hands to be setting down on the table (in groups of 3) in the first place, but apparently we do! Where did these come from? From the fact that moving blocks (that already exist) to and from spaces is a better metaphor for presenting multiplication as repeated addition or subtraction than a number line. So the "real" situation involves blocks moving from the table (to my hand) or from my hand (to the table). As such, if (-3)(-4) will involve some kind of "opposite" movement compared to -3(4) or 3(-4), then that movement is precisely borrowing or stealing or subtracting blocks FROM MY HAND to the table. A s a result, there will then be 12 blocks on the table (and a "hole" of -12 blocks in my hand). So, if -1(1) or 1(-1) means there are -1 objects on the table now (and +1 in my hand), then (-1)(-1) is the opposite of that, so that there is -1 object in my hand, and +1 object on the table. I don't know if this helps. But there you are.

[@ROHiT__]

REALLY NICE . KEEP ON THE GOOD WORK !

[@treena3956]

Thanks :) we'll keep them coming!

[@Abodroc94]

It's nice to see another axiom of mathematics!

[@Menny0_0]

pretty good. what about negative 3 in the bank?

[@talastra]

Please excuse all my thinking out loud here! :) And thanks for the video. I learned myself.

[@andreiinthedesktopworld1178]

Software?

[@thomasjefferson6225]

is this considered a rigorous proof?

[@bjrnvindabildtrup9337]

10:28 we can see that it HAS to be 1, it is proven to be 1. But how does (-1)x(-1) make +1? What is happening? What does (-1)x(-1) mean?

[@roysmith3198]

No math teacher knows. They have always told us multiplication is repeated addition. Multiplication is repeated addition OR repeated subtraction. number

[@talastra]

see my explanation in this thread (that refers only to addition and subtraction)

[@talastra]

@@roysmith3198 Very likely!. See my explanation in this thread (that refers only to addition and subtraction)

[@itgurukulgroup6654]

great

[@roysmith3198]

Multiplication is repeated addition OR repeated subtraction

[@simpleman283]

I do not understand how these people can make a video trying to teach this without saying that. Your comment should have been pinned.

[@luckygamer9197]

@@simpleman283subtraction is addition

[@simpleman283]

@@luckygamer9197 🤣

[@rocco3935]

the fundamental flaw with this argument is that 3 · 4 doesn't means adding the number 4 three times to itself but two times. Indeed here there are two additions 4 + 4 + 4 namely 4+4 is one addition and 8 + 4 is the other. there are no other addition. This make impossible to explain (-3)·(-4) in terms of repeated additions, but nevertheless it is still possible if one can find the true meaning of multiplication.

[@nrghexa825]

Soooooo Complex

[@SAMSSLOTCHANNEL]

So you can take 2 negatives and equal a positive so 2x2=4 and -2x-2=4 lol so why even use the negative

[@talastra]

I was motivated to think of the following thanks to your video; so, that's good. However, the video didn't explain the point. [Another similar video was completely unsatisfactory; yours is better and more productive] However, ultimately getting to (-1)(-1) = 1 is a tautology. Of course, it's mathematically the case, but it doesn't explain why it is the case in terms of multiplication as repeated addition and subtraction (as someone else commented). The root problem is that presenting multiplication as repeated addition or subtraction on a number line is a fail. Clearly, if I start at 4 and multiply by -2, then I subtract 4 twice [4-4 = 0, 0-4 = -4], and end up at -4, not -8. Whyyyyyyyyyyyyyyyyyyyyyyy? Below is my attempt at an explanation (inspired by the example of your video). However, I tried the explanation below to someone very bad at math, and it didn't seem to light up the understanding. So, maybe it's still a fail. Rather than thinking of multiplication as repeated addition and subtraction in terms of a number line (as the example above crashes on), let x(y) mean something like "put y blocks from your hand onto a table x times," then we see that the operation involves either moving blocks (in your hand) onto the table or (from the table) into your hand. So, if we first do 3(4), this moves 3 sets of 4 blocks (from my hand) to the table, giving 12 blocks on the table (as expected). Then given -3(4) or 3(-4), this will involve moving groups of blocks from the table to my hand, leaving zero blocks on the table (and thus convincing myself that I have somehow "reversed" what I did with 3(4) to begin with]. [Also, while -3(4) and 3(-4) are obviously mathematically interchangeable, it is not intuitive how "removing" -4 blocks 3 times or "removing" 4 blocks -3 times is actually the same thing, if it is. But set that aside for here.] However, if there were not already blocks on the table, then after -3(4) or -3(4), I will have borrowed or stolen or subtracted 12 blocks from the table to my hand, and there will be a "hole" or "deficit" of -12 blocks on the table (as expected). Hmm. No one told us that we could "steal" from an empty table in this way, but apparently we can! Also, no one told us that we apparently have a stock of blocks in our hands to be setting down on the table (in groups of 3) in the first place, but apparently we do! Where did these come from? From the fact that moving blocks (that already exist) to and from spaces is a better metaphor for presenting multiplication as repeated addition or subtraction. As such, the situation involves blocks moving from the table (to my hand) or from my hand (to the table). As such, if (-3)(-4) will involve some kind of "opposite" movement compared to -3(4) or 3(-4), then that movement is precisely borrowing or stealing or subtracting blocks from the HAND to the table, with a result that there will then be 12 blocks on the table (and a "hole" of -12 blocks in my hand). QED

