Why a negative times a negative is a positive | Pre-Algebra | Khan Academy
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- Опубліковано 12 вер 2024
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Use the distributive property to understand the products of negative numbers.
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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.
Well said
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.
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.
@@Dibu-Jitos A 10 year old is ready to deal with this and more complex topics if you really know what you are teaching
@@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.
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!
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!
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.
Bro i'm a bachelor degree
lol same
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
Wonderful explanation. Understanding it really only comes once you know "why" things are the way they are. 26 and I just learned "why" today.
You’re 27 now. Say Hi to me. I’m just getting yo know why now
@@michaelneftali8711 say hello to me to
@@michaelneftali8711 i am 55
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!
@@MsVamPireChicI think he use that to explain and it wasn’t anything important
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.
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.
EXACCTLY my teachers could never explain the "why".. they only knew the "how"..
@@mydogskips2 this is one of the most ignorant comments ever written
@@mydogskips2 this is one of the most ignorant comments ever written
@@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
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
DiazdelVivar Yes, I see what you did there, thanks. But how do you apply that in a mathematical situation?
is the same.. two denials is positive (-)(-)=+
when
very nice! thats a good way to explain it, thx
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.
DiazdelVivar A
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.
Xehemoth I know right. Same thing a with a negative divided by a negative.
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 :)
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(-)=(+)
@@chelseascott98 Thanks a lot!
It's simple:
Love +
Hate -
Love to love = Love
Love to hate = Hate
Hate to love = Hate
Hate to hate = Love
I didn't know Sal could draw amazing portraits
Thanks
He has beautiful handwriting, so it shouldn’t be a surprise.
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
great teacher, thanks you for posting such helpful videos.
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!!!
Math is the pure logic. It's all proven
Don't prove it mathematically. I can feel 5 ×-3 = -15 as (-3) +(-3) +(-3) +(-3) +(-3) = 15... How can I feel -5×-3? 🙏
(-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
View multiplying a positive with a positive as repeated addition. Multiplying a negative with a positive as repeated subtraction. Then multiplying a negative with a negative becomes repeated subtraction but of a negative number, so it's repeated addition again.
It's a proof... Not a reason...!
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".
You should study physics pal XD
watching this at 3.10am cause i know it'll be interesting...
Nurd
@@user-jq3qt8no7h no
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!
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.
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.
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! =))
Because Sal spent 1/3 of his life in college.
@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!!
yes please, share the phd work, would love to see how he explained this..
So that means if I had 2 negative covid tests, then I would have been tested positive?
Haha
*very funny*
*did laugh*
Can we just appreciate that he make such good drawings😀😀
Hey Sal, have you considered explaining quantum physics? or have you already done so?
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.
TARUN VERMA yes!! Finally someone said what I wanted to say
Can you explain this another way please
Are you saying when adding two negatives, since it is a spectrum, the other side is its positive equivalency ?
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.
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 =))
Finished my post grad and now i understand! Finally!
I continue to think there is something wrong with this explanation 🤔
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.
If someone likes this way of learning then consider this explanation aswell:
36 * 3 = 36 + 36 + 36 = 108
36 * (-3) = -36 - 36 - 36 = -108
-36 * (-3) = -(-36) -(-36) -(-36) = 36 + 36 + 36 = 108
The pattern of multiplication inverses.
METAINVERSION xd
Your explanation is the real reason, NOT his!
You, sir, are a hero and have saved many lives like mine.
Dumb ur actually muliplying -1 with -36 if u see it clearly .which give a positive no
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
This did not help me on WHY negative and negative is positive.
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 :)
Basically because any other answer will cause complications to the fundemental logic mathematics has created so far so they just put some digit that will cause the least flaws in maths.....negative numbers cant be multiplied because the only thing they apply to is temperature which cannot be multiplied or divided
but negatives are a quantity below negative to so if you multiply it it shouldnt be possible to to get the same quantity as a posititive
+Chelsea scott thats like saying 2 is -2
According to your logic, it doesn't decrease, it becomes positive (greater, not smaller) . I'm a collegue graduated and have actually never been thoughtful about this until now. according to this explanation, it's just to make all other math fit. It's not self-evident, but it becomes evident once you make all the other experiments with the distribution property.
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.
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.
-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?
View multiplying a positive with a positive as repeated addition. Multiplying a negative with a positive as repeated subtraction. Then multiplying a negative with a negative becomes repeated subtraction but of a negative number, so it's repeated addition again.
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.
Very useful for advance algebra 2. I hate doing things without understanding them.
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.
