Why a negative times a negative is a positive | Pre-Algebra | Khan Academy

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Use the distributive property to understand the products of negative numbers.
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[@noxure]

Biggest mistake of highschool math. They give definitions and formula's you have accept as gospel truth and then use them to solve abstract puzzle, like it's some sort of magical secret language. Mathematics is meant to make difficult things simple, not the other way around.

[@photonicsauce7729]

Well said

[@sunilj10]

noxure agree with you, we need logic not magic. Logic based foundation not magic based foundation. Unfortunately many educators don’t take the trouble or their own foundation is based on magic not logic.

[@Dibu-Jitos]

I don't agree: the brain of 12 old kid is not ready to deal with some abstract aspects of the logical process.. The brain needs to mature, it's a biological matter. Of course, there are many people in the world who are ready to understand abstract stuff, whatever it is.. but those people don't probably need to be explained things as the kind this video explains: they do by themselves at 5 or 6 years old.

[@ChechoColombia1]

@@Dibu-Jitos A 10 year old is ready to deal with this and more complex topics if you really know what you are teaching

[@dancewithgoli7817]

@@Dibu-Jitos I am a 12 years old and I can easily understand why negative Plus negative is positive but I just can express it using words.

[@its_a_me_marcus_g]

I’m a senior in high school and am now just watching this because of all my years in math, I’ve just been told that two negatives multiplied by each other make a positive, but I never actually understood why this is the case. Don’t know why it took me so long to look this up!

[@pihyv]

Glad to hear I'm not alone :D Once you understand something it becomes easier to combine knowledge and draw your own conclusions. Good luck to you!

[@russelsteapot8991]

You're so lucky to be young and learn about the whys. I'm pushing thirty, and the question of why negative numbers can equal positive numbers in some cases has just occurred to me in the middle of the night.

[@imed-eddinemorsli959]

Bro i'm a bachelor degree

[@vladimirprotein3275]

lol same

[@websurfer5772]

When are you ever going to need to multiply negative numbers in real life though? Actually, I guess I wasn't wording it right at first when I searched Google just now because I wrote it another way again and found the following on Quora: "Let us say you must pay someone ten dollars a month. This is the same as being paid negative ten dollars a month. Ten months ago (in negative ten months) you were 100 dollars richer. (-$10)x(-10mo.)=$100" That's pretty cool and now I want to be a miser with my money. lol

[@CaptTerrific]

videos like this are absolutely critical - I feel the reason so many people "hate" math, is that they feel they are bombarded with rules that they do not understand, and feel overwhelmed simply memorizing.

[@mydogskips2]

What's hard to understand, math is easy, just follow the rules that you are told, and you get the answer, simple. There's a reason why teachers don't generally explain WHY (in this case, why a negative x a negative = a positive), because understanding the reasoning and logic behind the rules is much more difficult, it requires far more critical and deductive thinking by the student. Sure, thoughtful and discerning people may want to know, but as my teacher once said, "Why does a negative times a negative equal a positive, because I said so," and that's all you really need to know.

[@botezsimp5808]

EXACCTLY my teachers could never explain the "why".. they only knew the "how"..

[@teabow.]

​@@mydogskips2 this is one of the most ignorant comments ever written

[@teabow.]

​@@mydogskips2 this is one of the most ignorant comments ever written

[@blessedafricarains6429]

​@@mydogskips2that is one of the most stupid things I've read if explaining the reasoning behind something is too difficult for the vast majority of students then maybe they shouldn't be taught that thing until they are ready, making them memorize math will only teach them bad habits that will cripple them when learning more advanced things

[@georgegummadi1507]

I didn't know Sal could draw amazing portraits

[@justitroyal7032]

Thanks

[@yogxoth1959]

He has beautiful handwriting, so it shouldn’t be a surprise.

[@bismarckspointyhat4422]

Wonderful explanation. Understanding it really only comes once you know "why" things are the way they are. 26 and I just learned "why" today.

