What do three negatives equal

Why a negative times a negative makes sense. Multiplying negative numbers review. Dividing negative numbers review. Next lesson. Current timeTotal duration Google Classroom Facebook Twitter. 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 It seems to me that the entire debt is a negative, so why would I suddenly classify any part of it as a positive? Make a large circle on a piece of paper. But here you have one negative and one positive number, so the sign of the answer will be negative.

Again, you have one positive and one negative number, so the sign of the answer will be negative. This time, you have two negative numbers, so the sign of the answer will be positive.

The answer is Again, you have two negative numbers, so the answer is positive. It is To begin with, consider the first part of the calculation. The fact that a negative number multiplied by another negative number produces a positive result can often confuse and seem counterintuitive. To explain why this is the case, think back to the number lines used earlier in this article since these help to explain this visually. In both of these examples, you have moved forwards i.

Negative signs can look a bit daunting, but the rules that govern their use are simple and straightforward. Keep these in mind, and you will have no problems. How much more money did they have 5 days ago? Here, the loss per day is one negative and going backwards in time is another. This aims not at the algebraic or arithmetic properties of numbers but more at the oppositeness of negative numbers.

Prerequisite knowledge: All contexts that build new understanding require students to understand the pieces of the context fairly well, so it is especially important to probe how students understand an idea when it is presented contextually. From Dr. Alex Eustis , we have this algebraic proof that a negative times a negative is a positive. First, he states a set of axioms that apply to any ring with unity. A ring is basically a number system with two operations. Each operation is closed, which means that using these operations such as addition and multiplication on the real numbers leads to another number within the number system.

Each operation also has an identity element or an element that does not change another element in the system when applied to it. For example, under addition, 0 is the additive identity. Under multiplication, 1 is the multiplicative identity. The full set of axioms required is below. From these axioms, we can prove that a negative times a negative is a positive. Prerequisite knowledge : While I went through and added the justification for each step of the proof that was missing, I needed a fair bit of fluency with the original set of axioms.

I also needed to not lose sight of the overall goal and to be able to recognize the structure of each part of the argument and match that structure to the axioms. This algebraic proof from Benjamin Dickman is much simpler than going back to a proof based on the axioms of arithmetic. From this, we can show that ab and — ab have opposite signs and therefore that a positive times a negative is a negative. Using the fact multiplication is commutative, a negative times a positive is also negative.

Prerequisite knowledge : The prerequisite knowledge for this proof is much less than the other one, but it does assume a fair bit of fluency with manipulation of algebraic structures. Using the number line again, and considering just -1 as a multiplier and p as some positive number:.