[@talastra]

I am not a mathematician (I've only suffered from a bad mathematical education), and I can't stop my amateur wanking here. Of course, I have no problem accepting the "it's just that way" command from a math teacher that multiplying two negatives gives you a positive. (Math teachers suffer from bad mathematical educations too). It's really more about abandoning this bad "metaphor" that multiplication is repeated addition and subtraction when illustrated on a number line. As an elementary school kid, I was completely flummoxed by negative integers because of this shitty metaphor that doesn't make sense. (Never mind that division isn't repeated subtraction!) Meanwhile, my curiosity is locked on how one could consistently distinguish between what is happening with -3(4) compared to 3(-4). Whether one is "adding" or "subtracting" blocks to/from the table is controlled by the sign inside the parenthesis. "Adding" -4 blocks to the table is adding them to the hand, while "adding" 4 blocks to the table is subtracting them from the hand. One is then "doing" this 3 times or "undoing" it 3 times (presumably reversing it). So 3(-4) is adding 4 blocks to the hand 3 times (leaving the expected -12 "hole" on the table], while undoing that, (-3)(-4) would be adding 4 blocks from the hand to the table 3 times [which is indeed the expected +12]. This is close to the desperate mnemonic I used to try to keep track of integer multiplication. If there were two minus signs, (-3)(-4), they would "combine" as a +. Lastly, this could be turned around, so that 3(4) didn't mean putting 3 groups of 4 onto a table, but stealing 3 groups of 4 from the hand. This would be -12 in the hand, but because this is a "reversal" of all the above, that would be the opposite of -12, or +12 (once again, the expected answer).

[@blvckbytes7329]

@@talastra Kind of reminds me of the same issue with how powers are repeated multiplications, but then how would you multiply a rational number of times? Really gotta look behind the curtains here.

[@jklimtsc]

This still doesn't answer the question of why, which is the claim of the video. It's a good explanation of negative numbers and multiplication but not a conceptual answer to the why does a negative times a negative equal a positive.

[@talastra]

Rather than thinking of multiplication as repeated addition and subtraction in terms of a number line (as the example above crashes on), let x(y) [x*y] mean something like "put y blocks from your hand onto a table x times," then we see that the operation involves either moving blocks (in your hand) onto the table or (from the table) into your hand. Nothing to do with a number line. So (just to work through all the cases), if we first do 3(4), this moves 3 sets of 4 blocks (from my hand) to the table, giving 12 blocks on the table (as expected). Then given -3(4) or 3(-4), this will involve in some way moving 4 groups of blocks 3 times from the table to my hand, leaving zero blocks on the table (as expected, and thus convincing myself that I have somehow "reversed" what I did with 3(4) to begin with]. However, if there were not already blocks on the table, then after 3(-4) or -3(4), I will have borrowed or stolen or subtracted 12 blocks from the table to my hand, and there will be a "hole" or "deficit" of -12 blocks on the table (as expected). But wait. No one told us that we could "steal" from an empty table in this way, but apparently we can! Also, no one told us that we apparently have a stock of blocks in our hands to be setting down on the table (in groups of 3) in the first place, but apparently we do! Where did these come from? From the fact that moving blocks (that already exist) to and from spaces is a better metaphor for presenting multiplication as repeated addition or subtraction than a number line. So the "real" situation involves blocks moving from the table (to my hand) or from my hand (to the table). As such, if (-3)(-4) will involve some kind of "opposite" movement compared to -3(4) or 3(-4), then that movement is precisely borrowing or stealing or subtracting blocks FROM MY HAND to the table. A s a result, there will then be 12 blocks on the table (and a "hole" of -12 blocks in my hand). So, if -1(1) or 1(-1) means there are -1 objects on the table now (and +1 in my hand), then (-1)(-1) is the opposite of that, so that there is -1 object in my hand, and +1 object on the table.

[@najlaabdala5170]

Check multiplication using math tiles.

[@simpleman283]

Multiplication is repeated addition, WRONG! Multiplication is repeated addition OR repeated subtraction. The X in multiplication is the symbol for groups. First number tells you if you are adding or subtracting & how many times.

[@luckygamer9197]

Subtraction is addition

[@simpleman283]

@@luckygamer9197 🤣

[@luckygamer9197]

@@simpleman283?

[@srinivasjayanthy5946]

It's not elegant

[@davidduran8601]

This is the worst explanation I've ever heard of.