India gave “0” to world. But, what does that mean?
While growing up, you’d have invariably heard this a lot of times. I did, a lot.
“Do you know India’s contribution to world? India gave “0” to the world.”
The kid-me would chuckle at this fascinating trivia. Has the world been so stupid that they invented all the numbers, but couldn’t come up with “0”, a seemingly innocuous digit. To my simple 8-year old mind, I narrated this along the lines of, “okay, all the other numbers had been there earlier, but nobody really thought about a number which holds no value”. The trivia was too amusing, the education system too rule-based, I packed it neatly in a box, and tossed it away into the stack of concepts that we memorise as facts, instead of understanding as, well concepts.
Until recently, when I was reading a book called Code, by Charles Petzold. There’s a section in the book which talks about base system of counting. After reading that section, I was like, ah, so Indian mathematicians invented this base system of counting. But then as I read more, I realised the far-reaching impact of this invention.
Let’s start with system of counting. Imagine a market in old times. I am thinking, before-christ, worshipping cats is cool, kings kill lions for sports kind of old times. Now, merchants needed to record their transactions, and a lot of transactions involved counting stuff. And, clay tablets were the in-thing. So, they would use clay tablets and draw lines to denote “quantity” of stuff. If they needed to record 5 sheeps, they would draw 5 lines. But, if you had to represent 1000 lines, you’d have to draw 1000 lines, and that’s a lot of effort honestly. So, some creative folks probably came up with an idea of representing large numbers using some other symbols. A circle can mean 20, a triangle can mean 5 and a line still means 1. Now, under this new system, if you have to write 37, you’d just draw one circle, three triangles and 2 lines.
Too simplistic? Here’s an actual clay tablet from Uruk, a Mesopotamian settlement.
If you think about it, roman numerals are kind of extension of same idea.
“I” means 1.
Five I’s make “V”, symbol for 5.
10 I’s make “X”.
“L” is 50.
“C” is centum.
“D” is 500.
“M” is 1000.
The rules are same too. You start with “I”s and keep simplifying by clubbing smaller numbers into bigger numbers. 37 in roman numerals is XXXVII. Pretty much the clay-tablet idea, right. Roman numbers were surprisingly effective for counting stuff, basic addition and subtraction. But multiplying and division became hard.
Enter Hindu-Arabic or Indo-Arabic number system
Meanwhile, mathematicians in India were evolving, what later came to be known as Hindu-Arabic or Indo-Arabic number system. It was brought to Europe by Arab mathematicians. One of the famous mathematicians was a dude called Muhammed ibn-Musa al-Khwarizmi. Cool name, latinised as, Algorithmi, led to another word called Algorithm, which supposedly are eating the world nowadays. He wrote a book on algebra around AD 825 that used Hindu system of counting.
Now, Hindu system of counting had some salient features, which are going to sound ridiculously obvious, but still pause and think about it.
- It had symbols to represent quantities 1 to 9 (१, २, ३, ४, ५, ६, ७, ८, ९), and all other quantities could be derived from these symbols. १ is same as 1, just different symbols.
- It was positional. That means, 25 and 52 were different quantities, based on the positions of symbols. In roman, L would always mean “fifty”.
- It had zero. Written as “०” in Devanagari script, this is one of the most important invention in mathematics.
There was no longer a special symbol required for counting “ten”. It was written as 10, and that was one of the reasons computers became possible and the world changed forever. Too fast? Let’s go a bit slow.
There are couple of breakthroughs in this system.
Numbers as symbols : First off, numbers do not have to signify “counts” of something. Numbers are abstracted out to just be symbol instead of relating to count of something physical. This is a powerful abstraction.
No special symbol for “ten” : In roman numerals, X means “ten”. In Hindu system of counting, “ten” is represented by 10. This is possible due to positional nature of the system. Read on.
Positional system : Hindu number system is positional, and there’s a formula to interpreting the numbers as quantities. For examples, 25 is just 2 and 5 placed next to each other. But, to signify quantity, it is interpreted as :
25 = 2 X [tens] + 5 X [ones] = Twenty-five of something
You can extend this idea to represent 358 as :
358 = 3 X [hundreds] + 5 X [tens] + 8 X [ones]
So as you go farther to the left, you increase the magnitude of the number from ones, tens, hundreds, thousands. And that’s why you don’t need special symbol for “ten”. You can write it as :
10 = 1 X [tens] + 0 X [ones] = Tens
A little diversion to Hindi counting
This idea of interpreting numbers based on their positions has always been there in Hindi and other Indian languages. For example,
45 in Hindi : पैंतालीस, roughly पांच + चालीस
68 in Hindi : अड़सठ, roughly आठ + साठ
There are few caveats to this straight-forward pattern. anything with 9 in unit behave slightly differently. The 9 translates to उन, which probably translates to “minus 1”. For example,
39 in Hindi : उनतालीस, is उन + चालीस. (40 minus 1)
49 in Hindi : उनचास, उन +पचास (50 minus 1)
Apart from these aberrations, possibly due to centuries of daily usage and influence from other dialects, Hindi counting is pretty formulaic as well. After 100, there are different names for powers of 10s (सौ, हज़ार, and so forth). Note that, Sanskrit counting is much more purer than Hindi counting.
