Why negatives multiplied equal positives

Your support is so much appreciated and enables the continued creation of great course content. Donations can be made via PayPal and major credit cards. A PayPal account is not required. Many thanks! The Astounding Power of Area 1. Course Home. Using piles and holes this looks like: Interpreting negative times a positive and negative times negative through repeated addition, however, is problematic. Most people agree we should stay with this idea. In the context of positive whole numbers it is repeated addition.

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Why a negative times a negative is a positive. Why a negative times a negative makes sense. Well said! Many people think they know what is a negative number,which actually is incorrect. Before everyone had digital cameras, people used film, and there was a negative of the film. If one took a negative of the film negative, one would obtain something that looked like the original photo.

I say we introduce negative numbers in the second grade. Then we can get rid of this subtraction nonsense, and reduce the number of basic operations that need to be taught. During a lecture the Oxford linguistic philosopher J. Austin made the claim that although a double negative in English implies a positive meaning and in French a negative one, there is no language in which a double positive implies a negative. Actually,Humans have adopted the convention,that whenever they encounter with a pair of elements,having opposite nature behaviour , it would be nice to name one of them as positive and the other as negative.

Consider in Physics. Why have we named electronic charge as negative and protonic charge as positive? We could have worked with Electricity by naming the electron charge as positive and the proton charge as negative! Nothing will change!

Except the terminology. So,in Mathematics too, we find a natural number quantity and a negative integer quantity as opposite.

Think that you just now have pennies. So you have ,with no complex confusion. You can have whatever you can by your owned pennies. Again,consider, a different situation. In this situation,it is far away of thinking what to buy,instead you are to think how to pay him back pennies.

So,here, we have a pair of opposites,both regarding pennies. So,if in the first case,you have pennies,then how much do you have in the second case? Yes,the opposite of what you had in the first situation.

So what is the opposite of the number ? Also one more thing I would like to include here,that the sign minus — , is a symbol, that we PUT between 2 numbers.

Then what does it mean by, for instance, -5? From what is 5 subtracted from? Does that literally make sense? Obviously, if you have something nothing that is 0 , and still need to give someone 5 things of what you have,then obviously you are forced to do the operation 0 — 5!

In short, without loss of generality, -5 is just a compact form for writing 0 — 5. John Allen Paulos makes the case in his classic book Innumeracy for using a debt model to understand not only addition with negatives but also multiplication with negatives, too. Gauss says otherwise. Such heartwarming nostalgia. I actually want to go back to school now. Thank you for this post. We start out learning about numbers, whole numbers, by adding and subtracting them.

We can picture and hold representations of them. Then we learn to multiply these positive numbers. Multiplication is a short-hand way of adding numbers. So far so good. You could also picture this, as recommended above, to think of this as a matter of directions. I cannot find a way to express this as an addition question.

Multiplication IS addition. Just as division IS subtraction. One teacher tried to explain it thus: two bad people leave town three times.

Lots of mathematicians throughout history quibbled with it or resisted it. A little late, but not fifty years late…thanks, Ben. My mind is slightly less boggled. That makes some sense, if we accept those rules. My inner 15 year-old is still balking somewhat. That seems like magic. Positive three, I can hold that in my hand. Is there any other world, other than directions vectors? Maybe that would help. It definitely resonates for some people though and is good to have in the repertoire.

Thanks Ben, that is an interesting quote! Great examples of real world negative values. I know you mathematicians must be shaking your heads slowly, sadly. In your example, your debt of 25 units is