A Table for Two

First things first: this isn't a story about a date. I want to share a story about how language is used to explain concepts; and how we tend to miss some subtle details about it.

For example, take mathematical tables. A table of two, if you will. Almost everyone I talked to (I and talk a lot!) agrees with the fact that multiplication is repeated addition. Starting from zero, if you add two and repeat it three times, the result is 6. You get the same result when you multiply 2 and 3. Note that the value of answer is the same whether you multiple 2 times 3 or 3 times 2. This simple rule is given a name 'commutative property'. It is as if the numbers can commute and move around, without changing the product.

So, it makes sense to derive the table of two as:

2 x 1 = 0 + 2?

2 x 2 = 2 + 2

2 x 3 = 2 + 2 + 2 ...?

But if you look at the way we communicate, or say the things out loud, it goes something like:

Two ones are two, two twos are four, two threes are six ...

If you take one word at a time, the table of two talks about taking two things at a time. Two ones are two. This mean, that if you add two ones, the result is two (1 +1 = 2). If you add two twos, the result is four (2 + 2 = 4). Similarly for two threes, two fours ...

2 x 1 = 1 + 1?

2 x 2 = 2 + 2

2 x 3 = 3 + 3 ...?

When you look at it this way, the table is a collection of two same numbers. Similarly, the table of 3 is a collection of 3 same numbers added together.?

Now, why do I want to share this??

Because we take for granted that (m x n) and (n x m) are the same. They are not. It is their sum or product that is mathematically same. Now the commutative property is the direct outcome because of the mathematical equivalence. I wonder how much more things I have taken for granted and never questioned.

#mathematics #tables #multiplication #DeepThought