ARITHMETIC, a story on Numbers

OVERVIEW

We deal with numbers in Arithmetic. There are various types of numbers. I will tell you a story of how numbers developed. Initially, when man used to live in jungles, they learnt to count the trees around them, in Nature. So, they counted: one, two, three, four. Ofcourse these words differ in different languages like in Hindi it is said ek, do, teen, char and in Kannada it is called ondu, eredu, muru, nakku etc etc. The ancients might have counted by using their fingers like we still do. 

Thus came the NATURAL NUMBERS which are the countable numbers: 1,2,3,4 …

Then came Arya Bhatta and he discovered ZERO. 

Think about it, I think this is a stupendous achievement. Because Zero means nothing. To discover nothing is next to impossible, if you think of it. How do you define nothing. It is actually the absence of something. If I now ask you to name the things around you, you can do it very easily. But how can you even think of nothing !!!. So we salute Arya Bhatta for this amazing discovery. 

And thus we now had the WHOLE NUMBERS which is Natural Numbers and 0 i.e. 0,1,2,3,4 …

And then people started adding these numbers. But when they learnt to subtract what happened was that sometimes there were no results i.e. 3 – 4 did not result in any number in the existing number system. 

So, they devised NEGATIVE NUMBERS: -1, -2, -3 etc and we came to have another new number system called

INTEGERS which is nothing but  the Natural Numbers & 0 & the negative of the Natural Numbers

At that time mathematicians might have thought that now they have really developed a sophisticated sytem. But there was more trouble coming their way, by way of division. When they tried to divide the natural numbers one by the other, again they did not have any solution inside their Number System which had to be expanded once again to get what we now know as

the RATIONAL NUMBERS which is any number expressed as p/q or p divided by q and we came upon a new category called FRACTIONS which populated the spaces between the Integers. 

But is this the end of the story ? No. Far from it. 

Mathematics grows by asking natural questions which crop up in the mind. Here the culprit was squaring a number. 

What happened was people came to square the natural numbers. So,
2 squared is 4 and
3 squared is 9 and 
4 squared is 16 and so on and so forth. 

Now mathematicians are a creative bunch. They think from all directions. So, in this case, they reversed the sequence and arrived at this:

1 is the square of 1
4 is the square of 2
9 is the square of 3
16 is the square of 4 etc etc. 

The mathematicians noticed that now the sequence did not have all the numbers. It had a sequence which was like 1,4,9,16,25,36… Where did the remaining numbers disappear.

The question that popped up was this
2 is the square of ?
3 is the square of ?
5 is the square of ?
etc. 

And this led to the addition of yet another category of numbers. Recall that so far we had started with the Natural Numbers, added 0 to it, added the negative natural numbers and also added fractional numbers in between to call the sequence as RATIONAL NUMBER SYSTEM. 

Now a new category of numbers IRRATIONAL numbers was added and it included those numbers which were missing in the sequence as shown above and we call them
square root of 3, sq rt 5, sq rt 7 etc etc. 

So, NOW, is it over or is it over ? asked the Mathematicians. 
Would you believe it there was one more type of numbers that they then discovered. These were numbers which were discovered from nature (remember Natural Numbers?) and they were PI, PHI and E. These numbers were called TRANSCENDENTAL NUMBERS as they did not come from any algebraic equations. Another peculiar aspect of these numbers was that the decimal part did not have any repetitions and continued endlessly. Since then computer geeks have been trying to find out PI to many many digits of decimals by using super computers and have still not found any repetitions. 

Now all these numbers together were named the REAL NUMBER SYSTEM. 

WHEW. ARE WE THERE YET. HAVE WE REALLY FINISHED WITH OUR NUMBER SYSTEM AND HAVE WE CAPTURED ALL THE NUMBERS THAT ARE POSSIBLE. 

Well the answer is again a BIG NO. 

Mathematicians designed yet another class of numbers. It emanated from the following thought. 

See if you squared a negative number you always end up with a positive number because (-)x(-)=(+)

But the mathematicians were not satisfied and they constructed another class of number 
by calling the square root of (-1) as a new number, i. They called it the IMAGINARY NUMBER, i. 
And this gave rise to a new branch of mathematics called COMPLEX NUMBER or IMAGINARY NUMBER SYSTEM. You will learn about this system when you study for your engineering entrance exam and let me assure you, although the name is complex, actually you can understand it very easily if your fundamentals or basic concepts in COORDINATE GEOMETRY is clear. 

So, now who can say that even more number systems will not be developed in the future. May be you yourself could become a mathematician and discover new systems that mankind will thank you for. Let us imagine such a future and begin our journey into mathematics. 

Vignettes from Nature