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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

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6  
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
2  
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
2  
Well, now your money system is fully grown up and decimal. –  Lubin Sep 19 '12 at 18:35
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2 Answers 2

up vote 2 down vote accepted

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

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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
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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.

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