Okay, welcome back everyone,

the point of today's video is to understand what we mean by sigma notation.

And we often use for

sigma this big Greek letter, sigma, which kind of looks like a big pointy S.

And as always in these lectures, the point is not necessarily to bombard you with

computations or to judge you on right or wrong answers.

But simply to demystify and explain notation,

which would otherwise be mystifying.

So what we're going to do in this lecture is work through three examples.

Each one of them an example of sigma notation, each one of these things here.

The sum from i = 1 to 4 of i squared,

the sum from i = 1 to 5 of 2i + 3 and the sum from j = 3 to 7 of j over 2.

These are all just fancy ways of writing numbers.

In fact, tell you the answers in advance, this first number equals 30,

the second number equals 45 and this last number equals 25 halves.

How on Earth did I know that?

Point of this lecture is to learn why that's true.

Let's dive in to do the first one.

We're going to compute the sum from i = 1 to 4 of i squared.

What I'm going to do is just do this myself first, and

then walk you through and unpack, how I did that.

Okay, so the sum from i = 1 to 4 of i squared is equal to

1 squared + 2 squared + 3 squared + 4 squared.

Okay, so suppose someone paid me to do this problem, I'd consider myself done.

I've worked it out, I've translated it from sigma notation to something someone

who knows arithmetic could do.

A person that is really a stickler for details, they're going to say well keep

going, but I'm going to say, maybe you need to pay me more money.

After we're done with that negotiation, we'll say, okay, from here,

it's arithmetic, that's just equal to dot dot dot.

The dots covering up the fact that I've done this in advance.

This is equal to 30, the answer we saw before.

But the real point of today's lecture is to understand this first equals sign.

How on Earth did I know that this funny symbol, the sum from i = 1 to 4 of i

squared is equal to 1 squared + 2 squared + 3 squared + 4 squared?

Okay, so let's walk through that.

The first thing to realize looking at this symbol is there's a bunch of things.

First, there's a counter in here, there's a symbol in here i squared.

That same person walking out to me on the street is pretty annoying by now says,

I'll give you ten bucks if you tell me what i squared is.

No deal, it's not really a fair question.

I know what i is, but actually I had some hints.

Down here at the bottom, I know how to start my range of i.

I know that I should start from i = 1, and at the top,

I know that I should finish when i is 4.

Okay, so let's do some scratch work and work that out on the side.

Here I have i = 1, here I have i = 2, here I have i = 3 and here I have i = 4.

So you'll notice that starting range, starting from 1,

that finishing range ending at 4.

And actually there's something here that's a little unfair,

nothing in the symbol tells me that I count by one,

going from the bottom of my range to the top of my range.

That's okay, that's sort of a cultural agreement.

You start from the low i, you end at the top i, and you count by one.

Okay, fine, so what do we do?

To each one of these i's we do what this thing in the middle tells us to do.

In this case, this thing in the middle, very bossy, tells us to square i.

So if i = 1 then i squared = 1 squared.

If i = 2, then i squared = 2 squared.

If i = 3, then i squared = 3 squared.

I think you get the hang of it.

i = 4, and i squared = 4 squared.

Okay, we've done the scratch work on the side.

What do we do next?

Well sigma, What you ought to think about this is equal to sum.

Meaning, we take all of these answers here and we add them up.

And that's how we get what we get up here.

We've computed on the side that 1i = 1i squared = 1 squared and so on, and

then we add them all up and get our answer.

For those of you who are sort of business minded you can think of this as a process.

Which can be broken up into parallel processes undertaken by different workers.

So one worker can compute what 1i = 1.

One worker can do 1i = 2.

One might i = 3.

One might i = 4, doesn't really matter when they do them.

They do them at the same time in parallel.

The end of the day, they compare their answers.

The last worker adds them all up, and we get the answer we get over here, or

over here.