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Number line

Posted in Mathematics, Methinks by ytelotus on June 20, 2013

What are Natural numbers, Whole numbers and Integers ?

Why do we need Negative numbers?

Why is -1 less than 0?  How can there be something smaller than nothing?

Is there anything rational about Rational numbers?

What is behind the random sequence of unending decimal part in Irrational numbers?

For answers, read on…

Introduction

We get imbued with the idea of numbers very early in life – so early that we consume it the way it is presented to us.  We first learn to count 0, 1, 2…  As we grow, more categories get added on – integers, rational numbers, irrational numbers, complex numbers etc.  We even learn about different systems of numeration apart from the decimal system (base 10),  such as binary system (base 2), octal system (base 8) and hexadecimal system (base 16) from the computing lingo.  This is how we come to traverse the Number Line.

Fig. 1 Number line

Fig. 1 Number line

Because the idea of numbers predated writing,  they happened to get their own symbols called numerals.  In the hands of different civilizations who individually worked on the numeric concepts, numbers found their diverse representation schemes.  The artifacts from those times give us a fair idea of the symbols used and the different numeration systems that were in vogue among civilizations.  As an example, the inscribed clay tablets found from the Babylonian era indicate the existence of sexagesimal system of numeration (base 60).  This is still being used by us for calibration of time, as in hours-minutes-seconds.  The dozen measure that groups numbers in units of 12 stands for the duodecimal system.

Many such systems of numeration came into being, but the one that became ubiquitous is the decimal system with base 10.  Though not used in the same sense as decimal system, the number 10 has a special meaning in all the numeral systems.  It stands for the respective base.
“10” =>  10 in Decimal system (base 10)
“10” =>   2  in Binary system (base 2)
“10” =>   8  in Octal system (base 8)
“10” =>  16 in Hexadecimal system (base 16)

Brief detour

Before we delve into the world of numbers under the decimal system and the categories thereof, let us attempt to get to a somewhat subjective understanding of measures.

Things around us exist in measures of nought (denoted by ‘0’) to infinity (denoted by ‘∞’).  Nought is the lack of something and infinity is the abundance of something without limit.  What connects the two is the infinite series of infinitesimals – the minutest immeasurable granules – acting as a filler between the two extremities of nought and infinity.

It’s important to understand that the scale of number line itself is relative to the viewer’s degree of vision.  As an example, if you take countable objects visible to the naked eye, the demarcation between any two consecutive numbers is very clear.
0 apple = no apple
1 apple = there IS  an apple
2 apples = there is ONE MORE apple

Now let us get to the microscopic scale of measurement.  Can you distinguish between 0 and the smallest indivisible measure next to 0?  If you are to say it is 0.000000000…1, after how many zeroes should you place that digit 1 there?  You cannot finitely define this.  Can you?  That opens up the world of infinitesimality as opposed to infinity that we have been so much familiarized with.  Just as in the case of infinity, there is no limit to infinitesimality as well.  While infinity stands for never-ending expansion, infinitesimality means incessant withdrawal to something deep within.  In this dichotomy lies the philosophy that we can neither contain nor discretize what is perceived as a continuum.  This is the very essence of the continuum of space and time – entities with unknown beginning or end.

Number Categorization

Back to the Number line.  Let us traverse it in the order that makes sense.

Natural numbers
The set of natural numbers is denoted by N as,  N = {1, 2, 3, 4, …}.  They are mainly used for counting.

Fig. 2 Natural numbers

Fig. 2 Natural numbers

Whole numbers
The addition of zero to the set of natural numbers gives the set of whole numbers denoted by W as, W = {0, 1, 2, 3, 4, …}

Fig. 3 Whole numbers

Fig. 3 Whole numbers

Integers
The addition of negative numbers to the set of whole numbers gives integers.  They are denoted by I or Z, as, I or Z = {…, -3, -2, -1, 0, 1, 2, 3, …}

The negative numbers can be interpreted in many ways depending on the contexts that made its existence essential.   It can mean undoing an action, or, the absence or lack of something : If you have 5 objects put together, removing 2 of them (-2) implies the absence of 2 objects.

In the context of pairs of opposites, negative numbers complement positive numbers enabling us to quantify the opposing forces in nature, such as, that of electrons and protons, opposite poles of north and south in a magnet, opposing forces of pull and push, centrifugal-centripetal forces etc.  If you represent the force of push to be something positive, say +10 Newtons, the same force applied in the opposite direction by means of pulling action is -10 Newtons.

