Multiplication

Ah, multiplication. The quickest and most useful numerical operation we have.

Where does multiplication come from? Why do most operations in high level mathematics and physics use multiplication?

The famous equation, known as the “Mass-energy equivalence”:

E = mc^2.

That is: E = m * c * c        (with c = 299,792,458 meteres / second, or 3.0*10^8 as shorthand)

Multiplication! The whole thing, straight through, just multiply together and you’ve got Einstein’s answer to the big question, “Does the inertia of an object depend upon its energy-content?” That’s a pretty small answer to a giant question!

How? Why?

Let’s look at what multiplication is in smaller numbers.

At it’s very basic, multiplication is the shorthand of addition.

5 + 5 + 5 + 5 + 5 + 5

That’s six 5’s being added together. To make our lives faster, early mathematicians proved that the grouping of addition into the number of ‘alike’ components is the same.

In other words, 6 * 5 = 5 + 5 + 5 + 5 + 5 + 5

6 * 5 could also mean 6 + 6 + 6 + 6 + 6

So 6 * 5 = 5 + 5 + 5 + 5 + 5 + 5 = 6 + 6 + 6 + 6 + 6 = 6 * 5

It’s shorthand!

That’s why we memorize the multiplication table. Not because your math teacher is boring, but because knowing the table will make your life that much smoother.

Now, we recognize that through grouping, it’s not right to say

“6 * 5 = 5 + 5 + 5 + 5 + 5 + 4 ” right?

We no longer have six 5’s. We have six 5’s and one four.

This is the concept of adding and subtracting “like terms”.

x + x + x + y = 3*x + y

y^2  + y^2 + y?

(It won’t be ‘3 * y^2’, because there aren’t three ‘y^2’

It is equivalent to ‘2 * y^2 + y’. That addition can’t be changed, but it’s alright – addition is cool too.)

But if I want to know how far I can get in a certain amount time, I need to find a way to relate time (hours) to distance (miles). These are two different things! We know we can’t group them – I cannot say that if I go 2 miles in 2 hours, I have gone.. 4 miles? 4 hours? 0 miles? 0 hours? None are correct, right? We need to account for both the miles and the hours if we will speak about it – and this is where multiplication comes into play.

So I have to convert a time to a distance.

Suppose you want to know how “fast” you are going, and you have gone 60 miles in 2 hours.

I would be able to say, 60 miles in 2 hours. That would be fine but it’s not the simplest thing we can say! Smaller numbers are better!

Now, we have what is called a ‘conversion table’. Let’s see the ‘grouping’ mechanism at work.

Distance = rate * time <—- I'll discuss this function later, but it's multiplication!

60 miles = rate * 2 hours

30 * 2 miles = rate * 2 * 1 hours

or 30 * 2 miles / 2 * 1 hours = rate

30 miles / 1 hour = rate

We shall say now, our rate was '30 miles per 1 hour' or '30 miles per hour'.

'Fast' is measured by different people in different places. Some use metres, some use minutes… essentially you can mix whatever 'distance' unit and whatever 'time' units. For the US, we use "miles per hour".

Miles per hour – we couldn't get that with addition. What does it mean? # Miles in 1 hour. We had to find the link between their units and now we can *group*

E = mc^2 let's us relate mass and energy – something that could not be done without the ability to mix units.

Take Newton's second law, a = F/m. The acceleration (a) of an object is proportional to the net force (F) acted on it and its mass (m). Imagine we could break down the F into a 10,000 tiny, individual, but identical pieces. The law would still hold for each force – that is to say, each force would cause an acceleration that was 1/10,000th of the total acceleration. They all add up together. So again, multiplication is a compact way of adding as many "unit" forces as you need, whatever you deem to be an appropriate 'unit'.

Where addition is '1 dimensional', multiplication allows for multiple dimensions to be taken into account without losing truth!