There has been a lot of talk about what base we should count in¹²³. The most compelling case so far in my opinion has been made by the best way to count in their video the best way to count. Though a bit on the nose title-wise, they make a great case for the use of binary.
Indeed binary is simple. The simplest base in fact. When it comes to arithmetics, binary is notoriously known for making calculation almost trivial.

That’s great on paper but there is a major issue with changing our way to count to any other base: change is hard. Most people won’t change their ways of doing stuff – like the old adage says: “if it ain’t broke: don’t fix it”.
But all is not lost. There are among us people willing ot learn and change their ways: children. But to teach them we have to starts with the basics. Counting seems like as great way to start.

Counting in general

Counting is the process of going from one number to the next. To do so we increment the number by one. In this process we first set the unit digit to the next digit of the base (e.g. from 4 to 5), unless it is the last digit of our base (i.e. 9 in decimal). In which case the units “rolls over”, meaning it goes to the lowest digit of the base (i.e. 0 in decimal), and we increase the next digit (usually to the left) following the same operations.
For example to go from 99 to 100 in decimal, the unit 9 is increased which rolls over which in turn increases the next 9 which also rolls over. This, in turn, increases the next digit from 0 to 1 – as there is an infinity of 0s to the left of the number they are made implicit.
Counting like this has the neat consequence that the next digit in a number represent a quantity b times greater than the previous one where b is the base. In other words, the first digit (i.e. unit) represents a quantity of itself times 1, then the next one of itself times b, the next itself times b², then b³, etc.

Counting in binary

This process of counting is thus applicable to base 2 otherwise known as binary. In binary, the digits are often represented by 0 and 1 – though it could be any 2 symbols, we just use those because of the convention of the decimal system.
Fig. 1 Shows and example of how to count in binary.

╔════════════════╗ 
║                ║ 
║       0 (0)    ║ 
║       1 (1)    ║ 
║      10 (2)    ║ 
║      11 (3)    ║ 
║     100 (4)    ║ 
║     101 (5)    ║ 
║     110 (6)    ║ 
║     111 (7)    ║ 
║    1000 (8)    ║ 
║                ║ 
╚╤═══════════════╝ 

Same consequence as before each digit (or bit) represent a quantity of power of 2. But as digit are either 0 or 1, each bit indicate the presence or absence of this power of 2 from the total. for example 5 in binary is 101, which is 1 * 2² + 0 * 2¹ + 1 * 2⁰ which when simplified is 2² + 2⁰ or 4 + 1 – indeed equal to 5.

Finger binary

To help children count to 10, we teach them to use their hands. The quantity of finger raised is equal to the number. We can use the same trick to teach binary. The idea is that each finger represents a bit, when down it’s a 0 when raised it’s a 1.
Fig. 2 shows the value of the finger for just one hand finger binary.

╔══════════════════════════════════════╗
║╭──────┬─────┬────┬──────┬─────┬─────╮║
║│finger│pinky│ring│middle│index│thumb│║
║├──────┼─────┼────┼──────┼─────┼─────┤║
║│power │  2⁴ │ 2³ │  2²  │  2¹ │  2⁰ │║
║├──────┼─────┼────┼──────┼─────┼─────┤║
║│value │ 16  │ 8  │  4   │  2  │  1  │║
║╰──────┴─────┴────┴──────┴─────┴─────╯║
╚╤═════════════════════════════════════╝

We can see that with only one hand it’s possible to count to 31 (16 + 8 + 4 + 2 + 1) – if one uses both hands it could go up to 1023!
Fig. 3 to Fig. 7 show how to count from 0 to 4 with finger binary.

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Oh that’s... really unfortunate.

You know what? Forget about it. Base 10 is alright⁴.