$\begingroup$There are different categories of numbers that we use every day.
Integers that written in decimal notation have $1, 2$ or $5$ as the leading figure, followed by none, one or more zeros. These are very common numbers, e.g. used in Bank notes: $1, 2, 5, 10, 50, 200, 1000, ....$
Other intergers, which are less common, e.g. your (approximated) height in $cm$, the (approximate) temperature of your body or the environment, or the (approximate) result of converting $200$ miles to kilometers.
Numbers that are not integers, such as $\$1.23$.
If you have $\$198$ (category $2$) bill in a restaurant, you'd probably tip $\$2$ to round it to $\$200$ (category $1$).
Is there a term for numbers of category $1$?
$\endgroup$155 Answers
$\begingroup$The first category is known as the 1-2-5 series, and it is an example of a system of preferred numbers. As the name implies, there is nothing mathematically distinctive about such numbers; humans just prefer them.
$\endgroup$2$\begingroup$The OEIS calls your first set a "Hyperinflation sequence for banknotes", though there have been many coins and banknotes with different denominations around the world, such as a $1935$ Canadian $\$25$ note.
Mathematically they are the numbers generated by $$\frac{1+2 x+5 x^2}{1-10 x^3}.$$
People have counted with other patterns, notably the Babylonian sexagesimal system which we still use for minutes and seconds and so often think of $15$ and $30$ as round in some contexts.
$\endgroup$6$\begingroup$I believe the most easily understood term for them is numbers with one significant figure.
Preemptive argument against pedantic people:
While it is true that "with one significant figure" can be used to describe a value and does not serve the asker's purpose in that case, it can also be used to refer to number and when it does, it does serve the asker's purpose. How clear one must be to ensure that everyone knows you are referring to numbers, not values, depends on the context the phrase is used in.
Secondly, one might say you need to specify the number system when speaking of significant figures, however, if you don't, in most contexts it can be assumed you are talking about decimal numbers.
$\endgroup$1$\begingroup$When I looked at these numbers, the first thing that occurred to me was "These are numbers that divide a power of the base (10, in this case)." Looking again, this description is more general than your case 1 (because it includes values such as 25), but seems to me to capture much of the idea of a number being "round".
Anyway, that's my 50 cents worth.
$\endgroup$$\begingroup$In regards to whatr marty wrote:
It is not that these numbers can be divided a power of the base 10. (well yes, but that is hot their definition)
The definition should be:
Ten to the power of (n-1), where n is the numbers of digits of the numbers divides the numbers in categorie 1.
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