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(TL;DR: Scalars are single numbers.)

I have so many scalars!

Opie, the open source snake

I’m sorry…that was lame…

Me

Last

time,

we covered some basic concepts regarding ** sets** on our journey to

understanding

**and**

`vectors`

**.**

`matrices`

Let’s do this!

# Today’s topic: Scalars

## What’s a scalar?

A scalar is a single number! This seems very simple (and it is). But we

need to know this to understand operations like *scalar multiplication*

or statements like *“The result of multiplying this by that is a*.

scalar”

We will use the notation from Goodfellow, Ian, et

al. and depict them in lower case,

italicised letters like this:

## How can we define the types of numbers our scalars should represent?

Say, we define our **arbitrary scalar**, , as a number from the set of

natural

numbers.

We would show this **set membership** like this:

The ‘’ symbol means

**‘is a member/is an element of/belongs to (some set)’** Pick the one you like most!

However, the whole statement is often read as ’ is a natural

number’.

The symbol ‘’ means

**‘is not a member/is not an element of/does not belongs to (some set)’**. Easy!

## Let’s bring this back to machine learning

What are the implications of defining our scalars as natural numbers?

Let’s start with an abstract example!

- Let’s say we start with the number , and we want to
`add`

some arbitrary number, , to it. - Let’s define as a natural number. That is,

*belongs to the set of ‘whole’, positive numbers starting with 1 and increasing with no upper bound*.

Here are some of the implications of our definition of

:

- cannot equal because , and

therefore, cannot take on the value of . - We can never get an answer where the first decimal place is

something other than**zero**. For example, there is no natural

number, , where .

Now here is my (crappy) attempt at intuitively bringing this back to machine

learning!

- Let’s say that our scalar, , is the value used to
**update the**after some iteration of training.

parameters in our model - Then we are restricted to making crude updates of
**at least one**

only! - Our algorithm may never converge and we might see the values of our

evaluation metric jumping about erratically as training progresses.

This might not be an academically rigorous explanation, but it’s

hopefully good enough to build some intuition.

## We’ll define our scalars as real numbers

We’ll make our universe of numbers into something larger where our scalars can take on more than just whole, positive values. We will

define our arbitrary scalars, , as coming from the set of real

numbers. That is:

## How can we represent scalars in R?

R technically has no scalar data type! From Hadley Wickham’s ‘Advanced

R’, 1st edition, we can find this

in the ‘Data Structures’

chapter:

Note that R has no 0-dimensional, or scalar types. Individual numbers

or strings, which you might think would be scalars, are actually

vectors of length one.

But in practice, we can emulate our real number scalar by doing something like this:

```
x <- 123.532
print(x)
```

```
## [1] 123.532
```

In the same section, we also find out that to test whether something is

a vector in R, one must use `is.atomic()`

:

```
print(is.atomic(x))
```

```
## [1] TRUE
```

Yes, we have ourselves a vector! How many elements do we have?

```
print(length(x))
```

```
## [1] 1
```

Hooray! We have ourselves a proxy for our scalar. Now what is the data

type of our scalar?

From the ** numeric** help page in the R documentation, we find this:

numeric is identical to double (and real). It creates a

double-precision vector of the specified length with each element

equal to 0.

Then from the ** double** help page, we find this:

All real numbers are stored in double precision format.

Let’s test it out!

```
class(x)
```

```
## [1] "numeric"
```

```
typeof(x)
```

```
## [1] "double"
```

Hooray!

# Conclusion

We now know that scalars are members of sets. We have defined our scalars

as coming from the set of real numbers.

On to vectors!

Justin

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