$N × 10^a$ notation

Question

Today our teacher taught us the $N × 10^a$ notation. He specified that $N$'s value must be $1 ≤ N < 10$ and that $a$ must be an integer.

I understood that much but I was wondering, why can't $N$ be greater than $10$?

Note: we learnt it as part of our IGCSE Physics class

Answer 1

This is because Scientific Notation's purpose was to show how large a number just by glancing at the exponent rather than counting all the zeros. Trying to calculate the rough size of a number is harder if $N>10$. For example, imagine trying to figure out how large $20000\cdot10^{13}$ as opposed to $2\cdot10^{17}$.

Answer 2

The only reason to require that $N<10$ is that if it isn't you could increase $a$ instead. So it's simply a convention giving a unique way tyo write every number using that notation. Without it, we could write $2\times 10^5$, $20\times 10^4$, $200\times 10^3$ and so on, all representing the same number.

Answer 3

It can be but the whole point of the $N \times 10^a$ notation is to show the order of magnitude with $a$ so we choose to make the leading digit in that range.

Answer 4

The convention that $1 \le N < 10$ means you can use scientific notation to specify the precision of a number. Then $1.0 \times 10^2$ represents a number known to be between $50$ and $150$ while $1.00 \times 10^2$ means between $95$ and $105$. You would lose that ability if you allowed $100 = 1 \times 10^2 = 10 \times 10^1$.