Fahrenheit / Celsius temperature converter
I don't know enough bash to offer you a technical review.

But, more philosophically: Have you considered writing this more in the style of a "program" than a "Press 1 to convert another"-type interface? Like:

~$ temperature c 30
86 ˚F
~$ temperature F 105
40 ˚C


My reasoning is that the above is easier and more flexible to use programatically, you don't need to write input loops at all and you can condense down to your core functionality:

#!/bin/bash

case $1 in
    [Ff] ) echo "$(( ($2 - 32) * 5 / 9 )) ˚C"; break;;
    [cC] ) echo "$(( $2 * 9 / 5 + 32 )) ˚F"; break;;
    *    ) echo "Usage: $0 c|f num"
esac