[@michaelneftali8711]

You’re 27 now. Say Hi to me. I’m just getting yo know why now

[@cxllmegeo3557]

@@michaelneftali8711 say hello to me to

[@raghukancharla6004]

@@michaelneftali8711 i am 55

[@MsVamPireChic]

Can someone explain to me why he did (3 + -3)? Where did that come from? I understand everything but that! How did he know to do that? I still don’t get it!

[@SN-ox5kd]

@@MsVamPireChicI think he use that to explain and it wasn’t anything important

[@Xehemoth]

You would think negative times negative would be a far greater negative. I still don't fully understand it. With that being said though, your explanation is by far the greatest explanation I have ever heard.

[@nickmichaels6242]

Xehemoth I know right. Same thing a with a negative divided by a negative.

[@chelseascott98]

You've heard multiplication referred to as "repeated addition" right? Instead of doing 7+7+7+7, you can just do 7x4. SO: Multiplying positives is repeated addition. Mult. a pos x a neg is repeated SUBTRACTION. (getting smaller) Mult a neg x a neg is repeatedly SUBTRACTING A NEG (getting bigger!!!) 0 - (-4) is the same as 0 + (+4) ... hope this helps :)

[@unoriginalusernameno999]

Hey think of this verbally. Mom I like to school---->(+) Mom I do not like to go to school----->(-) Mom I like to not go to school------>(+)x(-)=(-) Mom I do not not like to go to school-------->(-)x(-)=(+)

[@gwenchar7089]

@@chelseascott98 Thanks a lot!

[@XxMeTa_Zr0xX]

It's simple: Love + Hate - Love to love = Love Love to hate = Hate Hate to love = Hate Hate to hate = Love

[@DiazdelVivar]

you can see it like this; if I say: I went home + I didn't go home - I deny that I went home (-)(+)=- I deny that I didn't go home (-) (-)=+ (actually you're saying you went home, in the last one) and you could keep going, pair number of negative is positive

[@nickmichaels6242]

DiazdelVivar Yes, I see what you did there, thanks. But how do you apply that in a mathematical situation?

[@DiazdelVivar]

is the same.. two denials is positive (-)(-)=+ when

[@henrywang6931]

very nice! thats a good way to explain it, thx

[@zoeboy82]

You are doing the same thing the person in the video did. You are just assuming the value of didn't go home and denying going home. The only way to justifiably substitute an action or a thing with a integer is by having two exact opposites. Didn't go home is the opposite of going home. You can't enter denying going home without a complete opposite of the denying.

[@rayanrao4944]

DiazdelVivar A

[@classic8054]

I graduated college already and yet while I'm brushing my teeth this morning I asked myself why negative multplied by a negativr is positive and I can't think of it geometrically as in the area way of multplying so I looked upon it here and found this awesome vid! Now my mind found inner peace this morning. Hahaha

[@shorndrack]

watching this at 3.10am cause i know it'll be interesting...

[@user-jq3qt8no7h]

Nurd

[@KTC_008.]

@@user-jq3qt8no7h no

[@tuffcrowzer]

This is badass - I used to think that math was strictly memorization, memorizing what numbers and symbols did. But through seeing it again with a better grasp of vocabulary when you used terms like this equals this you prove why. I never questioned why in the past but here you taught us that we can prove it. So rad!!!

[@Captain_Vitorio_Muscovy]

great teacher, thanks you for posting such helpful videos.

[@ANascente]

I continue to think there is something wrong with this explanation 🤔

[@adityaagarwal636]

Can we just appreciate that he make such good drawings😀😀

[@sertaliozer1]

Why, my friend had his whole PhD based on this, Sal you've explained this phenomenon in 6 minutes where for him took three years! =))

[@vinaysharma5842]

Because Sal spent 1/3 of his life in college.

[@kiranrajashekaraiah3024]

@Ottoman TR Can you plz share his PhD work here. I still don't understand something here, and unable to find much stuff on the internet. It would be super helpful for me, thanks!!

[@vladimirprotein3275]

yes please, share the phd work, would love to see how he explained this..

[@emerlander]

Love your work, Sal. Thank you so much!

