unreasonably effective

Place-value number system like decimals for an example, is better than say Roman numerals. For example, a quick glance is sufficient to declare is bigger than , because has more digits (or symbols). On the other hand, it is not that easy to say which one is bigger: or , actually the latter is bigger even though it has less symbols. We can write any whole number, no matter how big, with only ten symbols: . Also, the number of symbols it takes to write bigger numbers grows much slower than the magnitude of the numbers. For example, takes a single symbol but a million does not take a million times more symbols, it takes only seven symbols: . Addition, subtraction, multiplication, division all basic manipulations have step-by-step, easy-to-follow procedures that require looking at a symbol or two at a time, even if the numbers involved are big. A patient child can go on adding huge numbers once she has learned how to add two digits.

Alright, but how did we come up with such a great way of representing numbers? In other words, is there a simpler, intuitive reason or urge out of which the place value system springs up naturally?

It’s natural to want to use only a few, say three symbols, to represent any whole number. We could count like: 1, 2, 3. For bigger counts, we can go on repeating same symbols: 1, 2, 3, 1, 2, 3, … However, we cannot tell the difference between say the first 2 and the second 2. So, we could arrange these in rows. First row: 1, 2, 3. Second row: 1, 2, 3. Now we can differentiate between the two 2’s like: 12 and 22 where the first symbol (from left) for each number represents which row it is in and the second symbol represents which number in that row, like two co-ordinates. For example, 12 is the second number in the first row. Similarly, we get three, four or more digit (symbol) numbers. This gives us a unique representation for every whole number.

Let’s add another natural urge: we want to deduce the magnitude of a number from its representation. Like from 21, we want to say how big the number is. We had to add a second symbol when we exhausted all the distinct symbols we had, like: 1, 2, 3. Then we needed two-digits. So, the extra digit (1 in 12 for example) should represent how many rows we already traversed. Like, for 12, we should say this number comes after a full row and then another number. But, for 12, we have not covered a single row yet, it is on the very first row. Likewise, for the number 21, we have not really covered two full rows, it is on the second row. What we actually need now is a symbol to represent “None” therefore zero. Then the numbers in the very first row can be written as 00, 01, 02 (our three symbols are 0, 1, 2 now) and the next row becomes 10, 11, 12. Now, for each number, the extra symbol (first from left) correctly indicates how many rows we have already covered. Since each row has the same count of numbers, we have a sense of the magnitude of the number. For example, between 02 and 10, we know 10 is bigger because to reach it we needed to count an entire row. For 02 we did not need to count an entire row.

We have thus reached at the place-value number system with two natural restrictions:

1. We wanted to use only a few symbols to represent any number.
2. We wanted to deduce the magnitude of a number from its symbolic representation say to compare.

The number of distinct symbols naturally becomes the base.