Question

Homework question: Mentally convert the base ten numeral 10 to a base five numeral. Image in comments.

Answer 1

Base 10 Base 5
0 0
1 1
2 2
3 3
4 4
5 10
6 11
7 12
8 13
9 14
10 ?

Answer 2

Here's another way of thinking of it.
In base 10, there are 10 different digits.
In base 5 there are five different digits.
The number you want to write in base 5 happens to be twice the number of digits in base 5 numbers.
What number is twice the number of digits in base 10 numbers?

Answer 3

A way to think about it is through the basis representation theorem, which states that an integer n in base 10 can be expressed through the formula: n = a_0(k^s)+a_s(k^(s-1))...
where k represents the base that you are inputting into your head. a_0, a_1,... a_n are constants which are put together to form a number. An example makes this more clear.

Like 10, lets say the number you want to represent is 5 in base 10 currently and convert it to base 5. number n is therefore 5 and base k is 5. 5 = a_0(5^0)+a_1(5^1) and the max number s you can exponentiate to is determined by how big your number is. a_1 = 1 and a_0 = 0, and the number is represented by (a_s)(a_s-1)..., therefore it is 10. Try it out and solve the equation in your head for base 10 numbers in any other base.