Multiplying and dividing negative numbers
Multiplying positive and negative numbers
Multiplying positive and negative numbers
- We know that if we were to multiply two times three,
- that would give us positive six.
- And so we are going to think about negative numbers in this video.
- One way to think about it,
- is that I have a positive number times another positive number,
- and that gives me a positive number.
- So if I have a positive times a positive,
- that would give me a positive number.
- Now it's mixed up a little bit.
- Introduce some negative numbers.
- So what happens if I had negative two times three?
- Negative two times three.
- Well, one way to think about it--
- Now we are talking about intuition in this video
- and in the future videos.
- You could view this as negative two repeatedly added three times.
- So this could be negative two plus negative two plus negative two--
- Not negative six. Plus negative two.
- which would be equal to--
- well, negative two plus negative two is negative four,
- plus another negative two is negative six.
- This would be equal to negative six.
- Or another way to think about it is,
- if I had two times three, I would get six.
- But because one of these two numbers is negative,
- then my product is going to be negative.
- So if I multiply, a negative times a positive,
- I'm going to get a negative.
- Now what if we swap the order which we multiply?
- So if we were to multiply three times negative two,
- it shouldn't matter.
- The order which we multiply things don't change,
- or shouldn't change the product.
- When we multiply two times three, we get six.
- When we multiply three times two, we will get six.
- So we should have the same property here.
- Three times negative two should give us the same result.
- It's going to be equal to negative six.
- And once again we say, three times two would be six.
- One of these two numbers is negative,
- and so our product is going to be negative.
- So we could draw a positive times a negative
- is also going to be a negative.
- And both of these are just the same thing with the order
- which we are multiplying switched around.
- But this is one of the two numbers are negative. Exactly one.
- So one negative, one positive number is being multiplied.
- Then you'll get a negative product.
- Now we'll think about the third circumstance,
- where both of the numbers are negative.
- So if I were to multiply--I'll just switch colors for fun here--
- If I were to multiply negative two times negative three--
- this might be the least intuitive for you of all,
- and here I'm going to introduce you the rule,
- in the future I will explore why this is,
- and why this makes mathematics more--all fit together.
- But this is going to be, you see, two times three would be six.
- And I have a negative times a negative,
- one way you can think about it is that negatives cancel out!
- So you'll actually end up with a positive six.
- Actually I don't have to draw a positive here.
- But I write it here just to reemphasize.
- This right over here is a positive six.
- So we have another rule of thumb here.
- If I have a negative times a negative,
- the negatives are going to cancel out.
- And that's going to give me a positive number.
- Now with these out of the way, let's just do a bunch of examples.
- I'm encouraging you to try them out before I do them.
- Pause the video, try them out, and see if you get the same answer.
- So let's try negative one times negative one.
- Well, one times one would be one.
- And we have a negative times a negative. They cancel out.
- Negative times a negative give me a positive.
- So this is going to be positive one.
- I can just write one,
- or I can literally write a plus sign there to emphasize.
- This is a positive one.
- What happened if I did negative one times zero?
- Now this might seem, this doesn't fit into any of these circumstances,
- zero is neither positive nor negative.
- And here you just have to remember anything times zero
- is going to be zero.
- So negative one times zero is going to be zero.
- Or I could've said zero times negative seven hundred and eighty-three,
- that is also going to be zero.
- Now what about two--let me do some interesting ones.
- What about--I'm looking a new color.
- Twelve times negative four.
- Well, once again, twelve times positive four would be fourty-eight.
- And we are in the circumstance where one of these two numbers,
- right over here, is negative. This one right here.
- If exactly one of the two numbers is negative,
- then the product is going to be negative.
- We are in this circumstance, right over here.
- We have one negative, so the product is negative.
- You could imagine this as repeatedly adding negative four twelve times
- And so you will get to negative fourty-eight.
- Let's do another one.
- What is seven times three?
- Well, this is a bit of a trick. There are no negative numbers here.
- This is just going to be seven times three.
- Positive seven times positive three. The first circumstance,
- which you already knew how to do before this video.
- This would just be equal to twenty-one.
- Let's do one more.
- So if I were to say negative five times negative ten--
- well, once again, negative times a negative.
- The negatives cancel out.
- You are just left with a positive product.
- So it's going to be five times ten. It's going to be fifty.
- The negative and the negative cancel out.
