Introduction to Real Numbers

Since before written history, man has kept count of things. Historically we have used different symbols to write down these counts for posterity. We currently use the Decimal System, also known as Arabic Numerals or Hindu-Arabic Numerals and some other labels. The decimal system involves the base 10 with numbers going from 0, 1, 2, ... 9, and then looping to 10 and onwards.

Romans, Greeks, Mayans, Chinese, and countless other human societies have had their own numbering system with different bases other than 10. There is no one best numbering system that fits all and even today we create new numbering systems to solve some practical problems that are better served by a new way to count things, most recently we created the binary, the octet, and the hexadecimal systems to help with computer science.

How and Why Number Systems Evolve?

Mathematics and how we define and denote numbers is a tool to interact with the outside world. All these different number systems have developed with that problem-solving attitude in mind. 0 implies notingness, or emptiness. By definition 0 does not exist as a count but it was deemed useful enough and was brought into existence as the symbol for nothingness most likely in Ancient India and later popularized by the early Islamic world.

Natural Numbers

Our numbering system started with the Natural Numbers that counted discrete items starting from 1. Natural numbers are all whole, complete numbers, 1, 2, 3, and onwards. One million, Five hundred and seventy two billion, three, all of these are natural numbers. But 0 is not a natural number. Neither are numbers like 3.5 or one half.

We represent Natural numbers in a set form as;

$\mathbb{N} = \left\{ 1, 2, 3, \ldots \right\}$

Whole Numbers

The first limitation of Natural numbers that we ran into was an abscence of zero. We expanded to Whole Numbers so that we could use zero to denote the absence of something. Whole numbers are all the natural numbers and zero.

We represent Whole numbers in a set form as;

$\mathbb{W} = \left\{ 0, 1, 2, 3, \ldots \right\}$

Integers

The next weakness of natural numbers was not being able to show deficit. If we want to show a container has three of something, we have a number 3. But what if we want to communicate that 3 is missing from that container. For this we needed negative numbers. As subtraction found it use, we got to Integers that included the negative numbers along with the positives and the zeroes to show surplus versus deficit.

We represent Integers numbers in a set form as;

$\mathbb{Z} = \left\{ \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots \right\}$

Rational Numbers

Then the need for partial numbers became more apparent, numbers that lie between the discrete numbers. How do we represent one half, two thirds, four and three quarters? So we got Fractions which are also called Rational Numbers.

A rational number is defined as $\dfrac{p}{q}$ where both $p$ and $q$ are integers. $\dfrac{1}{2}$ , $\dfrac{-3}{4}$ and $\dfrac{5486}{-153}$ are all rational numbers because they obey the $p$ over $q$ form.

What's Next?

This is where most of the common knowledge of numbering system ends. But it is marvelous that almost all humans now deal with fractions and negative numbers in their day to day life.

After rational numbers, we have irrational numbers, and transcendental numbers. Classifications that are useful to mathematicians and engineers to make great things but not directly useful for a layperson.

What are Irrational Numbers and Where did they come from?

At some point, we accepted the idea that our numbering system is not discrete but continuous. There is a number between 1 and 2. There is a number between 1.1 and 1.2. There is a number between 1.01 and 1.02, and on and on it goes. There are an infinite number of numbers between 1 and 2. We cannot count them all or write them all down. These infinite amount of numbers themselves are also infinitely long like $0.333333 \ldots$ .

First, think in the following terms. We have mathematical operations, addition, subtraction, multiplication and division that convert your input numbers into an output number. But we did not stop there. We invented more operations, roots like square roots and cube roots, the logarithmic function, and any of the trignometric functions. When we apply a square root, a log, or a trignometric function, sometimes the result is an infinitely long chaotic number, sometimes the result does not even exist as a real number which is the case with complex numbers.

The Greek Problem:

Consider a right angled triangle with a base and the adjacent side as equal to 1. What is the length of the hypotenuse? We know from Pythagoras Theorem that the length is $\sqrt{2}$ . It is a number that when multiplied by itself equals 2. That is the definition of square root of 2. The question is, how many decimal places are in that number. Since we live in a physical world, we think it should be finite number always. But then the Greeks developed the proof showing that the square root of 2 has infinite decimal places, it keeps going on and on. It is non-terminating.

Furthermore, they proved that $\sqrt{2}$ is not rational. It cannot be written like rational numbers as the fraction of two numbers in the form $\dfrac{p}{q}$ . The detailed proof is in the next lesson.

They further established that all square roots are not irrational. $\sqrt{2}, \sqrt{3}, \sqrt{5}$ and an infinite others are irrational. But, $\sqrt{4}, \sqrt{9}, \sqrt{16}$ are rational.

Real Numbers

For a recap, when we add new numbers to a previous set, we give it a new name. We had the Natural numbers.

\mathbb{N} = \left\{1,2,3,\dots \right\}

Adding a zero to this gave us the Whole Numbers.

\mathbb{W} = \left\{0,1,2,3,\dots\right\}

Adding the negative numbers to Whole Numbers gave us the integer set.

\mathbb{Z} = \left\{\dots, -3,-2,-1,0,1,2,3,\dots\right\}

Adding fractions to the negative numbers gave us Rational Numbers that include all integers and fractions of integers denoted by the set read as $\mathbb{Q}$ is equal to $\dfrac{p}{q}$ such that $p$ and $q$ belong to the integer set and $q$ is not equal to zero.

\mathbb{Q} = \left\{ \frac{p}{q} \;\middle|\; p, q \in \mathbb{Z},\; q \neq 0 \right\}

Adding all the irrational numbers that are represented by $\mathbb{Q}'$ to the rational numbers gives us the Real Number set which is shown as the union of all rational and irrational numbers.

\mathbb{R} = \mathbb{Q} \cup \mathbb{Q}'

The Irrational Number set alone can be read as all real numbers that are not in the rational number set.

\mathbb{Q}' = \left\{ x \in \mathbb{R} \;\middle|\; x \notin \mathbb{Q} \right\}

Activity: Match The Columns!

Reorder the Definitions to match them with the correct Set Names.

Set Names

Definitions

$\text{Natural Numbers}$

$\text{Whole Numbers}$

$\text{Integers}$

$\text{Rational Numbers}$

$\text{Irrational Numbers}$

$\text{Real Numbers}$

End of Lesson