Needed a confirmation - would multiplication be also considered REPEATED SUBTRACTIONS when dealing with negative integers as we consider it as REPEATED ADDITION when we deal with positive integers?
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.
Can we explain it through venn diagrams in a better way ?
The best explanation about it, thank you very much.
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?
Multiplying a negative number by another negative number is basically saying that you want the opposite of whatever you started with twice. So if you have -2 x -6 you want to increase the negative six value by 2 and then invert it
It's great that sal makes it's viewers think a different way as he did in this video...!! thanks a lott
Is there a way to prove that the product of two negatives is a positive without using the distributive property? I mean what if two negatives didn't make a positive, wouldn't the distributive property be not applicable for multiplication then? I think what I'm asking is I don't quite get the intuition behind it. Like, for example we can imagine 3 as moving "right" on the number line for 3 units and 3 x 5 would be repeating that "moving action" 5 times. I could imagine - 3 as being moving "left" along the number line 3 units but what are 5 negative repetitions (as in - 3 x -5)?
Or perhaps is the better way of looking at the concept of a negative - in terms of the number line - is flipping it? For example, - 3 is first flipping the number line, then moving 3 units. And -3 x -5 would be, (-1)x3 x (-1)x5, so we flip the number line, move three units, then we flip the line again repeat moving 3 units 5 times, which ends up being on 15 units on the "positive" direction.
Very nice ... using the distributive law to establish the facts about multiplying a negative number by another negative number.
Thanks for posting.
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.
this helped me so much in math class!
Mathematics is a brain and logic of the universe, and also our logic since we are the integral part of universe. So let me explain you through ordinary life:
if you do good thing (+) to a good person (+) that's something that creates good and positive outcome, so positive (+)
if you do bad thing (-) to a good person (+) that's negative thing, we all know that, so that is negative (-)
If you do good thing (+) to a bad person (-) you are helping the bad, the evil progress so that is negative (-)
If you do bad thing (-) to a bad person (-) you are destroying the bad, the evil, the outcome of this action is positive, so that is positive (+)
As you can see Mathematics teaches you how to behave, show you what is right or wrong, moral or unmoral. You don't need philosophers or gurus or politicians or econimists to tell you that, because some of them will persuade you that 2 or 3 is positive, that you should do good things to bad people in order to change them, which will lead to your regression, decline and even suffering.
And I would like to add that nowadays we need the fourth type of people even more than the first
If u do bad thing with bad person he could become more bad.
Brilliant explanation!, crystal clear
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.
I get it. thanks
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. :]
Awesome drawing of the philosopher! :D
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.
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!
I never thought it this way. Really great video!
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.
Im about to start college and never knew why. Thanks for this
Bad things happen to bad people and that is good or negative time a negative is a positive
A very short, yet educational video. Thanks
It's not "Because the maths say so. . ." It is because, as Khan mentions, if the number -2X-6 is going to be consistent with the operations we have previously define then it MUST be equal to 12. The way Khan has done it is generally the way one provides -a*-b=ab in most advanced math courses.
Hi Sir,
May you please tell me now do you make the videos?
simple intuition but never thought about it. thank you
Love your work, Sal. Thank you so much!
this is best example for how teachers truck you with there answers..
This is the only video that explains the logical reason, no weird metaphors, just logic
Negative times negative equals positive is the same thing as the enemy of our enemy being our friend
Great video! What software are you using?
Very great explanation and now u understand
Changed my mind actually I've managed to visualise why it works!
Good way of explaining it.Very clear.
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
I completely agree. Math should be taught like this in schools.
Genuine question as this is the first time I've actually thought about this, but wouldn't the power of zero present a problem for the distributive example? -2 (-1^0+1^0) is the same as -2(1^0+1^0), right?
M a senior mathematicien and i never had the idea to look this up. M glad that a random day i did
No dude your definitely right.I know math very well, done calculus and beyond but I only know it because I was fed into it, due to society. But looking into it in depth, it actually doesn't make any sense. -2 x -4. What in the world does that really mean? Okay start at -2...then what? I know it is 8 but why? I think the -2 represents the point of view from the bank(debt example) bank has 2 clients that it owes 4 dollars, 8 in total. +2 is our point of view. -2 would be the banks view. not sure
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.
Thank you so much for this ..
I have thought of it as a graph in my head. The first number is x and the second number is y. If both x and y are positive, then it graphs in the top right of the graph, and a line from the origin to the product is a positive slope. If x is negative and y is positive it graphs in the top left of the graph, and a line from origin to product has a negative slope; both are negative it is bottom left and the line is positive slope; y is negative and x positive bottom right with negative slope.
