Binary number system

Everyone has heard of the binary numbering system, and everyone knows that it is some kind of “computer language”. But why exactly is that? Why not use the familiar decimal system? What’s the point?

What is the binary system? It is a positional numeral system with base 2. But there is no “two” in it, only 0 and 1, and since there are only two digits, the system is called binary.

Modern digital language consists of zeros and ones, nothing else is needed. The most interesting thing is that the so-called machine code was used by humans long before the machines themselves, and possibly even before numbers appeared.

Why do we need the binary system?

The binary or binomial numbering system is convenient because of its simplicity. With a combination of zeros and ones, any number and any letter can be written, anything can be encoded in this way.

But the main thing is that there are only two values. It’s either “zero” or “one”. The signal is either present or not, the light is on or off, there is a hole or not (on a punched card), the sector is magnetized or demagnetized… The analogies are endless. The main thing is that encoding a signal is simple. There’s no need to create complex mechanisms or devices, just two states are enough.

For example, before humans learned to count and write, signals were transmitted using smoke from a fire or drum beats.

The binary system is simple, nothing simpler exists. Of course, there is also the ancient unitarial system, where there is only one value (for example, only 1), but with it, nothing can be encoded.

In any microchip, a transistor can be in two states: “closed” or “open” (0 or 1), current flows or not.

By the way, Morse code is also a binary code (dot or dash), as is the ancient signaling system — the “optical telegraph”. It’s just the fire of a campfire that can be turned on and off (fire is present or not) at night, and smoke can be used in the day.

Yes, the binary system is used because it’s convenient for encoding information with just two values. But is it convenient for calculations?

How to calculate

How can the binary system be used to write numbers? Just like the decimal system. The simplest example would be a combination lock, like the one on suitcases. Each disk rotates and can take a value from 0 to 9. Imagine that instead of ten digits, there are only two: zero and one.

Since the system is positional, it would look like this:

000000 — zero

Now it’s the number “zero”. To get “one”, you need to rotate the rightmost disc one step.

000001 — one

Now comes the interesting part: how does the number “two” look? Rotate the right wheel… And you get 0 again, because there are no other values. You need to do the same as in the decimal system: move the digit to the left. In the decimal system, this happens when the value exceeds 9, and in binary, it happens right after 1.

000010 — two

000011 — three

000100 — four

One hundred in binary is 1100100.

It’s very interesting how the multiplication table looks in binary:

Easy to remember, right? 0*0=0, 0*1=0, 1*1=1… And that’s it!
All mathematical operations are exactly the same

2+2=4

10+10=100

If you add it column by column, it becomes more obvious:

We add zeros, we get 0, we add two ones, we get zero (rotate the disk twice), and the one is carried over to the right.

As you can see, the math is the same, just the way of writing numbers is inconvenient, too many zeros and ones, which is inconvenient for humans, but for machines, it’s fine.

Just like with digits, you can do the same with letters. The Latin letter “a” will look like 01001010, the Cyrillic “а” — 000011100010111000011001, and even a space — 00010100.

The history of the binary system

It’s clear that humanity has been using binary code for a very long time. Signaling systems with smoke from fires and even the Chinese Book of Changes (700 BC) with its hexagrams have been known for a long time. But the binary code only gained its practical significance quite recently (if we exclude Morse code).

The great Leibniz worked on the binary system in the 17th century, but it was difficult to apply the binary system for any real purpose. At the same time, Pascal created his summing machine (a calculator) using the decimal system. It turned out that counting on such a “calculator” wasn’t that convenient.

Pascal’s summing machine (decimal)

And only in the 1940s, with the advent of the first electronic computing machines, did the binary code reveal its absolute usefulness and beauty. It became the machine language. It’s much easier to record information in it than with the familiar letters and digits.

The same can be done more easily with binary code

We all know the purpose of the binary system today, as everyone has a smartphone in their pocket. In fact, zeros and ones are used far more often than the decimal system, even if we, humans, don’t notice it. It’s no surprise that we’ve been using the binary system throughout history, but only during the machine era did we truly notice it.