- Your product is going to be positive.
- That's this situation right over there.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
Have something that's not a question about this content?
This discussion area is not meant for answering homework questions.
A company wanted to encourage sales of its "fuzzy wigets" by offering a $3 rebate to the first 1000 customers to send in the rebate forms. The company's financial computer input was set up so that the total rebates were calculated as a negative number and money was allocated for the amount needed.
-3 x 1000 = -3000 $3000 allocated for rebate use only.
Unfortunately, the printer had problems and no rebate forms were printed, so no rebates were given. In order to use the program to re-allocate the money, the number of customers not using the rebate program had to be entered as a negative number.
-3 x -1000 = 3000
$3000 could now be reallocated for other uses.
We know 10*3 = 10+10+10 = 30 and -10*3 = -10+-10+-10 =-30
So why is -10*-3 = +30 ?
If I lack 10 apples (-10) three times (3) then -10*3 = -30, I lack a total of 30 apples.
But if I lack 10 apples (-10) negative three times (*-3) then it means for three times I DON'T lack 10 apples (-3-10 > (-)the negation of 3*-10, meaning not -30, but 30!) therefor I HAVE ten apples for three times = 30 apples.
Hope this helps you picture it! (I answered that before I think but it was moved, so here it is again)
What is funny is, this is not easy to answer. Here is a guy at North Carolina University doing his best, and failing:
He was asked about multiplication of a negative and a negative and most of his answers are about a negative and a positive or about addition, not multiplication.
This is something that came about in Ancient Mesopotamia, Babylon, or somewhere like that. It is around 3000 years old... and we still don't have a great explanation that everyone accepts and agrees upon right off.
That may be odd, but Zeno asked a mathematical question, one of the famous Zeno's paradoxes, that could not be answered properly for 2400 years until Newton and Leibnitz developed Calculus!
Sometimes math is just put together a certain way because it consistently works that way. This is a proof by induction and a very deeply argued point of philosophy. This, sadly, leaves a few things to just be taken for granted or forces a person to become a philosopher so they can more properly argue about them.
Such as if you have: -3 x -3
the negatives cancel out to be: 3 x 3
and both of those math problems equal 6
You can watch him write it out in this video between 3:10 and 3:20.
*Hope that helped!*
Above sea level
Sea level ------------------------------
below sea level
So when you just say sea level it's like zero . You wouldn't say negative sea level or positive sea level.
Anything times zero equals zero.
It's also a neutral number, like Danielle C said.
(2) The result of multiplying by 0 is 0 by definition. So according to that definition, it can't be positive or negative.
But even though it is neutral, it is powerful! ( 1000 without any 0's would be 1 )
posted 11:20 ET USA
1 * 1 = 1. OR positive 1 * positive 1 = positive 1.
-1 * 1 = -1, OR negative 1 * positive 1 = negative 1.
1 * -1 = -1, OR positive 1 * negative 1 = negative 1.
-1 * -1 = 1, or negative 1 * negative 1 = positive 1.
positive = love
negative = hate
If you love love or hate hate, then you are positive.
If you love hate or hate love, then you are negative.
This also works for addition and subtraction for getting the signs correct.
Example: 6 + -4 =
The "plus negative" can be looked at as loving hate ("negative" or simply changing the sign to minus), so you are actually subtracting the 4 from the 6. The answer, of course, being 2.
I hope this helps.
If I say "Eat!" I am encouraging you to eat (positive)
But if I say "Do not eat!" I am saying the opposite (negative).
Now if I say "Do NOT not eat!", I am saying I don't want you to starve, so I am back to saying "Eat!" (positive).
e.g. -1 x -1 x -1 x -1 x -1 (There are five -1 is this problem, five is odd) = -1
e.g. -1 x -1 x -1 x -1 (There are four -1 is this problem, four is even) = 1
for example -9 x 5 =-45 because u hate to love so it = hate
-9 x -9 = 81 because u hate to hate so it = love
hate = - and love= +
I don't really get it either, but that's how it works.
In the context of multiplication, we talk about how many total we have if we have 3 groups of 5 apples -- that is 3x5=15 apples. Well, that doesn't quite work for negatives, unless we start to talk about owing again: we owe 3 people $5 each, so 3x(-5)=-15 we owe 15 dollars. But now, what about two negatives? There are "-3" people to whom we owe 5 dollars... maybe that means three people owe us? So (-3)x(-5)=15 so we actually GET $15. Not terribly sensical, but maybe someone else can elaborate further?
