Scientific Notation and 10

Question

When writing a number such as $65$ in scientific notation, is it ok to leave off the exponent on the ten? I.e. $6.5 \times 10$?

Answer 1

In the slightly unusual scenario where you want to write this number, I think it should be $6.5\times10^1$. Leaving off the $^1$ would cause me (and probably most others used to seeing scientific notation) to parse the expression as a mathematical expression rather than a number. If you're unhappy with that, just write $65$.

It's not that unreasonable to sometimes write a number like $65$ in scientific notation. For instance, if I were writing a table with numbers with a large dynamic range in a column, I might write them all in scientific notation to preserve a nice alignment. I'd even write $1.2\times 10^0$, if relevant. I've found that since I got a good bit of practice reading numbers in this format, I don't even notice any more and my brain simply acquires the number without effort.

Answer 2

The use of scientific notation makes writing very large and very small numbers much easier.
However scientific notation is also very useful when one is concerned with the accuracy of a numerical value and in particular to show which zeros are significant.

$200$ can be written in scientific notation as $2\times 10^2$ or $2.0\times 10^2$ or $2.00\times 10^2$ to give an indication of the accuracy to which the numerical value is quoted.

What about your value of $65$?
That is different in meaning from a value quoted as $65.0$ and $65$ by itself will clearly indicate that it is the $5$ which the least significant digit.
So using $\times 10^1$ is not necessary for $65$ however $60$ is a different matter if you really want to show whether or not the zero is significant.
It might be that with $60$ as part of a table with other values present there is no need to use scientific notation but to be sure you need to use $6.0\times 10^1$.