[@Clipster15]

I have been thinking about the rules/conventions of mathematics a lot lately considering that when moving to more advanced subjects, I forget and take for granted the basics - I can't have that. I came to the conclusion that there is nothing too special about this rule but rather is chosen as to be consistent with other mathematical operations. I then ended up using the distributive law, as done here. This gives me some confidence that I am not as stupid as I thought.

[@TheStatisticsExpert]

Fantastic lesson! Thank you very much!!

[@rhoadess]

I like to think of a negative sign as a kind of operator that reflects it in the opposite direction on the number line. So take the positive value 1 on the number line and place a negative operator on it, -(1) = -1*1 = -1 on the number line. Now take that same result and reflect it again -(-1) = -1*(-1) = 1 on the number line. We are back where we started. The result given by Khan is very intuitive, and mathematical.

[@SanketRaipure]

It's a proof... Not a reason...!

[@utlo]

A proof is a type of reason. Also the target audience of this video is unlikely to know the mathematical definition of the word "proof".

[@lampshadethisforshadowthat1050]

You should study physics pal XD

[@cybertoast24]

this helped me so much in math class!

[@AkshatSinghania]

When I asked this question , I was told that it is just a rule needing to be memorized My primary teacher couldn't appreciate my abstract thinking but my highschool teacher can so im doing great now in mathematics

[@cristianfcao]

Videos like this are VERY important! Most people (myself included) when studying math rarely have an intuitive understanding of the meaning of most of the mathematical procedures, which we apply just because we are told to. This makes us forget math as soon as we stop applying it regularly and makes it difficult to solve non standard math problems. I'd urge you Sal to make a whole list of videos explaining the "why" behind most basic mathematical procedures and rules, I think it can help a lot!

[@TheEGod.]

I never thought it this way. Really great video!

[@HuckleberryHim]

I completely agree. I always try to understand basic or even more advanced mathematical concepts. At times the depth of things completely escapes my comprehension; why is it that the formula for the vertex of a parabola actually works, for example? But most of the time, I understand it, and I receive much joy from being able to do so. THAT is math; it is deeper than the recitation of formulas. It is about the numbers as entities, acting logically to compose mathematics.

[@asmrclan8627]

M a senior mathematicien and i never had the idea to look this up. M glad that a random day i did

[@sertaliozer1]

you might be right, he actually finishes this year, i still don't understand how he was able to write some much about it in three years, i'm an Aerospace Engineer masters graduate - little things like this still fascinates me =))

[@TheJainmehul]

It's great that sal makes it's viewers think a different way as he did in this video...!! thanks a lott

[@pathetic_girl]

Eh, I just don’t really understand this one so much. They say either you’re a math person or you aren’t but for me it seems like the ladder. I am really good at some other stuff in math like long division and fractions, but when it comes to negative numbers I just don’t get it. But props to you guys for the one video I understood the most, doesn’t mean I understand it now, but I definitely understand it a bit more than before.

[@therealjuan6184]

If you don't understand this, then it just means you have to start from the beginning. Maybe watch a few videos on what numbers are and the number line.

[@eeriemyxi]

-2 * -5 = x -5 = x / -2 [Cancel out - on both sides] 5 = x / 2 5 / 1 = x / 2 [Cross multiplication] x = 5 * 2 = 10 maybe that helps?

[@xy0nix]

Brilliant explanation!, crystal clear

[@juanfleury]

The best explanation about it, thank you very much.

[@DissonantSynth]

Brilliant video. Thank you!

[@karanagarwal1106]

That was really helpful .. thank you..

[@chao1322]

Math is so interesting. I am truly amazed how ancient philospher managed to create math and be able to be consistent with all other mathatics rules.

[@maheshreddy5786]

Don't prove it mathematically. I can feel 5 ×-3 = -15 as (-3) +(-3) +(-3) +(-3) +(-3) = 15... How can I feel -5×-3? 🙏

[@ashleywebster5671]

(-5)+(-5)+(-5) So now we’re testing it with the addition rule; when the signs are the same you add and retain the sign. Answer will be : -15

[@jwhitdatman27]

Great video thank you for posting!

