Why a negative times a negative makes intuitive sense | Pre-Algebra | Khan Academy

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Use the repeated addition model of multiplication to give an understanding of multiplying negative numbers.
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Pre-Algebra on Khan Academy: No way, this isn't your run of the mill arithmetic. This is Pre-algebra. You're about to play with the professionals. Think of pre-algebra as a runway. You're the airplane and algebra is your sunny vacation destination. Without the runway you're not going anywhere. Seriously, the foundation for all higher mathematics is laid with many of the concepts that we will introduce to you here: negative numbers, absolute value, factors, multiples, decimals, and fractions to name a few. So buckle up and move your seat into the upright position. We're about to take off!
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[@rmsdudqkr]

can't believe i watched this as a senior in university :D

[@darrendias8863]

You're not alone I bet a ton of uni students don't know this.

[@jimmy0149]

@@darrendias8863 im in. 6th and learning this

[@sastashroud7646]

@@darrendias8863 learning while I'm working professional having 1 year experience i am really feeling ashamed now

[@Phagocytosis]

I love it when Sal draws in his videos (or has already drawn at the start of them)!

[@PedroGuilhermeSiqueiraMoreira]

I always thought of it like this: if a pool leaks some amount of water, lets say 2 liters per hour (or -2), after some period of time, lets say 1 hour (or +1), I have lost 2 liters (or -2) of water because (-2)x(+1) = -2, but if I I wanna know the leakage of, lets say, 2 hours ago (or -2) the answer would be +4, that means I had 4 liters more than the amount I have now, that means (-2)x(-2) = +4. Please, excuse my english.

[@yaswanthpakalapati9994]

Great explanation bro 😘😘

[@monzalesrenzrochem._stem1404]

Niceeeee.... I feel better

[@sastashroud7646]

Ayoooooo this is best comment

[@Hellsconsort]

Took a couple of goes to sink in but I got it finally. Thanks Sal.

[@michaeltrankvilitsky836]

Subtracting a negative from a negative is not like taking away someone's debt. When you take away someone's debt, you're left with zero. So a more accurate analogy is when a person pays you back the money he owes you. It's just phrasing, but it really impacts one's understanding. I think you should refilm this video with some examples people provided in the comments and use a coordinate plane to explain it better. But overall this is a great course, thanks a lot. You've done tremendous work to explain basic math, it's the most comprehensive math course for children I've ever seen.

[@motobaafeoke4900]

Good day, sir. I think he meant, if you wanted to "take away" someone else's debt, you would "give" them money. The action, not the result. Thank you.

[@amandar3905]

Exactly what I was thinking. I like the idea of giving examples.. like use apples or pennies. The video helped explain it better than a previous video I'd watched, but I think I need to find 1 more with examples.

[@Jose-tl6uy]

thanks this is great! its so weird how you can know something to be true, but not understand it...very grateful for this video and for the service the kahn academy provides!

[@Peter_1986]

It's because he repeats things several times that he is so easy to understand. Instead of speed-talking through everything and being like "lol the listener should know this already wtf" he actually takes the time to respect the listener and take the time to explain EVERY part, repeating it a couple times to make sure that the listener understands.

[@Sachica00]

When i see sign multiplication, i always think that the negative sign is just telling me to do the opposite of what i'm doing, and the positive sign tells me to do exactly what i'm doing. so, a negative times a negative is, instead of a working with a negative you gotta do the opposite which is a positive number. Great vid Sal, you're giving people arround the world what we really need: Education. Thanks.

[@cimmik]

That makes so much sense.

[@Owl90]

Thank you for making me understand this.

[@robokarizma0308]

Concept of debt is also nice way to get over this .

[@Mrtheunnameable]

Thank you so much. I know that -1*-1= 1 but I never understood why, and I've wondered about it every day in math class.

[@Nisstyre56]

I think it's better to view it as "scaling". So when you multiply a negative integer by a positive integer you are "scaling" the negative integer, and there is only one way you can scale it: down.

[@timmytim9054]

Thank you. You save millions of lives! Sal is love, Sal is life.

