We find it handy to count in the decimal system using 10 numbers from 0,9, known as base 10, before we have to put two together to make 10 to keep counting further — 10 fingers and toes and all. But it turns out you can represent all numbers equally using just two digits, known as base 2, a 0 and a 1. This is called the binary system and is credited to Gottfried Wilhelm Liebniz in the 1600s.

The binary system is handy because 1s and 0s can be represented by simply by on/off and reproduced in as simple means as pebbles in trays , the sign of a magnetic field eg positive/negative, or a gate as open or closed. This made them the choice for designing computers and is how digital information is stored and transmitted today.

When we write a decimal number we use positional notation with each successive position representing 10 to the next power. So 739 is understood as (7 x 10^2) + (3 x 10^1) + (9 x 10^0). 10 to the power 2 is 10 x 10, so is the number of 100s in the number. Anything to the power 0 is 1.

The same is true in binary, so 1101 is understood in decimal as:

1101 = (1 x 2^3) + (1 x 2^2) + (0 x 2^1) + (1 x 2^0)

1101 = (1 x 8) + (1 x 4) + (0 x 2) + (1 x 1)
     = 8 + 4 + 0 + 1
     = 13

Each 0 or 1 in a binary number is known as a bit — named by Claude Shannon as short for binary digit — and 8 bits is known as a byte.

Translating to decimal looks fiddly, but computers don't have to do that, they can just add, subtract or multiply the binary numbers directly. Amazing to think that the device you may be reading this on now is just incredibly efficient at manipulating 1s and 0s.

For a readable and visual introduction to the history and operation of computers — from binary, logic gates, transistors, circuits, and Moore's law through to software and AI — you could do a lot worse than my Dad's book The Computing Universe 😀

Binary

The evolutionary psychologist Robin Dunbar identified that an average person’s social network appears to follow simple scaling laws.

Our networks grow from our closest friends—whom we would seek personal advice or help from in times of severe emotional and financial distress—through our superfamily (or close friends), to our acquaintances (or clan), whom we might invite to a party, and to our most casual friends (or tribe).

At each level, the number of people typically increases by about a factor of 3. So you might have 5 close friends, 15 in your superfamily, about 50 in your clan and about 150 in your tribe.

The 150, with some variance, seems to be around the limit for the number of people we can keep track of, reasonably call part of our social network, and maintain as a cohesive group. This may be partly due to the limits of our memories and the effort it takes to maintain relationships.

There are many instances of the ~150 in action, for example, in challenges of transitioning companies as you exceed 150 employees while maintaining cohesiveness, the size of Neolithic farming communities and the basic unit size of Roman armies.

The number is actually 148, but 150 is catchier and seems fine between friends.

Numbers sketches