# $10-e$ interesting decimal expansion property

The number $$e$$ has several interesting properties but I noticed something that may just be coincidence; a neat coincidence however!

Here is the decimal expansion of $$e$$.

$$e=2.718281828459045235360\dots$$

And here is the decimal expansion of $$10-e$$

$$10-e=7.281718171540954764639\dots$$

Notice that there are a lot of similar digits in the beginning in fact if you look closely at the first 16 digits, each pair of neighbor digits in $$e$$ are the same as each of the other number’s corresponding pair of neighbor digits only they’re swapped. There are only two exceptions within those First 16 digits but it still seems like a significant coincidence, if it is one.

Here you can tell more clearly. $$\boxed{2.7}\boxed{18}\boxed{28}\boxed{18}\boxed{28}\boxed{45}\boxed{90}\boxed{45}235360\dots$$

$$\boxed{7.2}\boxed{81}\color{red}{\boxed{\color{black}{71}}}\boxed{81}\color{red}{\boxed{\color{black}{71}}}\boxed{54}\boxed{09}\boxed{54}764639\dots$$

The red boxes show the exceptions where it would work if we just add one to each of their digits.

Is there any way of explaining this besides coincidence? I haven’t been able to find any discernible patterns after these first 16 digits.

• $9.99999999\dots = 10$. Also, $(10-e)+e=10$. Note that the $n$'th decimals added to eachother always add to $9$ as a result. The boxes you highlight just happen to be the positions where consecutive digits add up to nine or add up to ten. It just so happens that the first several pairs do so. If i'm not mistaken, it is believed that $e$ is a normal number but has not been proven yet. Dec 29, 2017 at 16:39
• This is no coincidence! Add each of the pairs of numbers and you will get $99$. If you notice the entire sum, it will be $9.999999... = 10$ Dec 29, 2017 at 16:39

This works given that in $e$ we are getting several pairs of digits that add up to $9$, nothing more.

That is, let's take some random decimal number, making sure that we get pairs of digits adding up to $9$, e.g:

$$4.637236091881....$$

Subtract from $10$

$$10-4.637236091881....=5.362763908118...$$

And you get the same result!

• You’re right. Wow that should have occurred to me Dec 29, 2017 at 16:39

It so happens that some pairs of $e$ after the decimal place add up to $9$. The pairs highlighted in red don't work due to some carrying in later digits.