The idea verily extends to the relative measures of things :  We denote electrons by -ve charge and protons by +ve charge.  In a neutral atom, the number of electrons is equal to the number of protons.  The charge of a neutral atom therefore equals 0.  If we remove a few electrons from an atom, the atom gains positive charge in proportion to the number of electrons lost.  Similarly, if we add electrons to a neutral atom, the atom becomes negatively charged in proportion to the number of electrons added.  In this context, the charge of the atom whether +ve or -ve, is said to be relative to the charge 0 when the atom is in neutral state.

An understanding gained thus of the negative numbers helps us explain the position of these numbers in the number line shown in Fig.4.  In reality, is there a number less than zero?  Can anything be smaller than nothing?  No.  Then, what puts negative numbers ahead of zero or positive numbers in the number line?  The answer lies in their relatively complementing nature with respect to positive numbers.

Fig. 4 Integers

Fig. 4 Integers

A philosophical rub to this whole idea of number line with positive and negative numbers on either side of zero is that, when you are zero, what you see within i.e. -1, -2, -3 … is a mere reflection of what you see without i.e. 1, 2, 3, …  Considering that a number line is infinite in length, your zero can be a moving one, nailing it right at the point that you are at any given moment of time.

So far, so good.  But we’re not there yet.  The numbers covered thus far are all distinct numbers – the transition from one number to the other is abrupt.  The space between any two consecutive numbers has not been taken into account.  The next two categories of numbers are inclusive of this space taking you to the world of fractions and decimals.

Rational numbers
A rational number is a number that can be expressed as a “ratio”.  So, there’s nothing “rational” about this number if you had ever got misled by the use of this term.  They’re just fractions.  Perhaps, “fractional” number would have been a better name.  It is denoted by Q, as, Q = { p/q  | p, q E I and q is not 0}

Needless to say, any natural number, whole number or integer can be expressed as a rational number by virtue of the property of 1.  The number 1 divides all numbers without changing its value or property.  e.g. 3/1 = 3, -2/2 = -2, 0/1 = 0

Another way to look at rational numbers is, every rational number can be expressed as a terminating decimal or a periodic (recurring) non-terminating decimal.
Terminating decimals :  1/8 = 0.125 ,  97/8 = 12.125
Non-terminating periodic decimals :  1/7 = 0.142857142857142857…  (the pattern 142857 repeats itself in the decimal part) , 100/11 = 9.0909090909… (the pattern 09 repeats itself in the decimal part)

Irrational numbers
Any number that is not a rational number, something that cannot be expressed as a “ratio”, is an irrational number.  Again, there is nothing “irrational” about irrational numbers.

Famously falling in this category are the roots of numbers that are not perfect squres of any number e.g.√2, √3 etc.  Also, falling in this category are the surds – the n-th roots of numbers that cannot be expressed as the nth power of any other number e.g.  3rd root of 7 denoted by 3√ 7 etc.

Keep in mind though that the irrational numbers are not limited to roots or surds as they’re otherwise made to look.  For example, pi = 3.14159265358979323846… that is an irrational number cannot be expressed as the n-th root of any other number.  Such a number is called transcendental number.  The best definition for irrational number therefore is that, it is any number that is not a rational number.

Another way to look at irrational numbers is, an irrational number is a non-periodic non-terminating decimal number.  e.g. √2 = 1.4142136… is a non-periodic number that grows infinitely with no sequence of digits repeating itself systematically.  This property implies that the irrational numbers interpenetrate the space left behind by the rational numbers.

Convergence aspect of non-terminating decimals 
Though it appears that a non-terminating decimal number is an unending number, it is actually a recursively converging number.  Its beauty is that, while it extends the decimal part digit by digit from one end, it is actually limiting the scope of the decimal number each time the digit is added.  Here’s how :

Let’s take x = 2.  We can say, 2 ≤ x < 3.  We add a decimal part, say, 0.5  to this.  Now, x = 2.5.  By adding this decimal part, we increased the value of x above 2.  At the same time, we limited the scope of x from 3 to 2.6 (next higher number at that decimal position).  By limiting scope, we mean, now on, no matter how many digits you add to the decimal part of x, the value of x will never go beyond 2.6.  i.e. 2.5 ≤ x < 2.6.
Now, let’s add a digit 8 to the decimal part i.e. 0.08 to x.  x = 2.58.  The scope of x is now reduced from 2.6 to 2.59.  i.e. 2.58 ≤ x < 2.59.
Next, we add 0.003.  x = 2.583.  The scope of x is limited to 2.584.  i.e. 2.583 ≤ x < 2.584.

