Is '$10$' a magical number or I am missing something?

It's a hilarious witty joke that points out how every base is '$10$' in its base. Like,

\begin{align} 2 &= 10\ \text{(base 2)} \\ 8 &= 10\ \text{(base 8)} \end{align}

My question is if whoever invented the decimal system had chosen $9$ numbers or $11$, or whatever, would this still be applicable? I am confused - Is $10$ a special number which we had chosen several centuries ago or am I missing a point?

Solution 1:

Short answer: your confusion about whether ten is special may come from reading aloud "Every base is base 10" as "Every base is base ten" — this is wrong; not every base is base ten, only base ten is base ten. It is a joke that works better in writing. If you want to read it aloud, you should read it as "Every base is base one-zero".

You must distinguish between numbers and representations. A pile of rocks has some number of rocks; this number does not depend on what base you use. A representation is a string of symbols, like "10", and depends on the base. There are "four" rocks in the cartoon, whatever the base may be. (Well, the word "four" may vary with language, but the number is the same.) But the representation of this number "four" may be "4" or "10" or "11" or "100" depending on what base is used.

The number "ten" — the number of dots in ".........." — is not mathematically special. In different bases it has different representations: in base ten it is "10", in base six it is "14", etc.

The representation "10" (one-zero) is special: whatever your base is, this representation denotes that number. For base $b$, the representation "10" means $1\times b + 0 = b$.

When we consider the base ten that we normally use, then "ten" is by definition the base for this particular representation, so it is in that sense "special" for this representation. But this is only an artefact of the base ten representation. If we were using the base six representation, then the representation "10" would correspond to the number six, so six would be special in that sense, for that representation.

Solution 2:

The magic of the number 10 comes from the fact that "1" is the multiplicative unit and "0" is the additive unit. The first two-digit-number in positional notation is always 10 and also always denotes the number of digits.

Solution 3:

Yes, ten ( ..... ..... ) is a special number. Not magical but special because it is a very convenient base for species that have ten fingers.

Arguably we can use hands and fingers to encode 1024 numbers using the binary system, but that would be less robust across reading directions and some configurations/gestures are physiologically hard to do.

Solution 4:

I do not accept your concept of "1-0" as being a number.

The 1-0 you are using is a notation used on different numbers. So, as special the number 10decimal is, the notation 1-0 is not a special number.

To me, it is a special notation.

1-0 is the notation for the number 10decimal.
1-0 is the notation for the number 2binary
1-0 is the notation for the number 8octal
1-0 is the notation for the number 12radix12
1-0 is the notation for the number 13radix13
1-0 is the notation for the number 14radix14
1-0 is the notation for the number 15radix15
1-0 is the notation for the number 16hexadec

So, calling number 10dec a special number because the notation 1-0 is special would be akin to expressing the correlation

cows eat corn. cows are stupid.
Mary eats corn. And therefore, Mary is stupid.

However, you could say that the notation 1-0 denotes a number that is special within each radix. That is saying that every number is a special number in the set of all radix systems.

• There are innumerable radix systems.
• There are innumerable numbers.
• A radix system is denoted by radix(n)
• where n is a special number within the set of numbers in radix(n) because it is denoted by the notation 1-0radix(n)
• Therefore, every number is a special number within the radix denoted by that number.
• So is the notation 1-0-0 special, as is the notation 1-0-0-.......-0

The notation 1-9 is also a special notation, for all radix systems greater than radix(8), because it signifies the special occasion when the number mutates from 1-8 to 1-9 or from 1-A to 1-9

In fact, every notation member of the sets of all possible notations is special, by the virtue that that notation signifies a transition from a lesser value to a greater value, vice versa.

The notation A is also special notation, for all radix systems greater than radix(9). Because it signifies the transition from a numeral digit procession to an alphabetic procession.

Therefore, the number 10dec is indeed a special number not by the virtue of the notation 1-0, but by the virtue of the notation A. Because for all radix systems greater than radix(10), the value 10dec is always denoted by the special notation A. Where A is special because it is a consequence of the end of numeric digit procession into an alphabetic one.

That is like every parent in the world saying "My kid is special".