[@alphie10]

Brilliant!!! Makes sense funny thing is i actually thought about it many times but it always puzzled me in some ways but i can see it more clearly

[@uzairhussain4856]

a ≠0 By definition Multiplying a non zero number by - 1 reverses the direction of that number. That is -1×(+a) = -a -1×(-a) = +a. This explains the product of any two integers of any signs. (- 2) ×(- 2) =( - 1)(2)(-1)(2) =( - 1) (-1)(2)(2) = - 1(-1)(4) = +1(4) =4. (- 3)(2) = - 1(3)(2) = - 1(+6) = - 6. a × b has no factor of (-1) so it asks you to find the product ab without reversing the sign of product. In -a ×b factor (-1) occurs only once so its asks you to find the product ab and reverse its direction. In -a ×-b factor (-1) occurs twice so it asks you to find the product ab and reverse its direction two times. In -a× - b × - c factor (-1) occurs three times, so it asked you to find the product abc and reverse its direction 3 times. Subtraction of numbers also requires the definition on top. Subtraction of number means the addition of negetive of given number. That is a-b = a+(-b) It means If you want to subtract a number b from a. What you actually do is multiply the b with - 1 and then add the reversed b in a. Example: Lets a = 2, b = -3 -b= - 1(-3) = 3 a+b = 2 + (-3) = - 1 a-b = a + (-b) = 2 + 3 = 5. Another example:: a = 5 b = 3 Task : Subtract b from a. Subtracting (b) from ( a) by definition would be the addition of (negetive of b) or (reversed b) or - 1(b) into a. Thus first we need to find -b. -b = - 1(b) = - 1(3) = - 3 a - b = a + (-b) = 5 + (-3) = 2

[@swal593868]

Could never understand why multiplying 2 negatives became a positive number. Your wonderful answer had at last made it clear. THANK YOU:)

[@kerrywsmyth]

I wish Khan went one step further and showed how this works on a number line.

[@unnatigokhale6237]

thank u. its very simple way/ method to understand negative numbers multiplication on number line

[@ujjjawal1217gasj]

if khan academy were able to give education to evry indian child,we would hav been on top

[@miraym5728]

This video is a life saver..... Thank u sooooo much 😊😊😊😊

[@tmcdgreen]

Wow, that is a really good explanation!

[@chase_modugno]

I think it's pretty interesting that negative numbers are just a function of our imaginations since it's impossible to have less of something that doesn't exist, i.e. 0.

[@GamerPro-rs9gy]

Sal your video explains everything, you are the best math teacher

[@HellaPerformance]

Wow thanks, that actually makes a lot of sense.

[@vthegreat]

HOW DO YOU DO IT! THANK YOU SO MUCH!!

[@GameFrosts]

Brilliant example!

[@priteshprakash950]

thanks it helped me a lot

[@JasonCtutorials]

Wow nice drawing love it!

[@ThoseRejects]

sal's handwriting on the tablet has improved so much

[@earthygreen]

I still am having a difficult time conceptualizing -2 x -3. 2 x -3 makes sense to me. -3 + -3 = -6 I *don't* get -2 x -3. How do you add up -3 NEGATIVE 2 times? There's a space in my brain reserving a spot for this explanation and it won't let this go until that spot is filled.

[@ekinebobmanuel4551]

Actually like it or not math is quite prevalent in reality. We don't make up the patterns, we observe them. Sort of like how Pythagoras didn't decide that A^2 + B^2 was going to be equal to C^2, rather he derived that equation empirically after noticing it in every right triangle.

[@ya-rx8nd]

earthygreen Because it's negative 2 times like he showed in the video. A negative negative by definition becomes positive and put together the numbers add up.

[@christinamcphee9290]

Try this. -2 means you are subtracting 2 of something -3 is the amount of that something, so call that something debt. You have 2 debts of $3. So , you are subtracting or taking away those two debts, -2 ×-3.This. cancels the debt, giving you +6, or $6.

[@robokarizma0308]

It is like removal of Rs. 3 debt two times. So this converts the bearer to a lender.

[@rokkistart]

@@christinamcphee9290 best explanation Thanks.