[@Farhan-hp8kr]

Thank you for the great explanation 😊

[@SeraphisQ]

A very short, yet educational video. Thanks

[@nyaatarara]

This is the only video that explains the logical reason, no weird metaphors, just logic

[@salsspar2132]

simple intuition but never thought about it. thank you

[@lakshmananparthasarathysub9873]

Thanks a lot , I was kinda confused with maths recently and disturbed couldn’t do maths in real life or for real use now legit I get it

[@MrTsingo]

Very nice ... using the distributive law to establish the facts about multiplying a negative number by another negative number. Thanks for posting.

[@johnbrennan3372]

Good way of explaining it.Very clear.

[@manishchhetri]

Thank you so much for this ..

[@BrainFreezeMC]

Im about to start college and never knew why. Thanks for this

[@citydrums7525]

Great example. I was wondering why this was so outside of the simple shorthand explanation of two negatives multiplied equals a positive. Being able to see it makes it comprehensible.

[@StarTrailer-ChinaTrailer]

Thank you very much;

[@rohanthomas5434]

For curious minds out there, our familiar integers form a beautiful algebraic structure known as a ring. (-a)(-b) = ab is a provable property of rings. Sal presents the cream of this proof without using jargon.

[@ViniciusClaro2012]

Excelent!! Thanks a lot!!

[@vamosabv]

Deceptively brilliant! I recently asked a couple of (smartish) friends of mine why this is so (we're all graduates/postgraduates in Engineering). We were unable to come up with a good explanation.

[@DaFan86]

Awesome drawing of the philosopher! :D

[@Ghost572]

Changed my mind actually I've managed to visualise why it works!

[@bigdaddy793-1]

This video is gonna get me through online 6th grade!

[@nishanta4575]

Finished my post grad and now i understand! Finally!

[@aleksandrpopov8490]

you're the best. thank you

[@TARUNVERMA911]

This can also be explained through physical concept of vectors. Multiplication changes direction while addition does not. Multiplication gives equal importance to - and +, while in addition + has more importance (along with magnitude) while - is just secondary. (+) + (+) = (+) ,,,,,, (+) x (+) = (+) (same signs after =) (-) + (+) = (- / +) ,,,,,, (-) x (+) = (-) (can be different signs in addition depending on magnitude) (-) + (-) = (-) ,,,,,,, (-) x (-) = (+) (different signs in both) Multiplication is not effected by magnitudes of digits, but only by direction of signs.

[@nasser101]

TARUN VERMA yes!! Finally someone said what I wanted to say

[@kativat2068]

Can you explain this another way please

[@kativat2068]

Are you saying when adding two negatives, since it is a spectrum, the other side is its positive equivalency ?

[@Ahinorah94]

I completely agree. Math should be taught like this in schools.

[@O1NKery]

I get it. thanks

[@garudastan]

Well done. Thank you.

[@maulanabagussaputra1589]

Very great explanation and now u understand

[@bon12121]

Just what i needed.

[@giselyandrade582]

This video is awesome!

[@iamemil9782]

Incredible!

[@ernestoabrantesjr.2043]

Thank you! Now I know. I can now answers those who asked why negative times negative is positive 😂

[@TheThinker8878]

Hey Sal, have you considered explaining quantum physics? or have you already done so?

[@ngjade9711]

speaking from hindsight I understand the logical rationalization of why 5 x -3 must be -15, and -2 x -6 must be 12, and so far the math built up up from this foundation has made sense. But how did they know so sure back then?

[@whatsuptrends2936]

this is best example for how teachers truck you with there answers..

[@k.tanishnaidu4366]

Satisfied!

[@NGC-7635]

Multiplying two negative numbers and getting a positive number makes much more sense when you think of it terms of repeated subtraction instead of repeated addition. For example if you treat (-2)x(-6) as subtracting (-6) 2 times or subtracting (-2) 6 times, so you end up with +12 because what you are really doing is -(-6)-(-6) or -(-2)-(-2)-(-2)-(-2)-(-2)-(-2) which if you notice the minus symbols cancel out and you get from that is 6+6=12 or 2+2+2+2+2+2=12.

