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ZL THE LAWS OF COUNTING
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x#hhX"`DIf you had a set of marbles and you wanted to know exactly how much you had, you could lay them out and count them one by one. But you could also count the marbles by putting them into groups. If you had 15 marbles, you could count them as 3 groups of 5 or 5 groups of 3. In fact, when you say that 3 x 5 is 15, or 5 x 3 is 15, this is what we mean. That is, 3 x 5 means 3 groups of 5 things.
In this same way, by taking a large number of things and putting them into different groups, the Multiplication Tables were discovered. It must have taken a very long time. However, if you have a box of marbles and you divide them into 7 groups of 5, then from the Multiplication Tables you know that 7 x 5 is 35, so you know the number of marbles you have without having to count them one by one.
This is an example of how useful Mathematics is, it helps you to save time if you know the rules of Mathematics.
When it comes to counting, which we call Arithmetic, we notice three important things that we call laws of Arithmetic.
Earlier on it was stated that a group of 15 marbles can be arranged as 3 groups of 5 marbles, or 5 groups of 3 marbles. That is, 3 x 5 gives the same answer as 5 x 3. This is called the Commutative Law .
An easy way to remember the Commutative Law is that when it comes to multiplication, it does not matter which number comes first, second, third and so on because the answer will be the same.
As you may have suspected, a similar law exists for the addition of numbers. It is called the Associative Law and it means, for example, that 5+3+2 gives the same answer as 3+2+5 or any other combination of these three numbers.
Let us go back to our previous example with the box of marbles to discuss the last law of Arithmetic that we will be looking at. If we arranged our 15 marbles as 3 groups of 5 marbles each or 3 x 5, we can take each of the 3 groups of 5 marbles and arrange it as a little group of 3 marbles and a little group of 2 marbles. Click on the icon at the bottom of the screen to see this. We write this situation mathematically as 3(3+2). That is, that we have 3 groups of marbles where each of the 3 groups is arranged as 2 smaller groups of 3 and 2 marbles.
Therefore, when we check the total number of marbles by counting these samller groups we get 3 small groups of 3 marbles and 3 small groups of 2 marbles. Click on the icon again to check this. We write this mathematically as, 3x3 + 3x2. Looking at the mathematical term 3(3+2), we see therefore that the multiplication by 3 was distributed to each of the numbers inside the ( ). That is, 3(3+2)=3x3 + 3x2. This law is therefore called the Distributive Law.
The explanation of the laws of counting given above was done using only addition in the examples. What about subtraction ? In that case the laws are the same and this is easy to see using examples with marbles again.
For instance, what does 3(7-2) mean ? It means that for each group of the 3 groups you first put 7 marbles in the group, then take away or subtract 2 marbles. Therefore 3(7-2) is the same as having 3 groups of 7 and taking away 3 groups of 2. That is, 3x7 - 3x2 which gives 15 marbles. But if I have a box of 15 marbles it would be easier to group them as we first did, 3x3 + 3x2. We would get the same 15 marbles. In fact, to count the 15 marbles using subtraction you would first have to borrow some marbles so that you could get the 3 groups of 7.
You are right. It does not make sense to count the marbles this way because there is an easier, more direct way to do it using addition instead of subtraction and get the same answer. But this is so only because the direct way is easier. In Mathematics you will find that very often the direct way is not easier or even possible and you have to use an indirect way. This is very important in solving problems and in our problem solving later on, I will prompt you to find indirect ways to solve problems. In fact, in the problems solved by professional mathematicians the indirect way is almost always the only way.
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