Professor Sampler’s Notes: Decimal, Binary and Hex

Topics Covered:

  • The decimal system
  • The binary system
  • The hexadecimal system

A good decision is based on knowledge and not on numbers – Plato

Since humans don’t seem to get along too well with each other, scientists had to find symbols that are ‘almost’ universally agreeable to all. These groups of symbols are called numbers by third graders. Signal processing engineers call them numbers too, because that’s what they were taught in third grade.

But not all numbers are the same. The ones we use everyday fall under the decimal system. But that’s not what is used in electronic equipment. It’s sort of like how the alphabets are arranged on a keyboard – not in the order we were taught in school, right? Are we forever doomed to perform excellently at the task of making simple things as complicated as possible? Not really. There’s a method to this madness.

Ten fingers
Where do you think the decimal system came from?

One would have thought the engineers would use a number system that we all recognize, our beloved decimal system. After all, our bank accounts, SAT scores and batting averages run on this system. But no, they preferred two of the most complicated number systems known to man – the binary system and the hexadecimal system.

Numbers in the binary system: 0 1
Numbers in the decimal system: 0 1 2 3 4 5 6 7 8 9
Numbers in the hexadecimal system: 0 1 2 3 4 5 6 7 8 9 A B C D E F

Why on earth would anybody be so mean to the rest of the universe? Actually, they were being kind. These systems might seem like busy work to the rest of us, but they are brilliant and elegant solutions to complicated challenges inherent in the marriage of analog and digital systems.

E.g., hexadecimal systems allow us to use lesser characters to describe the same thing. The number 15 needs two characters: 1 and 5, and you’ll have to reserve two seats on the bus for both of them. But if you use the hexadecimal system, all you need is the letter F. The hexadecimal system is a great compromise between complexity and convenience in the digital world. You don’t think so? Imagine having to report a pathetic SAT score of 161. In hexadecimal it’s A1. That sounds about right.

But what about binary systems? 15 in binary is 1111, and our SAT score becomes 10100001. That means more seats on the bus. What gives?

The answer to that is Aristotle’s law of identity. A thing is, or it isn’t. All number systems were created for convenience, but the only number system that seems to agree with nature is the binary system. It’s almost as if they speak the same language, or made of the same ‘stuff’, if you will. But when have humans been known to bow down to nature? Only when we are forced to, and in this case, we are. Luckily for us, it makes us look good, too.

Electronic systems can communicate perfectly when they speak the same language. We already know analog systems don’t do this perfectly (though most of the time they do a pretty good job!), so if digital systems had the same imperfections then we might as well stick to analog. It has to be all or nothing.

By using the binary system, we are able to encode analog data in a digitized form, and at the same time force electronic equipment (read: electrons and waves, or the wave function) to behave as we want them to. Think of it as musical notation – everyone speaking the same language and getting along together, or so we hope.

Takeaways:

  • The binary system is a compromise between the strengths of conductors and semiconductors and how humans think.
  • The hexadecimal system, complicated as it seems, saves a lot of space and makes calculations easier for machines.

Links for further study:

Next: Professor Sampler’s Notes: Analog vs Digital
Previous: Professor Sampler’s Notes: Digitization