Addition with 'sed'

If you honestly want to use sed, then this is the way to go:

s/[0-9]/<&/g
s/0//g; s/1/|/g; s/2/||/g; s/3/|||/g; s/4/||||/g; s/5/|||||/g; s/6/||||||/g
s/7/|||||||/g; s/8/||||||||/g; s/9/|||||||||/g
: tens
s/|</<||||||||||/g
t tens
s/<//g
s/+//g
: minus
s/|-|/-/g
t minus
s/-$//
: back
s/||||||||||/</g
s/<\([0-9]*\)$/<0\1/
s/|||||||||/9/; s/||||||||/8/; s/|||||||/7/; s/||||||/6/; s/|||||/5/; s/||||/4/
s/|||/3/; s/||/2/; s/|/1/
s/</|/g
t back

Input:

1+2
100+250
100-250

Output:

3
350
-150

Your mission, should you choose to accept it, is to implement multiplication.

sed isn't the best option here, it doesn't do arithmetics natively (see Increment a number for how you could possibly do it though). You could do that with awk:

$ echo 12 | awk '{print $0+3}'
15

The best piece of code to use will depend on the exact format of your input and what you want/need to do if it is not numeric, or contains more than one number, etc.

You could also do this only with bash:

$ echo $(( $(echo 12) + 3 ))

or using expr in a similar fashion.

I tried to accept your challenge @Richter, this is what I did using part of your code:

sed 's/[0-9]/<&/g
s/0//g; s/1/|/g; s/2/||/g; s/3/|||/g; s/4/||||/g; s/5/|||||/g; s/6/||||||/g
s/7/|||||||/g; s/8/||||||||/g; s/9/|||||||||/g
: tens
s/|</<||||||||||/g
t tens
s/<//g
s/.*\*$/0/
s/^\*.*/0/
s/*|/*/
: mult
s/\(|*\)\*|/\1<\1*/ 
t mult
s/*//g
s/<//g
: back
s/||||||||||/</g
s/<\([0-9]*\)$/<0\1/
s/|||||||||/9/; s/||||||||/8/; s/|||||||/7/; s/||||||/6/; s/|||||/5/; s/||||/4/
s/|||/3/; s/||/2/; s/|/1/
s/</|/g
t back'

Input:

04*3
4*3
40*3
42*32
150*20
1*3
3*1
0*3
3*0

Output: all the correct results