Place-Value Notation: Some Pros and Cons

Just so there will be no confusion, I'm going to say that the set of whole numbers (represented by W) contains zero. The natural numbers or counting numbers (represented by N) do not contain zero. As far as I know, definitions vary and there is no agreed upon standard, so this is just my personal preference:

It's convenient that every whole number can be represented as a finite sequence of digits. For example, the number 123 is represented in decimal as - well - '1', '2', and '3' in that order. Each representation is unique. When I write 123, it's entirely clear to which number I refer. Likewise, there is no alternate way of writing 123, so long as we disqualify 0123 for its leading zero.

We've all been working with place-value notation our entire lives, so we may forget why it's so useful. Think about the most common operations that you perform on whole numbers:

Comparison
Addition
Subtraction
Multiplication
Division

Three of these operations can be carried out very simply with place-value notation. When performing addition, subtraction, or comparisons on paper, we only need to look at each column of digits once.

This is so efficient that computers and electronic systems perform these operations similarly to the way we do them by hand. Granted, they are using a base-2 system where the only possible digits are 0 and 1, but it is nonetheless a place-value numbering system.

Multiplication and Division are a little bit more difficult to do, but still manageable. In the multiplication algorithm , each digit must be multiplied with every digit of the other number. This is fundamentally more complex than addition.

Imaging trying to perform these operations with Roman numerals. If those hadn't been replaced with a place-value system (base 10 or otherwise), western civilization could not have advanced technologically as much as it has.

One thing about mathematics is that notation matters. If your concrete representation of abstract concepts does not mesh well with the operations you'd like to perform on those objects, not only will your calculations be slower, but you may overlook techniques that in an alternate notation are plainly obvious.

Believe it or not, some operations are actually quite difficult to perform with our place-value system. Consider the following operations.

Finding the least common multiple of 2 numbers
Finding the greatest common factor of 2 numbers

How are we supposed to find these values? Trial and error is a painfully slow technique. In fact, it is "exponentially" more difficult than addition.

Fortunately, there's another numbering system for the natural numbers only - prime factorization - in which these operations can be performed very quickly. Multiplication and division can also be performed more rapidly in factorized form.

What's more startling is that addition, subtraction, and comparisons, which are so simple in place-value notation, become utterly impossible with factorized numbers.

We'll look into this mysterious notation in the next post.