How can I tell whether a variable contains a valid number?

First, you have to define what you mean by "number". The most common case when people ask this seems to be "a non-negative integer, with no leading + sign". Or in other words, a string of all digits. Other times, people want to validate a floating-point input, with optional sign and optional decimal point.

Hand parsing

If you're validating a simple "string of digits", you can do it with a glob:

# Bash
if [[ $foo != *[!0-9]* ]]; then
    echo "'$foo' is strictly numeric"
    echo "'$foo' has a non-digit somewhere in it"

The same thing can be done in Korn and POSIX shells as well, using case:

case $var in
        printf '%s has a non-digit somewhere in it\' "$var"
        printf '%s is strictly numeric\n' "$var"
esac >&2

If you need to allow a leading negative sign, or if want a valid floating-point number or something else more complex, then there are a few possible ways. Standard globs aren't expressive enough to do this, but we can use extended globs:

# Bash -- extended globs must be enabled explicitly in versions prior to 4.1.
# Check whether the variable is all digits.
shopt -s extglob
[[ $var == +([0-9]) ]]

A more complex case:

# Bash / ksh
shopt -s extglob

if [[ $foo = @(*[0-9]*|!([+-]|)) && $foo = ?([+-])*([0-9])?(.*([0-9])) ]]; then
  echo 'foo is a floating-point number'

Optionally, case..esac may have been used in shells with extended pattern matching. The leading test of $foo is to ensure that it contains at least one digit, isn't empty, and contains more than just + or - by itself.

If your definition of "a valid number" is even more complex, or if you need a solution that works in legacy Bourne shells, you might prefer to use an external tool's regular expression syntax. Here is a portable version (explained in detail here), using egrep:

# Bourne
if echo "$foo" | grep -qE '^[-+]?([0-9]+\.?|[0-9]*\.[0-9]+)$'; then
    echo "'$foo' is a number"
    echo "'$foo' is not a number"

Bash version 3 and above have regular expression support in the [[ command.

# Bash
# The regexp must be stored in a var and expanded for backward compatibility with versions < 3.2

if [[ $foo = *[0-9]* && $foo =~ $regexp ]]; then
    echo "'$foo' looks rather like a number"
    echo "'$foo' doesn't look particularly numeric to me"

Using the parsing done by [ and printf (or "using eq")

# fails with ksh
if [ "$foo" -eq "$foo" ] 2>/dev/null; then
 echo "$foo is an integer"

[ parses the variable and interprets it as an integer because of the -eq. If the parsing succeeds the test is trivially true; if it fails [ prints an error message that 2>/dev/null hides and sets a status different from 0. However this method fails if the shell is ksh, because ksh evaluates the variable as an arithmetic expression.

You can use a similar trick with printf, but this won't work in all shells either:

if printf %f "$foo" >/dev/null 2>&1; then
  echo "$foo is a float"

You can use %d to parse an integer. Take care that the parsing might be (is supposed to be?) locale-dependent.