Effectively, by adding decimal digits, the range of x has been narrowed down from  2 ≤ x < 3 to 2.583 ≤ x < 2.584 – a clear convergence process. Fig. 5 shown below illustrates this process.

Step 1 :  We start with stretching the space between the numbers 2 and 3 to a larger scale.  Divide this space into 10 parts (because we follow base 10 method).  We now mark 2.5  as the mid-point on this new scale.

Step 2 :  Now, stretch the space between 5 and 6 on the second scale to a larger scale.  Once again divide this space into 10 parts.  We now mark 2.58 on this newer scale.  This process repeats as we keep adding each digit to the decimal part.

Fig. 5 Dive-in process of non-terminating decimals

Fig. 5 Decimal number 2.58….

Though we cannot attach much physical significance to the repeating (rational) or non-repeating (irrational) nature of non-terminating decimals, the non-terminating property of decimal numbers in itself is indicative of the deep dive of these numbers into the world of infinitesimality. 

The detour that we took early on was to set the premise for these non-terminating decimals.  It is the world of an unending division process of number space into smaller and smaller units.  An impossible task, whatsoever.  We can’t express something that is continuous in terms of something that is discrete.  Can we?

Real numbers
Real numbers is just a name given to the union of the set of rational numbers Q and the set of irrational numbers Q`.  It’s denoted as R = Q U Q`.

Fig. 6 Real numbers

Fig. 6 Real numbers

Real numbers are therefore the totality of all the numbers discussed so far – from infinitesimals to infinite numbers, from negative to positive numbers.

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4 Responses

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  1. sansha2100 said, on December 3, 2013 at 6:02 am

    Dear Friend, A great yogi named Shailendra Yogi, has beautifully explained the same in following words :

    The ancient sages, or rishis recognized the three states of creation, and introduced the presiding deities of these three states: Brahma – the creator of the world, Vishnu – the perserver and Shiva , the destroyer. If we look at this symbolism from a scientific point of view, we discover a totally different meaning to it.

    Matter is the basic building block of the world; wherever we look, we find matter in myriad of forms. On both microscopic and macroscopic scales, matter pervades the whole cosmos. Brahma word literally means Infinitely Vast. It symbolizes the vastness of matter, the foundation of this whole creation. Or we can say that the sages gave the name Brahma to the inherent consciousness of conscious matter.

    We’ve read in the scriptures that Lord Vishnu is bearing the whole creation, and also sustaining it. When we think about it, the entity which bears or contains matter is empty space, or void. Even when we study atom – the smallest unit of matter, we find that there is emptiness between the nucleus and the electrons revolving around it. This void, or emptiness is the basis for matter’s motion. The sages identified this ability of Void to hold matter and gave the consiousness of this conscious emptiness the name, Vishnu.

    When we break down matter into smallest units, we reach subatomic particles, and when even those are further broken down, we discover pure energy. Investigating this pure energy in order to find its true source, we find that energy originates from pure thought. This pure thought, which we can also call the consciousness of matter, originates from conscious emptiness. This pure thought is the link which brings matter and void together.

    The element, which is beyond both matter and void, which pervades both matter and emptiness and yet is the cause for their destruction, which is beyond everything and still ever-present with us, which is motionless still appears to be moving, which our consciouness experiences in different forms due to difference in our levels, that element which is Time or Kaal was identified as Shiva by the sages. Time is that element which we experience all the time, yet it always seems beyond all experiences. Matter’s time is said to be finite. The time of emptiness, which holds matter was also realized by the sages. But who can understand the time of Time, save Time Itself!

    Sometimes we contemplate about the time when there was no creation, no universe, no good or bad, nothing; what existed then? Those who have practiced even a little bit of yoga and have been successful in awaking their dormant consciousness realize that what existed then was in the womb of Time and how did it get there, no one except Time Itself can tell. May be that’s why sages envisioned Lord Shiva with both male and female principles, as the Ardhanarishwara. This leads us to conclude that Time Itself impregnated Time, and in due course took birth in the form of this creation. When Time’s time comes to an end, only Time shall remain…

  2. mathangi said, on December 18, 2013 at 5:42 am

    Awesome insights. Learnt all these decades back. Was a good refresher.

  3. ytelotus said, on December 19, 2013 at 3:50 am

    Thanks, Mathangi. When I studied these in school, I did not have them connected. Now when I connect them together, it makes more sense to me.


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