What That Mean "In Base 18"?

Question

i am a programmer who interest in math , lately in palindromic numbers , so if it's stupid question i am sorry !

i was reading about palindromic numbers in wikipedia , at some point it says

In base 18, some powers of seven are palindromic:

 - 7^3 =     111
 - 7^4 =     777
 - 7^6 =   12321
 - 7^9 = 1367631

What that mean , In Base 18 , Thanks in advance

Answer 1

For example, $12321_{18}=1\cdot 18^4+2\cdot 18^3+3\cdot 18^2+2\cdot 18+1=117649_{10} \\ =1\cdot 10^5+1\cdot 10^4+7\cdot 10^3+6\cdot 10^2+4\cdot 10+9=7^6$

Answer 2

We are used to the decimal number system, i.e. the base 10 number system. As we all learn in school: 546 is five hundreds, four tens and six units:

$$546 = (5 \times 100) + (4\times 10) + (6 \times 1) = (5 \times 10^2) + (4\times 10^1) + (6 \times 10^0).$$

The base 10 refers to the fast that we write our numbers as a combination of powers of 10. In the base 18 number system, we would have:

$$ 546 = (5 \times 18^2) + (4\times 18^1) + (6 \times 18^0).$$

Things get a bit messy though. In base 10, we need 10 symbols: $0, 1, 2, \ldots, 9$ for each position. We only go up to 9 because 10 units is one ten, ten tens is one hundred, ten hundreds is one thousand, etc. In base 18 we would need 18 symbols for each position, e.g. $0,1,2,\ldots,9,A,B,C,\ldots,H$. An example of a number is base 18 could be:

$$1H2E = (1\times 18^3) + (17 \times 18^2) + (2\times 18^1) + (15\times 18^0).$$

Of course, the famous example is binary, which is base 2.