Addition and subtraction of natural and integer numbers

Addition and subtraction in natural numbers is easy to understand. If you have a set of things and you receive more, then you will have more than before when you count again. In the opposite, if you give out from what you have, you will have less than before when you count again.

The point of learning how to add and subtract is to avoid the need of having to count from scratch every time we receive or give things to others.

Adding natural numbers

The first step to learn to add is to start with small figures. If you have one thing and you add two, then you get a total of three. If you have seven things and you receive one more, then you have a total of eight.

Try with these. Even if you are old, it can be a good mental exercise.

1 + 3 = 4
7 + 2 = 9
3 + 5 = 8
7 + 9 =
2 + 5 =
6 + 6 =
3 + 9 =
1 + 5 =
2 + 2 =
6 + 5 =
2 + 6 =
7 + 8 =

Once we have some practice with that, the next step is to engage in more complex additions, like the ones below:

12 + 6 =
30 + 99 =
24 + 8 =
16 + 8 =
33 + 41 =
7 + 10 =
9 + 93 =
25 + 18 =
22 + 50 =
25 + 31 =
23 + 15 =
185 + 16 =

Which gives me the perfect chance to talk about adding: Traditionally, we were taught that we should first add the units, then the tens, then the hundreds, etc., until we finish. When trying to add 193 + 45, for example, most people first add the units and continue to the left: five plus three makes eight, nine plus four makes thirteen, so we need to add a hundred to the next column. Lastly, one plus one makes two, adding up to two hundred thirty-eight at the end.

Although this is OK and perfectly valid, the bad news is this kind of reasoning is opposite to what is natural for our brains. Our minds are used to thinking from the larger number to the smaller one. That said, my advice for teachers and students is to also develop the ability of adding from left to right, like our brains work, particularly in mental exercises. In our example, this would be: A hundred ninety plus forty makes two hundred thirty, plus three, plus five, it adds up to two hundred thirty-eight.

That's a lot simpler for all of us, and it really helps when trying to add mentally when we need it. Practice it. Every adult should be able to add numbers of three figures mentally without problems.

Subtracting natural numbers

To subtract natural numbers, I would say there is not much difference. If you have a number of things and you give some away, you will have less at the end. We learn to subtract to avoid the need of having to count from scratch once we have given our things to others.

How to subtract? We start with small figures again, removing the real object from a set in case we are teaching this to somebody for the first time.

4 - 3 = 1
7 - 2 = 9
5 - 3 = 8
9 - 7 =
5 - 3 =
6 - 6 =
9 - 6 =
11 - 5 =
12 - 2 =
6 - 5 =
12 - 6 =
11 - 8 =

Next, we do the same with larger numbers, like these:

12 - 6 =
99 - 30 =
24 - 8 =
16 - 8 =
41 - 33 =
17 - 10 =
92 - 9 =
25 - 18 =
62 - 50 =
45 - 31 =
63 - 15 =
185 - 16 =

In these examples, we can mention that we were traditionally taught to subtract starting from the units and going on to the left, just like we did with the addition examples. On occasions in which there are not enough units in a said position to do a full subtraction, and then we borrow a unit from the left to raise ten to the one we were trying to subtract, going on with these operations until we finish the subtraction.

For example, if we try to subtract 16 from 185, then we would start by trying to subtract six from five. As it is not possible, then we borrow a unit from the left and the five becomes fifteen. Fifteen minus six makes nine. Then we subtract the next column, one from seven. It is seven and not eight now because the eight lent one to the five in the previous step. As a result, we get a six. The final subtraction is one hundred sixty-nine.

Yet, again, carrying out a subtraction this way goes against our natural way to see numbers. Our minds are used to subtracting large figures first, so it would be good to also practice subtracting this way: One hundred eighty-five minus ten makes one hundred seventy-six; this, in turn, minus six, makes one hundred sixty-nine. Practise it, as every adult should be able to subtract numbers of three figures mentally without issues.

Adding and subtracting integers

What happens when we try to add or subtract positive numbers and negative numbers? Here the concept of debt I mentioned in the previous page helps a lot, as there are four ways to see operations with integers:

+ You may have a positive balance and you add more to it, which would be an addition
+ You may have a debt (a negative number) and you ask for more, increasing your debt: another addition
+ You may have a positive balance but you give away a few items, which would be a subtraction; or,
+ You may have a debt (a negative number) and you pay some of the debt, which would be a reduction of the debt: a subtraction

In short, you always add numbers of the same sign, and subtract numbers of opposite sign. Try with these examples:

12 - 15 =
1 + 4 - 3 =
- 12 + 7 =
7 + 19 + 34 =
-4 - 7 =
21 - 53 =
31 + 36 =
49 + 28 - 21 =
21 + 10 - 37 =
22 - 35 + 3 =
22 + 48 - 15 =


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