But why “tens” and “powers of tens”?
Why this fascination with “tens” and “power of tens”? One credible hypothesis is that, as homo-sapiens, when we started counting, we used the first tool available to us. The ten fingers on our hand. That’s probably why we are so comfortable with “tens”. And our whole notion is counting is built around “tens” of things or “powers of ten”s of things.
Let’s count things using our hands. Try to remember that words (one, two, three) are the counting and digits (1,2,3..) are mere symbols.
Notice that when we reach nine, and have to count ten, we denote it by 10, where the 1 on the left denotes that we have completed our hand once. Let’s continue counting.
Notice how, twelve is represented as “two more than full-hand-counted”. In other words, 12 is (1[times-hands-used]+2[extra fingers]). We continue this till nineteen. To count twenty, we have used our hands twice, so we write twenty as 20.
But what about Micky Mouse and friends?
Meanwhile, in the Disney world, Micky Mouse, Donald Duck and Goofy needs to learn to count too. The twist in the story is that they have eight fingers, four in each hands. Let’s see, how counting would evolve in Disney world as they use their hands to count stuff.
One thing that would be clearly different is number of unique symbols. You have 8 fingers, so, probably would have 8 unique symbols (0,1,2,3,4,5,6,7).
Try to remember that words (one, two, three) are the counting and digits (1,2,3..) are mere symbols.
Everything is same till seven. But the fingers run out as they reach eight. So you use 10 to represent eight. 1 on the left denotes that you have used your hands once. Let’s continue counting.
Okay, time for a little breather. You are using 13 (symbol) to represent eleven (count) of something? Counting stuff is strange in Disney world. Remember, you have eight fingers on your hands.
13 translates to, going through counting on your hands once |||||||| and count three more |||. That is a total of eleven.
There’s an easier formulation to this.
13 = 1 X [eights] + 3 X [ones] = Eleven of something
or,
13 = 1 X [8¹] + 3 X [8⁰] = Eleven of something
So, in a nutshell, Hindu-Arabic number system decoupled the notion of counting from the unique symbols we use to denote the numbers. 13 can mean different quantities depending on how many fingers you have(or how many unique symbols you use). The number of unique symbols is also denoted as base of the number system. Humans are comfortable with base 10. Donald Duck is comfortable with base 8. Another intuition for base is unique states of each digit, e.g. when we are using base¹⁰, we have 10 unique symbols for each digit (0,1,2,3,4,5,6,7,8,9).
Onwards to Binary system
Okay, time for another inferential leap. What if we use base-2, which is when we only have two symbols available. 0 and 1. Notice that everything we talked about above, applies to base² as well. Base² is also known as Binary system.
Well, the counting works, more or less the same way.
11 = 1 X [2¹] + 1X [2⁰] = Three of something
A pretty impressive feat is that mathematical operations carry over across different bases seamlessly. Just like we saw for counting, you can write algorithm which can perform multiplication in base¹⁰ and same algorithm can perform multiplication in base⁸ or base² as well.
And the reason, we decided to build computers on binary system is because you can represent a base² system more easily. For example, all you need is two distinct states. Well, you can use a light bulb to do that. When the bulb is on, it means 1, when bulb is off, it means 0. This is much more accurate and error-proof than having e.g. base³ where you need to have a bulb with 3 states; on, off and low-brightness. Then too, it’s error prone. How low is low-brightness? When do you term bulb as low brightness vs full-brightness. Binary system, is much easier to emulate in physical objects. For example, the bulb is either on or off. Simple.
Today’s computers are built with transistors, millions of them stacked together, where each transistor is capable of representing two-states, on and off.
In closing
There’s whole lot more of things that had to happen to make computer possible, e.g. logic, information theory etc. If you are curious, you can read Code by Charles Petzold.
This article is about what Indo-Arabic number system did for computers. So, next time when you are tempted to say, “India’s contribution to the world is 0”. Stop yourself and say, “India’s contribution to the world is Indo-Arabic numeral system.” and that stuff made telegrams, communication, computers and internet happen. You’ll probably pique a 8-year old kid’s curiosity to find out what that means.
References :
- Code, by Charles Petzold
- Clay tablets : https://www.bbc.com/news/business-39870485
- Numeral system : https://en.wikipedia.org/wiki/Numeral_system
- Roman Numerals : http://turner.faculty.swau.edu/mathematics/materialslibrary/roman/
