Why do we even multiply

What kind of absurd question is this? May be you are a maths genius but there are many people who do struggle with this question. I have come across this question many time in my life.

Let me put it this way… what is the result of 2 apples multiplied by 5 oranges? Now this is definitely a strange question. rt? If you think pure maths and numbers, the answer is 10, but somehow this question itself is not making sense. But then what is 2 or 5 if not attached to some object or phenomenon? May be this absurd question is making some sense now.

This time I came across the puzzle once again because I was reading Cosine Similarity. Yes the famous AI phenomenon. It is based on the vector dot product. A and B below are vectors.

The similarity here is the value of thita… and the numerator is base and denominator is hypotenuse.

In numerator we are multiplying the vectors. It is called dot product. But why multiplication, why not addition? I can also add the vector componets (i-hat and j-hat). Sorry you will have to go to the books if you are now aware of i-hat and j-hat.

That’s when the question came if multiplication is just repeated addition then why use multiplication here, followed by when multiplication makes sense and when not, followed by the thought 2 apples x 5 oranges. :)

While the above question does not make sense but give 2 apples each to 5 people does make sense. Now the 10 is valid. What happens if I said distribute 2 apples and 2 oranges each to 5 people. Something very interesting happens here. Its not simple 4 x 5 = 20. It works like 2 apples x 5 = 10 PLUS(+) 2 oranges x 5 = 10. The answer is still 20 but the internal mechanics needs to be appreciated. That’s the fun of maths :) .

Now look at vector dot product example below and things will start making sense. Each dimension (i-hat and j-hat) is moving independently and getting added finally to give you the result. It is a product but with twist of addition.. ha ha

This picture is from The Organic Chemistry Tutor YouTuber. What a fantastic guy.

But the question remains why do we even multiply?

Look at this equation

Work = Force x Displacement.

We are multiplying two forces to unlock third force. Or say we are multiplying two dimensions to build third dimension. Or say two dimensions multiply to produce a new dimension.

Hmmm…… simple but not simple :)