Makes toomuch sense
"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
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?
What's a real-life example that only involves multiplying two negative numbers?
Brilliant video. Thank you!
From now on I shall Watch at least one of your videos a day.
Okay, so this does make sense, but I get a bit lost as to why you just magically add in the 5(3+(-3)) in, I guess this does make a bit more sense, but multiplying a negative number by a positive number makes sense to me in thr way multiplication is.
When you take something like 5 x 3, it can be writte out as 5+5+5, nothing difficult
When you say -3 x 5, it's the same idea, -3 + -3 + -3 + -3 + -3, equals -15,
and multiplication states it can be written in either order, a x b or b x a, and get the same answer, but how would this idea work with negative numbers?
-4 x -2
So obviously the first thing that came to mind, is "how do you add something negative times" I thought maybe subtracting, but -4 - -4, jusr equals 0, then I thought about the idea doing it normally for some reason, but that equals -8, the only reason I can think of making this work, is since -4 x 2 = -8, then adding a negative, makes a double negative, making it positive, but if anyone has an idea of the way i described a negative times positive, lmk!
@khanacademy at the end of the video he says he’ll make a few other videos to explain/further prove why this is true. Where are those?
That was really helpful .. thank you..
Thank you very much;
Yeah possibly but with multipication i realised you start at 0 unless otherwise stated for example if it said 6*6 your staring at zero and adding sixes one after another. however if it said like 2+ 6*6 youd be starting on the 2 on the number line. that makes sense. Negative numbers to me now are actual physica entities we just put say we posses those things so a sum of -2*-2 really means originally someone owned that item and labled it as there like money.
Why does multiplying less than nothing, whatever tf that is, by more nothing give you something?
Good things happen to good people or positive times a positive is a positive
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
Okay, let's do it step by step with what we already know: 2*-6 = -(2*6) = -(12) = -12.
I take the minus out, multiply and add the minus after the multiplication, I didn't change any sign.
Now we do the same with 2 negative numbers: -2*-6 = -(2*-6).
Took one sign out, so we can worry about it later. Now, we already know from the first example that (2*-6) = -12. So, -(2*-6) = -(-12) = 12.
Yep. Still don't understand.
Plz explain multiplication using number line
I have a doubt in here suppose if we have to substract 2-(-3) , and connecting to analogy given in the video X have positive two then how would it make a scenes, his uncle is paying debt as (-3)which he doesn't have?
Ok, so I've had a question for awhile, and it's probably just be me looking at something all weird, but I'd like to ask anyway. What is the true significance of "0" and "1"? (Particularly "0") I mean, I understand that it, as well as 1, have an ability to be kind of "absolute", but why is that? Why is it that, "If we can make it balance out to 0," it's good? In relation to graphing, couldn't "0" be considered anywhere on a plane of infinity, as long as all the numbers move along with it in tandem? I guess my REAL question is: Where does "0" fit in real space? DOES it fit in real space? Or is it just an imaginary "fixed point" to designate an immutable location?
Thank you for the great explanation 😊
I want to know the real life logic behind it, not mathematically.
+ Orange Flash The best real life exemple of negative numbers is always finance.
To multiply by a negative number is like removing something. For instance, you have $100, and you removed 4 times your money. So you lost $400 and, as you already had $100, you are now $300 in debt. This equation in (-4)*100 + 100 = (-300).
Now, instead of having $100, you had a debt of $100. To multiply by a negative number now means you are removing your debt, not your money. If your debt was removed 4 times, it makes sense that you gained $400. So we have (-4)*(-100) + (-100) = 300.
Orange Flash yo when u decrease (-) something's negative quality it gets positive...
@@Igor_054 This really was the eye opening explanation I was looking for! Thank you!
Math is always imaginary. Its all about dimensions ... so picture the number line. Zero is the start point. Positive means moving right and negative moving the opposite, left. Now positive times negative will go the opposite direction. Negative times negative will go the opposite, the positive direction.
Think about a circle. The opposite of 0 is 180... the opposite of 180 is 360 which is back to 0... hope that makes sense
Love +
Hate -
Love to love = Love
Love to hate = Hate
Hate to love = Hate
Hate to hate = Love
This video actually does not give the answer to the question "Why a negative times a negative equals to a positive?". It only shows the fact that a negative times a negative equals to a positive. Yet, this is a good way to help young children imagine and visualize a multiplication between two negative numbers. Keep up the good work, and thanks for this beneficial video!!!
Well done. Thank you.
That is the most interesting thing I've ever heard
This video is gonna get me through online 6th grade!