# Real numbers

## Prerequisites

All four operations for the integers (+,-,*,/)

Pencil and paper algorithms for all four operations

## What is a real number?

Real numbers can be derived by infinite series of fractions such that the difference between fractions arbitrary distances apart tends to zero. This is hard to put into words, so let's illustrate a few examples:

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
nth term 1/1 1/2 2/3 3/5 5/8 8/13 13/21 21/34 34/55 55/89 89/144 144/233 233/377 610/377 987/610 1597/987 2584/1597 4181/2584

No matter what subsequence you choose, the differences between the terms tend to zero:

n 2 3 5 7 11 13 17 19
nth term 1/2 2/3 5/8 13/21 81/144 233/377 2587/1597
Difference 1/6 -1/24 -1/168 -19/336 335/6032

As long as the numbers in the row labeled n (in the second table) always increase, the differences will always get closer and closer to zero.

Here's an example of something that isn't a real number:

n 1 2 3 4 5 6 7 8 ...
nth term 1/1 1/1+1/2 1/1+1/2+1/3 1/1+1/2+1/3+1/4 1/1+1/2+1/3+1/4+1/5 1/1+1/2+1/3+1/4+1/5+1/6 1/1+1/2+1/3+1/4+1/5+1/6+1/7 1/1+1/2+1/3+1/4+1/5+1/6+1/7+1/8 ...

This is because you can do this:

n 1 2 4 8 16 ...
nth term 1/1 1/1+1/2 1/1+1/2+1/3+1/4 1/1+1/2+1/3+1/4+1/5+1/6+1/7+1/8 1/1+1/2+1/3+1/4+1/5+1/6+1/7+1/8+1/9+1/10+1/11+1/12+1/13+1/14+1/15+1/16 ...
Difference 1/2 1/3+1/4 1/5+1/6+1/7+1/8 1/9+1/10+1/11+1/12+1/13+1/14+1/15+1/16 1/17+1/18+1/19+1/20+1/21+1/22+1/23+1/24+1/25+1/26+1/27+1/28+1/29+1/30+1/31+1/32 ...

The differences don't tend to zero, for they're always above 1/2. This in fact shows that the starting sequence blows up to infinity.

Two real numbers are the same if the difference between their terms tend to zero, or intuitively, if they tend towards each other:

n 1 2 3 4 5 6 7 8 ...
Real 1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 ...
Real 2 1/2 2/3 3/4 4/5 5/6 6/7 7/8 8/9 ...
Difference 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 ...

## Representing Real Numbers

Look at the place-value chart here:

... 1000 100 10 1
... 4 3 2 1

Notice how it stopped in the ones place? What if it didn't stop? What if we have something like this instead:

... 1000 1000 10 1 . 1/10 1/100 1/1000 ...
... .

You would get all nonnegative real numbers. Why?

It's fairly easy to show that all numbers of this form are nonnegative real numbers (e.g., 48957.667458428982... can be represented as (40000,48000,48900,48950,48957,489576/10,4895766/100,48957667/1000...))

To see that all nonnegative numbers can be written in this way, if you have a nonnegative real number, it's possible to perform the divisions in each term and only keep some of the digits, as illustrated here:

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
nth term 1/1 1/2 2/3 3/5 5/8 8/13 13/21 21/34 34/55 55/89 89/144 144/233 233/377
Decimal form 1 .5 .66... .600... .6250...

The decimal forms will converge as every digit will stabilize.

## Math of real numbers

Addition, subtraction, multiplication and division all use the same method as integers. However, there are two things that should be noted.

Addition, subtraction and multiplication work by starting at the last digit. But in real numbers, there sometimes is no last digit. So to multiply 0.583436946468... by 3.525534690266..., you have to do something else. You have to do 0.5*3.5, 0.58*3.52, 0.583*3.525 etc. to see what happens at infinity.

In division, you don't stop at the ones place and take the remainder. Instead, you keep going forever.

## Reals vs Rationals

All rational numbers are real numbers. This can be shown by performing the division described by the rational number. For instance, 3/7=0.428571. Are all real numbers rational? No. For we can do this:

Number: .6 4 7 1 1 1 1 1 1 6 4 8 3 3 2 3 5 1 1 1 1 2 3 5 6 8 9 1 1 1 1 0 2 8
1/2 .5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1/3 .3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
2/3 .6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
1/4 .2 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3/4 .7 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1/5 .2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2/5 .4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3/5 .6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4/5 .8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1/6 .1 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
5/6 .8 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
1/7 .1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2
2/7 .2 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 8 5
3/7 .4 2 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 8
4/7 .5 7 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 1
5/7 .7 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 1 4
6/7 .8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7
1/8 .1 2 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3/8 .3 7 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
5/8 .6 2 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
7/8 .8 7 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1/9 .1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2/9 .2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
4/9 .4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
5/9 .5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
7/9 .7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7
8/9 .8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8
1/10 .1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3/10 .3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
7/10 .7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
9/10 .9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1/11 .0 9 0 9 0 9 0 9 0 9 0 9 0 9 0 9 0 9 0 9 0 9 0 9 0 9 0 9 0 9 0 9 0 9 0 9 0 9
2/11 .1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 8 7
3/11 .2 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7
4/11 .3 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
5/11 .4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
6/11 .5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
7/11 .6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

As the list contains all rational numbers from 0 to 1 (exclusive), the new number is between 0 and 1 (exclusive), and the new number isn't on the list (as it differs from each listed number in at least one position - the shaded digits), the new number is irrational. This also shows that there are more real numbers than natural numbers (if there are as many of the latter as the former, a list could be made containing all real numbers, then the same trick can be applied to arrive at a contradiction)

## The number line

The number line is a visual that can be used to grasp the real numbers. It works by picking a point, called the origin, which will represent 0, and a unit length in some direction.

Addition on the number line can be done as follows: Say you want to do 5+3. Start with two copies of the number line, overlaid such that all numbers map to themselves. Slide the second number line such that its 0 is laid over the first number line's three. The second number line's five will map to the first number line's 5+3, or 8.

Subtraction on the number line can be done as follows: Say you want to do 5-3. Start with two copies of the number line, overlaid such that all numbers map to themselves. Slide the second number line such that its three is laid over the first number line's zero. The second number line's five will map to the first number line's 5-3, or 2.

Multiplication on the number line can be done as follows: Say you want to do 5*3. Start with two copies of the number line, overlaid such that all numbers map to themselves. Keep the origins on both number lines fixed and stretch the second number line such that its 1 is on the first number line's 3. The second number line's 5 will map to the first number line's 5*3, or 15.

Division on the number line can be done as follows: Say you want to do 5/3. Start with two copies of the number line, overlaid such that all numbers map to themselves. Keep the origins on both number lines fixed and stretch the second number line such that its 3 is on the first number line's 1. The second number line's 5 will map to the first number line's 5/3, or 1.6.