Something I found while reading about base systems, I can’t really wrap my head around it. Maybe it has to do with fractions?
In: 41
Uh, what do you mean?
Base 2 is not base ten.
000 = 0
001 = 1
010 = 2
011 = 3
100 = 4
101 = 5
110 = 6
111 = 7
How is that “base ten?”
edit: Man, this is an incredibly stupid joke.
It’s a joke, because every number written in its own base comes out as “10”, one-zero. It’s just not always ten.
What’s 2 in base two? 10
What’s 3 in base three? 10
What’s 16 in hex? 10
So if you ask what the base is of the number system you’re using, and they answer in that number system, the base is always “10”
The joke is, how do you write “2” in base 2? You write it 10. (One two and no ones.) Thus, it is base “10.”
How do you write “3” in base 3? You write it 10. (One three and no ones.) Thus it is base “10.”
Etc.
In hexadecimal you don’t write 16, you write 10. In binary, you write 10 to represent 2. Regardless of the base, it’s always represented as 10 internally.
The choice of base in a counting system influences how we write that system. This means that “base 10” is kind of meaningless, because any counting system would describe itself as “base 10”.
For example, consider an alternate world that always had a base 11 counting system. That is, instead of “10” they had a special symbol for 9+1. Maybe it would be &. Then when they add 1 to &, they have reached the base of the counting system. That means they place a 1 in a new column to the left and start the first column over with a zero. Hence &+1=10. People in this world would just say that they have a “base 10” counting system and that people from our world are using “base &”. If we started talking about “base 11” they would think we were talking about what we think of as base 12, and so on.
The way out of this is an unspoken agreement to always use (the traditional) base 10 when describing the base of a counting system. This is fair and intuitive in a world where everyone is originally taught to count in base 10, but it would lead to a lot of confusion if we ever encountered folks who considered some other base to be natural.
You can convert any rational number from any natural number base to any other exact. Decimal does not have a special place.
It is not a way easy to convert from a fraction like you can with 1/10 = 0.1, 1/3 =0.33333 and it continues forever. The denominator has to have the same prime factor as the base to convert like 1/10 = 0.1. So conversion will be easier if two bases have the same prime factors. It is simpler between base 10 and base 20 than base 10 and base 9
It is possible to have a number base that is not a natural number like a base of pi. The number 10 in base pi will be pi in the base then and you cant write it as a fraction.
So there is nothing special about base 10, you can convert between any base you like.
Base is the amount of numbers in the first digit before you start the next set of digits. 0-12 = base 12, but 12 would be written as 10 in that system. It really is a shame we don’t use base 12.
Any base uses 0-(Base minus 1) as their first digit. Base 10 is 0-9. Base 2 is 0-1. Base 16 is 0-15 (represented by F).
Because of this, the base would always be the first 2-digit number representation. For base 10, 10 is written 10. For base 2 (binary), 2 is written 10. For Base 16 (Hex), 16 is written 10. and so on.
It is a joke, because written we normally write things you would end up writing 10 for the base number in that base for all bases.
16 in base 16 is written 10
8 in base 8 is written 10
2 in base 2 is written 10
Of course this only works for the way we write things down normally, with the least value digit on the right and each digit representing a value that is the base to the positions power and where 1 means one of something and 0 means none, but since we mostly do it mostly works.
to put it in more ELI5 terms.
The number to the right says how many ones you have.
The number left of it says how many of the base you have. In our normal decimal system it says how many tens you have.
In any other base it would say how many of that base you have.
10 always means one of the base this number is written in. If you write out the base in its own system it will always look like ’10’.
You have to read it as “base one zero”, not “base ten”.
Regardless of what number you pick as your base, in that base, it will always be written as “10”. In binary, you’d say “base two”, but in that binary code that nmumber two is written as “10”. Same for all other numbers.
That’s not coincidence, but basically the definition of a base. The number your number system is “based on” is defined as the first number for which you need a second digit, making the first digit a “1” (i.e. how often you counted to the base) and resetting the last digit to “0” (how much your remainder counts). In base 10, “10” simply means “one times ten plus zero times 1), in base 2 the same “10” means “one times two plus zero times one”.
A mindblowing example (for beginners) is hexadecimal (base 16). You write all numbers from 0 to 9 exactly as you know from base 10. Then you continue from 10 to 15 writing letters, i.e. A, B, C… until F. Finally, when you reach the sixteenth number, you exhaust the pool of available digits in base 16, so you need to add a new digit, so the number sixteen is written as “10” in hexadecimal (one times 16 plus zero). So in Hexadecimal, 9+7=10, but 10+7=17.
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