# What That Mean “In Base 18”?

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

-
It's like binary, but with $18$ instead of $2$. See en.wikipedia.org/wiki/Radix –  Qiaochu Yuan Sep 19 '12 at 16:45
Perhaps this will help: math.com/students/converters/source/base.htm. You typically work in base 10 numbers, but you must be familiar with base 2. Base-n can be any base you want in theory (mathforum.org/dr.math/faq/faq.bases.html). HTH ~A –  Amzoti Sep 19 '12 at 16:49
Thanks Guys For Links –  Synxmax Sep 19 '12 at 16:59
It was just a side comment illustrating that strange bases show up in 'real' life. There were 12 pennies in a shilling, and 20 shillings in a pound. Along with imperial units, it made for interesting primary homework. –  copper.hat Sep 19 '12 at 18:26
Well, now your money system is fully grown up and decimal. –  Lubin Sep 19 '12 at 18:35

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

-
Thank u , Thank u , Thank u i got it now 111 = 1 * 18 ^ 2 + 1 * 18 + 1 = 343 = 7^3 –  Synxmax Sep 19 '12 at 17:16
@Synxmax: correct. –  Ross Millikan Sep 19 '12 at 17:18

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.

-

The decimal point dividing by ten is what most people are used to, but dividing by 12's or 18's has quite an advantage. Dividing by 5 or 10 would no longer terminate; those would go on forever. But those are more than outnumbered by 3, 6, 9, 12, and 18, which would terminate quickly. Base 18 is compatible with the Mayan calendar, which is 20 days times 18 months per year. One thing base 18 gives us is rotational multiples of 7 and its square, 2D (D being the 13 digit), such as 2D0 and D02 (and 7's cube 111). Also note that other multiples of 111 include 10,101 and 100,010,001 (just like in decimal). As for fractions expressed in digits, base 18 gives them as follows: ...... 1/18 = 0.1 1/12 = 0.19 ...... 1/9 = 0.2 ...... 1/8 = 0.249 ...... 1/6 = 0.3 ...... 1/4 = 0.49 ...... 1/3 = 0.6 ...... 1/2 = 0.9 2/3 = 0.C ...... 3/4 = 0.D9 ...... 5/6 = 0.F ...... 7/8 = 0.FD9 ...... 8/9 = 0.G ...... So there you have it.

-