Unlimited numbers, limited digits
How was an infinite number system created? A child’s musings
This will be an investigation into the number system becoming infinite. Just to warn any mathematics enthusiasts, this is an outcome of childish inquiry. So please do not expect any genuine mathematics here.
As a prologue to this journey, I have the habit of creating a story in my mind as to how the scientific community reached various conclusions. Whether this story has anything to do with historical reality, I never cared. You might find resonances of this habit in the journey we will be taking together shortly. After all it’s just for fun.
In order to board the train the only ticket you need to have is a knowledge of numbers 1, 2, ….. all the way upto (say) 100. A knowledge of addition and subtraction is more than enough. So you see it is originally going to be a kid’s ride; maybe a few grown-ups, like me, will enjoy it as well.
So let’s board the train……
The journey starts on a fine evening, maybe a decade back. My father and I were resting on a mat spread out on the verandah (an open cemented space in front of the house), both looking up at the night sky.
My father was also my tutor back in those days. He had this beautiful habit of teaching me whatever he wanted through recreational gossips; no pen or paper. Just a little child and his dad chatting. He taught me Maths, Grammar, Moral Values, Ramayana, Mahabharata and what not…..all in the course of such gossips.
So on this fateful day, he just told me - the true revolution of number system came with the introduction of 0. Previously men could only count upto 9. With 0, you have 10, 100, 1000 and so on. You have an unlimited source of numbers.
Great. But one question came to my mind. Why was the entire credit given to 0? You can as well put as many 9s as you want after 1 creating 19, 199, 1999 and so on. However, the question that rose in my mind got lost soon. I did not pay much attention to it then.
Years flew past and then came a time when I was introduced to the chapter of number system for the first time. There 123 was broken down (rather defined) as 1×100 + 2×10 + 3. What a strange way to define 123. Surely multiplication is a concept introduced much later to the ability of man to understand the number 123. How can one define 123, a very primitive number, with such ideas that are logically much more advanced? Anyways, that confusion was also brushed aside for the time-being. But it remained somewhere in the recesses of my mind.
A few more years pass by and now I am being tutored by a very brilliant mathematics professor, the best one I have ever met. One day he just narrates a story as to how the process of counting might have started. He goes thus:
Suppose you have 7 cows (say). Each day these cows go to graze on the field by themselves and by dusk return to their sheds. Pretty well trained cows, no doubt. Now you have developed an intelligent way to keep track of these cows. Everyday they leave their shed, you pick up a pebble and put it in a bag for each cow that leaves. When they come back, for each cow that goes back in the shed you pick out a pebble from the bag and drop it. If there are no pebbles remaining in the bag good news for you. All matched. If there are pebbles remaining inside the bag you are in trouble. Go find the cows. In the rarest of occasions, if there are no more pebbles in the bag yet there are cows remaining to enter the shed…… You know what that says. Just don’t tell your neighbours. So that’s how the first idea of counting might have begun.
An extraordinary story by an extraordinary professor.
Now all the old questions come back from the closets of my mind.
- How can 0 get all the credit for making the numbers unlimited?
- How can 123 be defined as 1×100 + 2×10 + 3?
And I take the story by the professor forward……..
You, the owner of the cows, grow smarter. In order to keep a count of the number of cows in the shed you divide it into compartments.
If there is one cow in the first compartment you create a symbol for this entire picture or concept. The symbol is 1. Two cows in the first two compartments….you give the symbol 2. You keep going thus all the way till 7. No cows, of course it’s sad. But you have to have a symbol. So here comes 0.
So far so good.
Over the years you bring two more cows. That made you create two more symbols, 8 and 9.
Now the comes the twist. Your economic conditions get even better. You buy more cows. You are pretty happy. But just then something strikes your mind. You need more symbols!
Not just that. You understand that as you keep buying more and more cows you need to have an ever increasing number of symbols. Your memory starts failing. No you can’t keep going this way. Should you then stop buying any more cows? Should your financial status get limited by your memory?
Nope. That’s not the nature of man. So you develope this brilliant idea.
You keep ten compartments in your shed. Do not think of ten as 10 as yet. Just the idea that it is a certain maximum number of compartments you decide to keep in the shed. Then you build up a new shed with ten more compartments.
Now follow carefully……
Your first shed had 9 cows. One more cow and that shed is filled up completely. At this point instead of creating one more symbol for this picture you explain it thus…..
ONE shed is completely filled and the SECOND shed is empty.
Notice the shift in the language. Previously you were speaking of one compartment, two compartments etc. Now you have one shed. Both ONEs are symbolised by 1. But there is a difference in scale. Previously it was 1 compartment; now it is 1 shed. So the same symbol, used in different scales, convey different pictures.
The difference you make clear by appending to the statement “one complete shed” the portion “and next shed empty”. 1 followed by a 0. Hence 10. Now the next cow enters a compartment in the next shed and you have 11. One full shed and one compartment of the next shed. Notice how the two 1s now convey two different meanings. Same symbol, different meanings. That’s the key.
Once this jump is made you keep going like 12, 13, 14, …… all the way to 99. 9 complete sheds and 9 compartments of the next (tenth) shed. Just like the previous instance, you compile 10 sheds to make up a mega-shed and your scale increases. This allows you to go to 100. I am sure you have got the trick by now. Use just the digits from 0 to 9 and keep increasing your cow-based asset.
So that’s the way I finally solved my long standing doubts.
- 0 is not the real key though it does play a vital part. 1 complete shed and the next shed is empty. This emptiness is where 0 comes. And it is what differentiates 1 compartment and 1 shed.
- 123 being defined as 1×100 + 2×10 + 3. The ×10 might as well stand for the shed having 10 compartments. The ×100 might be a representative of a mega shed having 10 sheds or 100 compartments.
So my doubts being cleared, I was now pretty happy. A beautiful thought it seemed to me.
Just a passing note. Consider how the binary system goes. 0, 1, 10, 11, 100,….. There the same concept applies. Just this time the inventor’s memory started failing just after having memorised the symbols 0 and 1. So his sheds have two compartments each. His mega-shed has two sheds each. So on….. Same goes for any other crazy number system out in the world.
My dad and professor both played the vital role in this beautiful and stupid little journey. If I stretched it a bit too long or made it technical and boring, I apologize.
Thanks a lot for your patience, if you have read till this point. And just in case you enjoyed the ride, I hope I can come up with such pleasant trips in the future as well.
Thanks a lot dad. Thanks a lot professor. Thank you too. Let’s be grateful to the cows for their co-operation.