Math | CS | bciyh | Philosophy

Binary Systems

It is no secret that at the bottom of all computing abstractions lies the Binary.

I believe that Binary Systems lie at the heart of the Universe and moreover, that we have Quatum Mechanics because the universe is Binary.

Grokking this system is probably the most important thing that you will need to do today. So I say put down whatever you are doing. It can wait. Getting Binary is priority.

Distinguishing

Think about this…

        What Makes One Thought Different From Another?

If you think about it hard, I think you will find that at a very minimum, the difference between any two thoughts can only be the difference between a state being is one of two positions, call them On & Off.

The plain idea, in order

Binary is simply counting with two symbols instead of ten. We write numbers in base ten because we have ten fingers; a computer “counts” with switches that are either on or off, so it uses base two: 0 and 1. A single such switch is a bit; eight of them make a byte, which can hold 256 different patterns (2⁸). That is the whole foundation on which every photo, song, program and message is built.

Why it matters

Underneath every abstraction, numbers, text, colour, sound, instructions, is a pattern of bits. Grasp binary and the machine stops being magic: you see why a byte maxes out at 255, why colours are written like #FF8800, why “kilobyte” is sometimes 1000 and sometimes 1024, and how the same pattern of bits can be a number, a letter, or a command depending only on how it is read.

An everyday analogy: a row of light switches

Picture a row of light switches on a wall. Each is up or down. With three switches you can make eight distinct patterns (down-down-down, down-down-up, …). Add a switch and you double the patterns. That doubling with each new bit is why small numbers of bits reach huge numbers so fast: ten switches already give 1,024 patterns, thirty-two give over four billion.

A worked example: reading and writing binary

Each position is a power of two, doubling from right to left: 1, 2, 4, 8, 16, 32, 64, 128. To read a binary number, add up the positions that are “on”:

 1 0 1 1 0 1
32 .  8 4 . 1   ->  32 + 8 + 4 + 1 = 45

To go the other way, subtract the largest power of two you can, repeatedly: 45 − 32 = 13, 13 − 8 = 5, 5 − 4 = 1, 1 − 1 = 0, which lights up the 32, 8, 4 and 1 positions: 101101. Counting up in binary, 0, 1, 10, 11, 100, 101, is the same “carry when you run out of symbols” trick you learned in base ten, just with the roll-over happening at two instead of ten.

Misconceptions

• “Computers understand binary because it’s natural.” It is a practical choice: a switch reliably tells two states apart far more cheaply than ten voltage levels. Base two is engineering, not metaphysics.
• “Binary and hexadecimal are different worlds.” Hex (base 16) is just shorthand for binary, one hex digit stands for exactly four bits, which is why programmers use it.
• “A byte is always a character.” A byte is 256 patterns; what they mean, a number, an ASCII letter, part of a UTF-8 character, is a matter of interpretation. See Unicode in C.

A note on the big claim above

My intro makes a bold leap, that the universe is binary and that is why we have quantum mechanics. I will be honest that this is my own speculation, not established physics. There is a real and fascinating idea nearby, John Wheeler’s “it from bit” and the information-theoretic view of physics, and quantum computing really does rest on the qubit, which (unlike a bit) can be in a blend of 0 and 1 until measured. But that is almost the opposite of a simple on/off universe. Take my opening as enthusiasm, and the arithmetic below as solid ground.

How it connects

• Mathematics: number bases and powers.
• Unicode in C and Tokens: how bits become text.
• Computing and bciyh: building up from the bottom.

The truth revealed: every towering abstraction in computing rests on the humblest of facts, a switch that is either on or off, and learning to read those switches is learning to read the machine itself.

Sources and Links

• Binary number - Wikipedia
• Code, by Charles Petzold - the classic book on this idea