Why does plus times plus equals plus?

Question

and minus times plus, equals minus; and minus times minus equals plus? where do these rules come from?

Answer 1

Assuming by "plus" and "minus", you are referring to "positive" and "negative"... they are fundamental rules for mathematics. To understand them, you need to understand the definition of a "negative number", where subtraction of a positive number is the equivalent of addition of the number's negative form:

5 - 3 = 5 + (-3) = 2

You can think of negative values as the "explicit absence of" a value or number of items. With this definition, it should be clear why the multiplicative rules are as you stated.

(positive) * (positive) = (positive) ...is just the obvious, since any positive number of items added multiple times will always result in a positive number of items. Add 3 apples to a basket 4 times, and you will have 3*4=12 apples. Mathematically:

3 + 3 + 3 + 3 = 12

(positive) * (negative) = (negative) ...you can use the "explicit absence of" idea I stated above to visualize this one. If you have 2 holes into which a ball would fit and consider that number to be -2 (negative 2), and you had three rows of them, you will have 3 * (-2) = -6 holes, into which 6 balls would fit. Mathematically:

(-2) + (-2) + (-2) = -6

(negative) * (negative) = (positive) ...is derived directly from the 2nd rule above. If a single negative number changes the sign of the product, then two of them should change the sign right back. You can also use the "explicit absence of" idea. If you were counting pairs of holes as negative values (-2), but you were to fill up 4 of these pairs with balls (the filling of which would be considered as "negating the hole", or creating a filled hole), then you would have (-2) * (-4) = 8 holes filled with balls. Mathematically:

-[ (-2) + (-2) + (-2) + (-2) ] = -(-8) = 8

I hope this makes some sense, and that I didn't just create even more confusion. ^.^

Answer 2

Multiplication is defined as a*b where we have a groups of b. If we have 2*3, then we have 2 groups of 3, or 3 + 3 = 6.

Once we get into negative numbers, it gets a little complicated. Let us first think of what a negative number is. It is the opposite of a positive number, like 4. An opposite is a number such that when it is added to the original number, we get 0. So the opposite of 4 is -4, because 4 + -4 = 0. So if we have a*b and a is a negative number, we can say that if c is the opposite of the negative number, thus the positive, then c + a = 0. We can add or subtract 0 to any number and still have the same number, so,

a*b = (a - 0)*b
a*b = (a - a - c)*b
a*b = -c*b

Thus a times b is the opposite of c (which is the opposite of a) times b. c is positive, and we have defined a to be negative, so we have a negative (a) times a positive (b) equals the opposite of a positive (because c and b are both positive), or a negative.

If we have 2 negative numbers a and b, the let us say that c and d are their positive opposites. Then, as above:

a*b = (a - a - c)*(b - b - d)
a*b = -c*-d

or a*b = -(-(c*d))

Thus we have a*b equals the opposite of the opposite of c times d. Because c times d is positive, the opposite of it is negative. Because we have the opposite of that, the expression must be positive.

Answer 3

Good morning! I shall attempt to shed light on your three questions by using a simple way of looking at multiplication. It may not match what you will find in textbooks, but I think my method will work better.
First, please recall or note that multiplication is in fact, addition in disguise. For example, 3 X 4 is actually 3 times of the numeral "4" (don't we all read the sentence in English as such too?). So, the answer is obtained by adding the 3 repeats of the numeral "4" together, ie. 4 + 4 + 4 = 12. Did you see "3 times (of) 4" in the abovementioned? Hehe.
Well, since multiplication is commutative, ie. the terms can be swapped, 3 X 4 can be thought of as 4 X 3 too. So, repeating the above method of adding the numerals "3" up that appear "4 times", we have 3 + 3 + 3 + 3 = 12. That's for your first question.

However, when you have a negative in either one of the terms, say "-3" or "-4", as in your second question, you are then in fact just adding (-4) + (-4) + (-4) up, or, in the second way of seeing things, just adding (-3) + (-3) + (-3) + (-3) up, so the answer now becomes -4 -4 -4 = -12, or -3 -3 -3 -3 = -12.

For your third question, having two negative terms, -3 and -4 multiplying together, is just one step more involved than the second question case, because the additional negative sign now reverses the action caused by the first negative sign. Recall that the difference between your question one and question two is due to the single negative sign, which caused us to change/reverse the original "neutral", or "positive" sign in front of the +4 or +3 to be added up together, that's why we ended up adding -4 or -3 together instead. Please note that in maths, when we write "7", it actually means "+7", not "-7".