[@moWEEEED]

very nicely done, that was a better way to explain it than Khan's way

[@TheSpecialAviator]

loved it

[@DODEJX1705]

thank you

[@GreenFi5h]

I have to say the reason that the statement, "There are minus three cows in a field" (I assume you mean -3 cows, and not there are 3 less cows in the field), is because negative numbers are part of the integer number system. In this example, it would have no bearing in real life because it is impossible to have negative anything (less then nothing). That is why we use natural numbers for things that have a bearing in real life, like the cow example, and integers for things like finance.

[@vokvovkov]

you can also think of multiplying by a negative as rotating by 180°, and multplying by a positive as rotating by 0: rotating two times by 180° is like rotating by 0°!. this can also give an intuiton of complex numbers, which introduce the other angles: multiplying by i is rotating by 90° (anticlockwise)

[@Anigeo94]

loved it 8/10

[@Austrahlian]

Can you do a video on why positive numbers multiply to equal positive numbers, aswell as odd numbers. And why odd numbers and even numbers multiply as even numbers. I know the basics, I just wouldn't mind a more in depth understanding.

[@effortless35]

You can use numbers for things that have nothing to do with counting stuff. For example it could mean position along a road, with one direction being positive and another negative. So if you go 5 miles in the positive direction and I go 2 miles in the negative direction we'll be 5 - (-2) = 7 miles from each other. Once you are comfortable dealing with negative numbers you can find them useful in many ways and they start to feel just as "real" as positive numbers.

[@puhteeah4876]

I figured this out with a friend AND THEN was sent this video

[@skinzfan8926]

i like'd this! your videos should have more overlapping themes with philosophy.

[@pjtrusci]

- (-3) - (-3) = 6. the negative outside of the parenthesis is basically -1 and you multiply that -1 times a -3. They cancel.

[@NazaMusicProductions]

Also to the point of requiring to prove what a negative number is in order to prove this equation, you don't need to prove something that has been proven to prove something else. For example when you ask where did someone get their apple, they'd say from the shop. Not from the shop whom of which got it from the factory and they got it from the farmers and the farmers got if from the tree and the tree got it from growing and so on...

[@dk0r51]

00:31 "...you want to have a slightly deeper intuition than just having to accept it so that it is consistent with the distributive property..." 04:33 ::Applies distributive property despite intending to give an explanation independent of the property.

[@G0d0fC0d4939]

Your English was solid

[@corycsweet]

Another way to think about it is when you are subtracting negative numbers, lets say: A - (-B), by intuitive sense, and the logic that was just presented, we know that when you subtract a negative, it is the same as adding a positive, therefore: A - (-B) = A + B Now lets subtract A from both sides - (-B) = B And now lets rewrite the left term -1 x -B = B, or -1 x -1 x B = B Divide B from both sides (assuming B is not 0) -1 x -1 = 1 That's what you are really doing when you subtract a negative #.

[@danbdatsme]

you are my hero

[@BlackbodyEconomics]

Yep

[@ThePeterDislikeShow]

Better yet, I find these videos help me sleep when nothing else works.

[@neogauntlet1008]

i think the illustration of taking away someones debt makes sense

[@HellaPerformance]

Will you now please explain the intuition behind why a negative exponent produces a fraction?

[@NazaMusicProductions]

If we were to do this in the current day we would take hours, days, weeks just to prove one thing that seems not too hard to prove.

[@melissacarl2002]

OH YEAH!

[@beast1346]

what app are u using to write all this down

[@aag1234567]

I the ancient philosopher mathematician aprove this .

[@Mrtheunnameable]

It's part of the graph.

[@NazaMusicProductions]

The point you have just given there was that he needed to say "what actually a negative number means." But this is not the case in the video, he is proving that a neg x neg = pos and neg x pos = neg. Now what a negative number is and for your information this video extended from a previous video of why neg x neg = pos which he gave hint for at the start of the video by not having a proper intro.

[@banditontherun1024]

Well if you define even numbers as 2k, and odd numbers as 2k +1 (where k is some integer) you will see: ExE is 2k x 2k. That is 4k^2. But 2k^2 is also an integer (an integer sqr is still an integer). So now you have 2 times some integer, so 4k^2 must be even. ExO is 2k x (2k+1). That is (4k^2 +2k). That also equals 2(2k^2 +k). Well 2k^2 is an integer, and an integer plus another integer (k) is also an integer. So the result is an integer (2k^2 +k) times 2. Again this has to be even.

