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### Course: Arithmetic (all content) > Unit 4

Lesson 8: Multiplying & dividing negative numbers- Multiplying positive & negative numbers
- Multiplying numbers with different signs
- Why a negative times a negative makes sense
- Signs of expressions
- Dividing positive and negative numbers
- Multiplying and dividing negative numbers
- Multiplying negative numbers
- Why a negative times a negative is a positive
- Simplifying complex fractions
- Simplify complex fractions
- Multiplying negative numbers review
- Dividing negative numbers review

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# Multiplying positive & negative numbers

Learn some rules of thumb for multiplying positive and negative numbers. Created by Sal Khan.

## Want to join the conversation?

- what would -5 x 1 be? 1 x any other number is that number but i do not know if that works for negatives.(251 votes)
- Yes, it works for negatives, too. 1 is a pretty cool number.(211 votes)

- what happens if you have a negative times a positive times a positive or a negative times a negative times a positive?(21 votes)
- Good question. A negative times a negative times a positive will be a positive⏤the first two negatives cancel each other out to make a positive, so when you multiply them by a positive you will be multiplying a positive times a positive. Likewise a negative times a positive times a positive⏤the first two (neg. times pos.) will make a negative so when you multiply that by another positive you will end up with a negative. Does this make sense?(68 votes)

- Would two negatives being multiplied the same thing as thinking there both positive? Example: -a*-b = a*b true or false?(20 votes)
- stan is absolutely right, if you multiply or divide any two negative numbers, the answer will be positive. so yes your example is true. if you still cant figure it out, try to find a lesson about multiplying and dividing positive and negative integers(12 votes)

- Why is negative times a negative a positive while positive times a positive is not negative?(14 votes)
- A simpler algebraic proof

Using the fact multiplication is commutative, a negative times a positive is also negative. Similarly, we can prove that a negative times a negative is a positive. Since we know that −ab is negative, and the sum of these two terms is 0, therefore (−a) × (−b) is positive(6 votes)

`What is going on in the comments?`

(12 votes)- What’s going on is that people are just posting random things and being annoying(10 votes)

- If a=2

And the problem I'm solving for is -2^a

Then is this problem supposed to look like -(2 x 2) or (-2) (-2)?

I'm so confused.(12 votes)- If 'a' is a positive 2 and we have a negative 2 we are multiplying it as -2 x 2 which also looks like -(2 x 2). If you had two negative numbers your example would be (-2)(-2).(7 votes)

- If you multiply six positive numbers,the products sign will be ?(6 votes)
- If there are an even
**amount**of negatives, it's positive, if there are an odd**amount**of negatives, it's negative.(5 votes)

- how would we multiply fractions(7 votes)
- To multiply fractions, you need to multiply the numerators with the numerators and the denominators with the denominators, for example, 3/5 × -15/6 = (3 × -15)/(5 × 6) = -45/30, then you need to divide the numerator by the denominator using the short division method, therefore, -45/30 = -3/2 [since it is an improper fraction, you convert it into a mixed fraction].

Alternatively, you can directly divide the numerators with the denominators and then multiply the result, like, 3/5 × -15/6 = (3 × -15)/(5 × 6) = (1 × -3) × (1 × 2) = -3/2.(5 votes)

- how to solve binomeals(5 votes)
- Binomials, you ask?

Here's how to solve them:

For example, lets have the straight-up maths question (x + 7) (x + 12).

First, you multiply the two unknowns into x^2, then you multiply the first 'x' with the '+12', turning it into '12x'. After that, you multiply the '+7' with the unknown in the second set of brackets, making it '7x', and finally, you multiply the '+7' with the '+12', making 84.

But we're not done yet. You now need to chuck them all into an equation. So, the x^2' comes first, then the '+12x', then the '7x', then the '84', so the equation is as follows:

(x + 7) (x + 12) = x^2 + 12x + 7x + 84.

Of course, you can simplify it (slightly). Combine the only set of like terms, and voilà! Your equation turns into x^2 +19x +84. Pretty simple, innit?

Sadly, life has its ups and downs, or should I say positives and negatives? It's the same with maths, or binomials, in our case. Here's another question:

(x + 4) (2x - 6)

Oh! what's that? It's a wild minus sign! (I really need to get friends...). You may have noticed that in the previous question, I put the respective signs in front of their respective numbers, regardless of if they're positive or negative. That helps me (and possibly you) distinguish the sign of them and not get jumbled up between the mind-boggling positives and negatives. Now, onto the question. It's the same procedure as before, but now with an extra challenge - the negative sign, the step-by-step is as follows:

2x x x = +2x^2

-6 x x = -6x

4 x 2x = +8x

-6 x 4 = -24

Now to get them into the equation: (x + 4) (2x - 6) = 2x^2 - 6x + 8x -24, which simplifies to 2x^2 + 2x - 24.