So, here, as mentioned, the second negative sign reverses the action of the first negative sign, so we end up removing what we did for question 2, ie. the negative signs in front of -4 or -3, and end up with +4 or +3, just like in question one, which makes the answer 12 again, just like in your question one.

Hehe, was that longwinded? Well, I also teach it alternatively using number lines to students. But, since number lines are hard to draw here, I used this method. Hope you feel better now. Do feel free to ask again/more if you need to. Cheerio!

Answer 4

Its easier to understand these idea if you think of them in terms of rates of change.

Lets say you run an orange grove, and during the harvest, your workers are bringing in 100 crates of oragnes a day. Thats PLUS 100 crates per day. Over ten days, you're going to bring in 10 * 100 = 1,000 crates, a positive influx of crates.

Now lets say you start losing trees to orange canker at a rate of two trees a day. Thats MINUS two trees a day. After ten days, you've lost 10 * -2 = -20 trees, an overall loss.

Now suppose that during this bout of disease, you wanted to estimate how many trees you had last week. Thats 7 days ago, so it's MINUS 7 days. You had -7 * -2 = 14 trees more than you do now. That's a positive ammount of trees; you ahd more then than you do now.

Hope that helps!

Answer 5

Break the problem down, and this should be clear. Multiplication is just repeated addition.

Examples:
4*3 = 4 + 4 + 4 = 12
-4*3 = -4 + (-4) + (-4) = -12
-4*-3 = -(-4) - (-4) - (-4) = 12

All of the basic arithmetic operations can be broken down into addition, if you want. Subtraction is just addition with a negative number, multiplication is just repeated addition, and division, is just multiplication by a reciprocal. Even powers and logarithms are just repeated mulitplications/division, so they too can be reduced to addition problems.

Once you break it down, you see that the rules you're asking about arise naturally.

Answer 6

Simple alegebra my good friend. and it could simply be represented as: 3x +1 = 5x +9 (where x being the height) then lets do wat we do by isolating x so lets subtract 1 from both sides then we get 3x = 5x +8 then we subtract 5x from both sides and then we get -2x = 8 so then we divide by negative 2 inorder to get x which give us x = -4 and then thats the answer which means that the hieght is -4 You can check this by subing in -4 where height is mentioned so then it will look like this for the first equation: 3 * -4 +1= 11 and the second 5 * -4 +9 = 11 and since 11 is = to 11 then we have the rite answer....XD!!!

Answer 7

That's an awesome question, to be honest! I will attempt to answer it as best as possible. To begin with, I believe an easy way to conceptualize this is to think of money, since the idea of having "negative money" isn't as foreign as having "negative apples". In other words, when thinking about this use money as the subject as opposed to the usual "apples and oranges" that are used in arithmetic examples.

So, first let's look at multiplying positive numbers. If you have 3 dollars, and someone says they'll give you twice your money, you will have 6 dollars. +3*+2=6. That's easy.

Now for multiplying postive and negative numbers by each other. Say that you have your three dollars, and someone charges you twice that amount to perform a service. Again, double your money is two, but relative to you it's a negative two because your being charged money and relative to you you're losing 6 dollars, thus -6 from your total money.

Finally, multiplying a negative by a negative is probably the most difficult to understand intuitively. Let's say that you've paid your six dollars and now have a total of -3 dollars. So now someone again offers you twice you money to do something. At first you'd think: easy, multiply by +2, but we know that that isn't right since a positive by a negative is a negative. It also makes sense because to "double" and to "increase" don't necessarily denote moving in the same direction on the number line. If someone says they're going to "double your money", and you've got negative money, you're effectively losing money. Think about it for a second remembering that you've got to think about it as relative to you! So, in order to "increase" your money you don't "double" your money, you've got to "NOT" double your money (IE -2), thus -3*-2=6. Now you have +3 dollars.

I know that this explanation might be hard to follow, but I hope it helps!

Answer 8

I love math and am very good at it. I'm not quite sure but i Iook at it this way, - times - equals + because the two- combine to make a +. good luck finding the real answer, that's just my idea.

Answer 9

They are general mathematical rules ... best bet is to contact a math professor in college. He or she should be able to explain the nuances concerning your multiplicative conundrum

Answer 10

they are your basic multiplication rules.. I would just take them as they are given.. there is really no point in questioning them

a good way to remember is same is positive different is negative..