[@anushab2895]

Can we express -2 *-3 as -(2*3) which is opposite of (2*3)=6 and hence -(2*3) is -6

[@davidvorick]

Anusha B (6*6) = 2(3*3)? ===> you can only factor addition like that.

[@ya-rx8nd]

I think it makes perfect sense unless you're one of those that wants to oppose a negative negative # being positive. It's just the number sign reversing or cancelling out, thus the combined value equals positive.

[@Gregorovitch144]

Whilst it is true that our convention of calling an electron's charge negative and a proton's positive is arbitrary - they are opposite polarities and we could have reversed the convention - we do know that the overall charge of , say, a hydrogen atom is zero, i.e nothing, but also that proton and electron have charges of equal magnitude but of opposite polarity. Therefore one polarity must be more than nothing and the other must be less than nothing.

[@tomrose4560]

I think its easier to visualise multiplying two negatives in the following way: -2 * (-3) = remove two groups of (-3). If we start with 6 zero pairs (6 black discs representing 6 lots of "+1" and 6 white chips representing 6 lots of "-1") We can then remove 2 groups of (-3), leaving just the 6 black discs, i.e. +6 Another way is to start with an equation: -2 +2 = 0 Then multiply each term by a negative number, say (-3) -2*(-3) +2*(-3) = 0*(-3) -2*(-3) - 6 = 0 -2*(-3) = 6

[@tintran7421]

Tks a lot that makes me sense

[@essboarder23]

I am averaging his videos are about 7 minutes. There are 3113 videos. So (7x3113)/60 = 363.18 So an estimate would be 363 hours.

[@LivingSkits]

but that negative of the electron is something. Just because it has negative in its name, doesn't mean it's less than nothing.

[@ya-rx8nd]

It makes perfect sense unless you're one of those that wants to oppose a negative negative # equaling positive. The 2 negatives reverse or cancel out thus the combined value equals positive. Example: -2(-3) (0) - -3 - -3 = 6 I put that zero there to show it's starting from nothing. What's minus a negative #? It goes the other way and the only other way. Positive. Negative negative times is positive.

[@karismydog]

Plus at 4:50 you say that we know that -(-2)=2 thus neg x neg = pos, then what's the whole point of doing these calculations.

[@jantunez4350]

simply its somewhat hard and easy

[@wilkes982]

could be his next video?

[@crissilovesyouxo]

taking away someone's debt..makes sense...

[@tactics6659]

Woah

[@Gregorovitch144]

So what do mean when we say there are minus three cows in a field? Outside of "I have four apples, you take two, how many have I got left?" type questions (i.e. taking a smaller number away from a bigger number) and the special case of financial accounting we do not really have an intuitive grasp of a what negative numbers mean. I believe the ancient philosophers asked the cow question and concluded negative numbers were merely useful device only with no basis in reality. .

[@m4rcus4ur31ius]

That's how subtraction works because "it's like taking away someone's debt ... which is the same thing as giving them money".

[@williamyang5698]

I don't really get why you needed the brackets for the neg times neg

[@genopsyism]

This seems like a recursive definition. -(-2) is still a multiplication of two negative numbers... it's the same as -1*(-2). At 3:35 you simply state that it should be positive because that's how subtraction works. What am I missing here?

[@91jgphonecall]

negative means opposite. whats the opposite of a pos 3, -(3)? its -3. whats the oppsite of a negative 3, -(-3)? its 3.

[@alphie10]

7!!

[@essboarder23]

On KZbin, he has 3113 videos

[@abhishekpandey9049]

sal from where do you get these interesting faces !!!!!!!!!

[@kevinjohn7013]

Whenever you multiply two numbers together...the second number in the order always refers to the number of times that you add the first number...So, 4 x 5 is equal to 4 + 4 + 4 + 4 + 4 = 20. Another easy example, 6 x 3 is equal to 6 + 6 + 6 = 18. So, if the second number is only referring to the number of times that you add the first number; then how can the second number ever be negative? The first number can be negative because it indicates the direction that you are moving on the number line, so... -4 x 5 is equal to -4 + -4 + -4 + -4 + -4 = -20. Or, -6 x 3 is equal to -6 + -6 + -6 = -18. But how can you take -4 x -5 times? Or 4 x -5 times? You can't, "negative 5 times" doesn't exist. It can only be "5 times."