[@rorangarrowson5]

Thank for this. I really wish I was shown this by my teacher. I never understood this before. Not I get why to a point.

[@sliver170]

Sal is such a good artist!

[@D-Dev]

Great explanation.

[@Waranle]

"I'd urge you Sal to make a whole list of videos explaining the "why" behind most basic mathematical procedures and rules" that course is called Number Theory and it is out there my friend

[@Veganphobic]

From now on I shall Watch at least one of your videos a day.

[@lkjhggdddr]

Negative times negative equals positive is the same thing as the enemy of our enemy being our friend

[@nikanm17]

That is the most interesting thing I've ever heard

[@JAMHuiMusicVideos]

THANK YOU

[@HoomanPlays]

They should really teach this part in school instead of just making us memorize the rulesets.

[@ashraf2168]

Well explained 🎉

[@robertthrelfall4820]

In simple terms -1 * -1 = 1 is to preserve the =distributive law. Without the distributive law algebra as we commonly use it would not be possible. We have -1*(1-1) = -1(0) =0, so -1*1 + -1*-1 = -1 + 1 = 0.

[@ivanrascon]

Bad things happen to bad people and that is good or negative time a negative is a positive

[@jayrewhernandez868]

This is great... the video gave me a complete understanding, of how and why negatives work the way they do.. thank you.. Ps Can you help me understand commas..lol

[@belakun1244]

If Billy has five negative apples, and Sammy has three negative apples for each negative apple that Billy has, how many positive apples does Sammy have?

[@_matthewhalloran]

Found this as a senior in high school looking for this answer… I wonder what else I am overlooking

[@WillyKZY]

the title made me think for abit haha. very good explaination (y)

[@LiciaCOG]

That's awesome, gosh what other simple math things have I not taken out the time to try and understand... aww makes me happy, i keep telling people math always makes sense, so stop saying it doesn't, we just need to find a way to make it make sense to you

[@conorspalvieri5948]

I’m going into third year engineering physics and had to watch this. Why don’t I know this.

[@Migueldeservantes]

Yu Da man! thanks

[@IHeartViHart]

I had another way of thinking about it early on in the video. Decided Sal's way (using distributive property) is better after hearing it, but other commenters might be interested in the way I thought about it. One other identity that we need to keep mathematics constant is a/a = 1, and if a = -b, then we can say (-b)/(-b) = 1 --> (-b) * 1/(-b) = 1 --> (-b) * (-1)/(b) = 1, therefore a negative times another negative must equal a positive. :]

[@chardyP]

khan, you are amazing

[@nevinofstedal8679]

Why am I only now understanding this as a 20 year old...

[@Technikplayzzz]

good drawing skills sal and this vid helped me understand better 😁👍👍👍

[@NickWright]

On that last example at 05:28 I'm lost on how you ended with -12 + 12 = 0 in that breakdown and then that suddenly becoming +12 in the conclusion? I mean I understand 2 negatives multiplied equal a positive number. But I did not follow this breakdown apparently lol.

[@wallymayer5998]

I feel so enlightened

[@charlies3524]

Im sure they find out onther way but this is one smart way to prove it :D

[@michaelmacias3499]

Got to say that Khan Academy videos give you some of of the best explanations. I remember asking this question to my math teacher, why is it that a negative × a negative = a positive. Couldn't explain. Now I'm 30 years old and finally understand why a negative × a negative = a positive lol 👍

[@theShashwat0024]

Research paper of Mohit Gaur Reasonably explains about negative × negative gives the positive answer.. Its very easy why it is so.. based on the original indian texts about Mathematics.

[@calhlin]

Thanks Salman! :)

[@Ghost572]

I do understand negative numbers now but it took me a whole night of deep thinking to come to any conclusion, this video wasn't very good it just said yes this is this and thats it!

[@SoThisIsMyCat]

Very very nice video compliments

[@dumbasffff]

Wow. just wow.