Why do two minus make a plus

I add two sandbags down two , subtract five puffs of hot air down five , subtract one sandbag up one , then add seven puffs of hot air up seven. You can read the calculation as "Four add negative two, subtract positive five, subtract negative one, add positive seven", and think to yourself "Four, down two, down five, up one, up seven" or equivalent.

Happiness Model Here is a teacher's description of how she explains positive and negative numbers to her classes: "I believe that adding and subtracting with negative numbers makes sense.

We talk about how you feel if someone gives you a positive thing, or if someone takes one away. We talk about how you feel if someone gives you a negative thing, or if someone takes one away. Now how would it be if someone gave me a detention negative? How would I feel? Adding something positive or taking away something negative improves the situation up the number line we go. Adding something negative or taking away something positive makes the situation worse down the number line we go.

Again, before too long, the consensus is that subtracting a negative is the same as adding a positive. This is mind-blowing for some students and I tell them that I can understand why they might find this counter-intuitive but this is because they have always believed that subtraction makes smaller. However, the activities have shown that subtracting a negative will make a number larger: like it or not, we cannot argue with this beautiful mathematical logic!

Notice that we cannot 'take away' negative 3 because we do not have three red tiles in our model. We can overcome this though, by adding enough zero pairs:. Now we can take away the negative Some students like this approach, others less so.

If you subtract a negative number, the two negatives combine to make a positive. Instead, you can think of it as turning one of the negative signs upright, to cross over the other, and make a plus. For clarity, you would never write two negative signs side by side without brackets.

So if you are asked to subtract a negative number, it will always have brackets around it so that you can see that the use of two negative signs was intentional. Once you have the numerical answer, then you can apply a very simple rule to determine the sign of the answer:. As a side issue, this goes some way to explaining why you cannot have the square root of a negative number there is more about this in our page on Special Numbers and Concepts.

The square root is the number that is multiplied by itself to get the number. You cannot multiply a number by itself to get a negative number. To get a negative number, you need one negative and one positive number. The rule works the same way when you have more than two numbers to multiply or divide. An even number of negative numbers will give a positive answer. An odd number of negative numbers will give a negative answer. 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.