This is a bit of a grey area. You need to recall that R will always invoke a print method, and these print methods listen to some options. Including 'scipen' -- a penalty for scientific display. From help(options):

‘scipen’: integer. A penalty to be applied when deciding to print numeric values in fixed or exponential notation. Positive values bias towards fixed and negative towards scientific notation: fixed notation will be preferred unless it is more than ‘scipen’ digits wider.

Example:

CopyR> ran2 <- c(1.810032e+09, 4) 
R> options("scipen"=-100, "digits"=4)
R> ran2
[1] 1.81e+09 4.00e+00
R> options("scipen"=100, "digits"=4)
R> ran2
[1] 1810032000          4

That said, I still find it fudgeworthy. The most direct way is to use sprintf() with explicit width e.g. sprintf("%.5f", ran2).

The "e" is a symbol for base-10 scientific notation. The "e" stands for $\times 10^{\rm exponent}$. So -1.861246e-04 means $-1.861246 \times 10^{-4}$. In fixed-point notation that would be -0.0001861246.

This notation is pretty standard. Even Microsoft Excel understands it, not just R.

It's the textual version of 10 to the power of. e.g.

          -02
8.131 x 10     == 8.131e-02

2.2e-16 means $ 2.2 \times 10^{-16}$. It is a shorthand convention for scientific notation hanging over, I think, originally from Fortran programming language.

To print the number you can treat it as a string and use sub to reformat it:

changeSciNot <- function(n) {
  output <- format(n, scientific = TRUE) #Transforms the number into scientific notation even if small
  output <- sub("e", "*10^", output) #Replace e with 10^
  output <- sub("\\+0?", "", output) #Remove + symbol and leading zeros on expoent, if > 1
  output <- sub("-0?", "-", output) #Leaves - symbol but removes leading zeros on expoent, if < 1
  output
}

Some examples:

> changeSciNot(5)
[1] "5*10^0"
> changeSciNot(-5)
[1] "-5*10^0"
> changeSciNot(1e10)
[1] "1*10^10"
> changeSciNot(1e-10)
[1] "1*10^-10"

It's not ugly, but exactly what's expected. If you type

$2x-10$

then you expect that there is some space around the minus sign, because it denotes an operation. When you type $1e-10$, TeX interprets it in exactly the same way, because it can't read your mind: the two expressions are formally the same, only two symbols are different.

If you want that an expression that's normally interpreted as a polynomial should be treated in a different way, then you have to properly mark it.

One solution might be

$1\mathrm{e}{-10}$

because in this case the braces around -10 tell TeX to enter a subformula and so the minus sign is initial, so not interpreted as a binary operation, but as a unary operator.

You could make a definition, such as

\newcommand{\expnumber}[2]{{#1}\mathrm{e}{#2}}

and input the number as

$\expnumber{1}{-10}$

but there's a much better alternative, the package siunitx.

\documentclass{article}

\usepackage{siunitx}
\sisetup{output-exponent-marker=\ensuremath{\mathrm{e}}}

\begin{document}

\num{1e-10}

\end{document}

This package offers many more features than just printing numbers in the desired format; consult its documentation to find them.

Note that siunitx is not understood by MathJax, so with it you must stick to the “hand made” solution. You can still say, in it,

$\def\num#1{\numx#1}\def\numx#1e#2{{#1}\mathrm{e}{#2}}$ 

and a formula such as $\num{1e-10}$ will be printed in the way you want.

The R expression

exp(1)

represents e, and

exp(2)

represents e^2.

This works because exp is the exponentiation function with base e.