“Base” is the number of distinct integers you have in play. In Base 10, there are ten of them. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. You can think of the numeric representation 10 as “1 ten, and 0 ones.”
In Base 2 (binary) the only two digits available are 0 and 1. The first four binary numbers are 0, 1, 10, 11, which represent zero, one, two, and three. In Base 2, “10” means “1 two, and 0 ones.” But, “Base 2” can’t be written in binary, there’s no concept of 2! Indeed, the way we reflect two in binary is 10. Which means, when we’re talking in binary, “Base 2” is written as “Base 10.”
This holds true for EVERY base. In Base 4, we have the digits 0, 1, 2, and 3. So if we want a value of four, we need to write it as 10. “1 four, 0 ones”. So, when we’re talking in Base 4, the way to say “Base 4” is ALSO by saying “Base 10”!
The trick behind it is that numbers written don’t have context-free meaning. You can’t communicate what “10” means without knowing how many distinct digits your conversational partner is working with. Most people have centralized on base 10, but there’s no inherent advantage to doing things that way. Indeed, it’s kind of awkward in lots of ways. Consider Base 12 (the digits of which are most often 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, as an aside). In Base 12, you can easily divide your base numbers by 1, 2, 3, 4. That’s SUPER handy, since we obviously break things up into groups of 3 and 4 pretty often in our daily lives, but that’s pretty painful in Base 10 because you immediately run into the need for fractions.
“Base” is the number of distinct integers you have in play. In Base 10, there are ten of them. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. You can think of the numeric representation 10 as “1 ten, and 0 ones.”
In Base 2 (binary) the only two digits available are 0 and 1. The first four binary numbers are 0, 1, 10, 11, which represent zero, one, two, and three. In Base 2, “10” means “1 two, and 0 ones.” But, “Base 2” can’t be written in binary, there’s no concept of 2! Indeed, the way we reflect two in binary is 10. Which means, when we’re talking in binary, “Base 2” is written as “Base 10.”
This holds true for EVERY base. In Base 4, we have the digits 0, 1, 2, and 3. So if we want a value of four, we need to write it as 10. “1 four, 0 ones”. So, when we’re talking in Base 4, the way to say “Base 4” is ALSO by saying “Base 10”!
The trick behind it is that numbers written don’t have context-free meaning. You can’t communicate what “10” means without knowing how many distinct digits your conversational partner is working with. Most people have centralized on base 10, but there’s no inherent advantage to doing things that way. Indeed, it’s kind of awkward in lots of ways. Consider Base 12 (the digits of which are most often 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, as an aside). In Base 12, you can easily divide your base numbers by 1, 2, 3, 4. That’s SUPER handy, since we obviously break things up into groups of 3 and 4 pretty often in our daily lives, but that’s pretty painful in Base 10 because you immediately run into the need for fractions.