[@fkthewaat]

Look, when you multiply a positive with a negative you do this: Ex. 3x -5= (-5+-5+-5) which = -15 So when you multiply a negative with a negative: -3 x -5 = So you multiply -5 "-3" times.

[@yooy3086]

whats the software or thing you use?

[@DavidMcAfee]

I am pretty sure he uses paint (the program that comes prebundled with Windows), however just has "invert colours" selected under options.

[@moeezk7]

I know this comment is 8 months old, but incase you are still wondering, he used smoothdraw 4. You can find this on google.

[@reeyea7405]

-(-2) = +2 how?

[@handleh]

- means taking away and here we are taking away the negative and its same like + which is giving away

[@TheMightyPatapon]

It just doesn't not make any sense!

[@peterf9360]

Pre-algebra huh. I would guess most do not understand this at the calc iii level. Or would struggle explaining

[@LivingSkits]

Nothing can be less than nothing. You are just making an assumption by the deletion of other possibilities.

[@t_apk4477]

superb .not so good

[@paulsha9582]

ok i understand negative x positive this is just a negative added so many times. But i am a confused on negative x negative .You use the idea of taking away someone's debt .Now i am thinking that negative x negative is like repeated addition out of debt. So something like -$20 now if you add -2 x -2=4 they will now be in -$16 debt is this how this is working ? Now if this is right -2 x -2 = 4 so if they was $4 in debt then they have zero debt now .But how can you say 4 ? confused here!

[@RaulBillieJean]

I still don't know why negative plus negative is positive

[@RaulBillieJean]

I mean negative times negative is positive..

[@alexzhou2343]

hi

[@gamegod-oj6io]

you didnt explained how -1 * -2= +2

[@websitr3986]

Can any of you critics show me the real proof that negative times negative is positive?(or give me a link with this proof?)

[@justinlequire]

If I owe somebody $100 and someone pays that debt 1 time, do I have $100 or zero dollars. I had a debt of $100 and it was paid for me once. These videos explain how it SHOULD work based on everything we know, but there are no real life applications for multiplying a negative by a negative. This is because negative numbers can't really exist in real life. They're just positive numbers in another direction. Zero degrees in temperature is just an arbitrary measurement and anything below zero is simply a potive degree that is below what we have already established as zero. You.can owe the bank money, but you can't pay it a negative amount of times. You certainly can't tell them that you haven't paid them at all, so your account should be at $0 since you've paid them back zero times. Also, if I only have 2 cupcakes and you would like 3 of them, there is not really a way that I can give you 3 anyway and be left with a negative cupcake. I could give you 2 and say that I owe you 1, but really I just gave you the only 2 that I had and a promise. Math doesn't work on promises and negative numbers only exist to hold people accountable for when they agree that they owe somebody.

[@nocturnalvisionmusic]

*"negative numbers are just positive numbers in another direction"* Never heard it like this before but very nice way of stating things :D Thanks @Justin Bear Attack ;)

[@darkqueen3597]

There is still a question that.... At the last moment he multiply -(-2) that also indicates -1*-2😑

[@biscoole]

comment number dy/dx (3x)

[@TheAngryCanary]

Like khan says, you are subtracting a negative number. the act of subtraction and a negative number are two different concepts. All rational numbers should be considered vectors. And since a negative scalar does not make sense... which is what people intuitively sense... then -2 * -2 would simply have to be rewritten to mean what it really means. which is -2 * -1 * 2 and the middle -1 really means (flip the vector) so... -2 (flip the vector) * 2 = +4 We need better notation methinks

[@essboarder23]

so I would just multiply that by the average length of his videos

[@HellaPerformance]

The graph had to come from somewhere.

[@hansanderson4768]

Please cease the color coding.

[@JohnJames-nl8ji]

He's subtracting an extra time

[@Moh_Sa]

فسر الماء بعد الجهد بالماء As in the Arabic saying You still didn't give the "intuitive" meaning behind the product of two negative numbers.. In the last line when you said = -(-3)-(-3) equals 6 means you just multiplyied -3 by a -1 You didn't explain it you actually went in a circle 😅