You also might come across some specials ones like (x + 3)^2, which if they are all represented in unknows, equals to a^2 + ab (a x b) + ab + b^2, which simplifies to a^2 + 2ab + b^2. In the opposite case, however, like (x - 3)^2, it's literally the same thing but instead of the second addition symbol in the simplified version, it's a subtraction symbol instead. You might be wondering, 'why isn't the last addition symbol negative as well?' Well, allow me to explain. With binomials, at the end of the equation with the two numbers, you multiply them together. But if the binomial is a duplicate of each other, you just square the last number, and put it in the equation, and their sign will always be positive (unlike me) because two positives make a positive, while two negatives also make a positive. Keep in mind that this only works for 'squared' binomials.

Here's another special example of a binomial - (x + 6) (x - 6). This marvellous binomial is nicknamed 'the difference of two squares', and once you memorise the formula, it's really a cake walk to do these types of questions. (a + b) (a - b) = a^2 - ab + ab - b^2. The two 'ab's cancel each other out, so you're left with the first number's square minus the second number's square.

That's the end of my hopefully educational and fun lecture, and I hope anyone who reads this has a wonderful week. Please consider upvoting as this took me half an hour to make, and thank you for staying this long to read what I have to say about binomials! I live in Australia by the way, so some of my spellings may be different from the rest of you.(11 votes)

- Children

pay attention to the math and stop saying dumb things(9 votes)

## Video transcript

We know that if we were
to multiply 2 times 3, that would give us positive 6. And since we're going
to start thinking about negative numbers
in this video, one way to think about it I
had a positive number times another positive
number, and that gave me a positive number. So if I have a positive
times a positive, that will give me
a positive number. Now, let's mix it
up a little bit, introduce some negative numbers. So what happens if I
had negative 2 times 3? Well, one way to think
about, and we'll talk more about the intuition in this
video and in future videos, is, well, you could
view this as negative 2 repeatedly added three times. So this could be negative 2 plus
negative 2 plus negative 2-- not negative 6--
plus negative 2, which would be equal to-- well,
negative 2 plus negative 2 is negative 4 plus another
negative 2 is negative 6. So this would be
equal to negative 6. Or another way to think about
it is if I had 2 times 3, I would get 6. 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 in which we multiply? So if we were to multiply
3 times negative 2. Well, it shouldn't matter. The order in which
we multiply things shouldn't change the product. Whether we multiply 2
times 3, we'll get 6, or if we multiply 3
times 2, we'll get 6. And so we should have
the same property here. 3 times negative 2 should
give us the same result. It's going to be
equal to negative 6. And once again, we say
3 times 2 would be 6. One of these two
numbers is negative. And so our product is
going to be negative. So we could write
a positive times a negative is also
going to be a negative. And both of these are
just the same thing with the order in which we're
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 will get
a negative product. Now let's think about
the third circumstance when both of the
numbers are negative. I'll just switch
colors for fun here. If I were to multiply
negative 2 times negative 3-- and this might
be the least intuitive for you of all. And here I'm just going to
introduce you to the rule. And in future videos,
we'll explore why this is and why this makes mathematics
more all fit together. But this is going to be,
you say, well, 2 times 3 would be 6, and I have a
negative times a negative. And one way you
can think about it is that the
negatives cancel out. And so you will actually
end up with a positive 6. I actually don't have to
write a positive here, but I'll write it here
just to reemphasize. This right over here
is a positive 6. 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 encourage 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
1 times negative 1. Well, 1 times 1 would be 1,
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 1. I could just write 1,
or I could literally write a plus sign
there to emphasize that this is a positive 1. What happens if I did
negative 1 times 0? Now, this might say,
wait, this doesn't really fit into any of
these circumstances. 0 is neither positive
nor negative. And here you just
have to remember anything times 0
is going to be 0. So negative 1 times
0 is going to be 0. Or I could have said
0 times negative 783, that is also going to be 0. Let me do some interesting ones. What about-- I'll pick a new
color-- 12 times negative 4? Well, once again, 12 times
positive 4 would be 48. And we're in the circumstance
where one of these two numbers right over here is negative,
this one right over 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 4 twelve times. And so you would
get to negative 48. Let's do another one. What is 7 times 3? Well, this is a bit of a trick. There are no negative
numbers here. This is just going to be 7 times
3, positive 7 times positive 3, the first circumstance,
which you already knew how to do
before this video. This would just be equal to 21. Let's do one more. So if I were to say negative
5 times negative 10, well, once again,
negative times a negative, the negatives cancel out. Then you're just left
with a positive product. So it's going to be 5 times 10. It's going to be 50. The negative and the
negatives cancel out. Your product is
going to be positive. That's this situation
right over there.