# Packing Decimal Numbers Easily

While desktop computers have tons of computing power and storage, some small CPUs don’t have a lot of space to store things. What’s more is some CPUs don’t do multiplication and division very well. Same can be said for FPGAs. So suppose we are going to grab a bunch of three-digit decimal numbers from, say, a serial port. We want to store as many as we can, and we don’t want to do a lot of math because we can’t, it is slow, or perhaps it keeps our processor awake longer and we want to sleep to conserve power. We want a way to pack the numbers as close to the theoretical maximum as we can but with little or no math.

The simple approach is to store the numbers as ASCII. Great for processing since they are probably in ASCII already. If they aren’t, you just add 30 hex to each digit and you are done. That’s awful for storage space, though, since we can store 999 in 10 bits if it were binary and now we are using 24 bits! Storing in binary isn’t a good option, if you play by our rules, by the way. You need to multiply by 10 and 100 (or 10 twice) to get the encoding. Granted, you can change that to two shifts and an add (8x+2x=10x) but there’s no easy way to do the division you’ll have to do for the decode.

Of course, there’s no reason we can’t just store decimal digits. That’s call binary coded decimal or BCD and that has some advantages, too. It is pretty easy to do math on BCD numbers and you don’t get rounding problems. Some CPUs even have specific instructions for BCD manipulation. However, three digits will require 12 bits. That’s better than 24, we agree. But it isn’t as good as that theoretical maximum. After all, if you think about it, you could store 16 distinct codes in 4 bits, and we are storing only 10, so that 6 positions lost. Multiply that by 3 and you are wasting 18 codes.

But there is a way to hit that ten-bit target without doing any math. Its called DPD or densely packed decimal. You can convert three decimal digits into ten bits and then back again with no real math at all. You could implement it with a small lookup table or just do some very simple multiplexer-style logic which means it is cheap and easy to implement in software or onboard an FPGA.