+ x + = +
- x - = ?
I can never remembering what combinations equal what. Any story thingys that help you remember? I know the previous person said "Does my saying work; If a Bad thing happens to a bad person (-x - -x) its a good thing (answer would be a positive)?" But I need every other combination and the answers to it you know:
Negative times positive equals what?
Positive times negative equals what?
The order in which they are multiplied does not matter.
The expression was -2 * -3.
Let me first explain how you would represent -2 * 3 on the number line.
-2 means go left 2 postions. Start at zero and go left two postions and then go left two more postions and then go left two more positons. You are now at -6.
Ok, now lets do 2 * -3. Two is positive so it means go right two postions. The minus on the second number means opposite, So, start at 0 and go the opposite of right two postions and go opposite of right right more postions and go opposite of right two more positions. And again you are at -6.
Now lets do -2 * -3. The negative on the two means go right. The negative on the three means go opposite.
So start at zero. Go the opposite of right two positons and go the opposite or right two more postons and go the opposite of right two more postions. You are now at 6.
So -2*-3 = 6 and you have shown it on a number line.
I hope that helps.
how do you do 3 plus itself -2 times?
example: 8 x -4 just get the answer which is 32 and then you can remember that (pos x neg = neg), so you get -32
hope this helps
Here's an example: Multiply (-3)(-4)(-2)
First multiply -3 by -4 = 12
Then multiply the answer 12 by -2 = -24
So the answer is -24.
Hope this helps.
-10 x -4 would be +40. then you multiply +40 by -2 and you'll get -80
multiplying three (or any odd number of) negative numbers will always have a negative answer
Hope this helped
It's kind of saying like : I will not not do something...
Interesting Question. :)
while - X - = +.
I don't understand that.
Can you please explain it?
In general, we're just students here...
the people making the videos are employed to do so..
Negative x's positive = negative
Same for division
hav any tips?
(negative number)+(negative number)+(positive number) = ?
then the answer can be either positive or negative depending on the particular numbers. If you are multiplying:
(negative number) x (negative number) x (positive number) = (positive number)
then the answer will always be a positive number.
Would it equal +1/8 or -1/8?
If you have an even number of negative factors it's positive.
If you have an odd number of negative factors it's negative.
but I have a question when you multiply a negative and a positive do you get a negative
I think there should be a video bout it.
in simpilar words can somebody explain the formula?
(if that makes ANY SINCE)
Then -2(3-1) -2x3= -6 -2x1= -2 subtract the two results= -8 now I have 27 and -8 here is where I get stuck, is the answer 19? Need some help this question is part of a assignment which I need to hand in by 15/10/2012 after two days. If someone can point which videos or section I need to go to would be helpful. Thanks
Now let's consider multiplying (-3)x(2). Draw out a number line and put a dot at 0. (-3)x(2) tells you that you add -3 two times. So from zero you go to the left three places to get to -3 (you've added -3 once), then left another three places to get to -6 (you've added -3 twice) so the answer is (-3)x(2) = (-6).
What if that 2 was negative (i.e. (-3)x(-2) = ?)? Then you would add -3 to itself two times _in the opposite direction_. So on the number line we start at zero, and instead of moving left, we move right 3 places to get to +3 (-3 has been added -1 times), then we move another 3 places to the right (-3 has been added -2 times total) to get to +6. That is, (-3)x(-2) = 6, and a negative multiplied by a negative is a positive (the negatives "cancel each other out").
Does that make sense?
Share a tip
When naming a variable, it is okay to use most letters, but some are reserved, like 'e', which represents the value 2.7831...
Thank the author
This is great, I finally understand quadratic functions!
Have something that's not a tip or thanks about this content?
This discussion area is not meant for answering homework questions.
At 2:33, Sal said "single bonds" but meant "covalent bonds."
For general discussions about Khan Academy, visit our Reddit discussion page.
Here are posts to avoid making. If you do encounter them, flag them for attention from our Guardians.
- disrespectful or offensive
- an advertisement
- low quality
- not about the video topic
- soliciting votes or seeking badges
- a homework question
- a duplicate answer
- repeatedly making the same post
- a tip or thanks in Questions
- a question in Tips & Thanks
- an answer that should be its own question