Bases of Integer Numbers System

In any system, there are two factors. 1] The value of the symbol we use ( e.g. we know what 6 stands for) and 2] at what position the symbol is placed(e.g. at hundredth place). In our decimal system, that is base 10 system, the right most (unit) place has vaule of symbol∗100symbol*10^0, the next (tenth) place to the left has a value symbol∗101symbol*10^1, and the next symbol∗102symbol*10^2. The nth place will have value symbol∗10nThus  1370=1∗103+3∗102+7∗101+0∗100.      symbol*10^n\\Thus ~~1370=1*10^3 + 3*10^2+7*10^1+ 0*10^0. ~~~~~~\\
. In decimal system, there are ten symbols, 0,1,...,9, each with their own value, and several locations, with weightage 10 times the location just at the right. In above example,3 has weightage of 100 while 7 to its right has weightage of 10. To specifically indicate that we are using decimal system, we may write 1370101370_{10}. Note that if there were no 0 it was not possible to give correct weightage to 137.

In place of 10 symbols, we can have eight symbols, 0,1,2,3,4,5,6,7 and the weightage of locations changing by power of 8. Say 13708=1∗83+3∗82+7∗81+0∗100=1∗51210+3∗6410+7∗810+0∗1=68910 in decimal system. This is octal ,the 8 based system where13708 to us means 68910       1370_8 =1*8^3 + 3*8^2+7*8^1+ 0*10^0 =1*512_{10} +3*64_{10}+7*8_{10}+0*1= 689_{10}\\~ in~ decimal ~system.~This~ is~ octal~,the ~8~based~system~ where 1370_8~to ~us~means~689_{10}~ \\~~~~~~\\
So we can have systems with different bases. Binary system has only two symbols, 0 and 1, and the weightage changes by power of 2. 1001102=1∗25+0+0+1∗22+1∗21+0=64+4+2=7010.  1001102  to us means only 7010  The hexadecimalsystem, is 16 based system. 100110_2= 1*2^5+0+0+1*2^2+1*2^1+0 =64+4+2= 70_{10}.~~100110_2~~to~us~means~only~70_{10} \\~~\\ The~ hexadecimal system,~is ~16~ based ~system.~
The extra six symbols are (Cap or small both are OK.) A16=1010, B16=1110, C16=1210, D16=1310, E16=1410, F16=1510.a137016=10∗164+1∗163+3∗162+7∗161+0∗160=66033610a137016 a big number of 66033610 for us!!    A_{16}=10_{10},~ B_{16}=11_{10} ,~ C_{16}=12_{10},~ D_{16}=13_{10}, ~ E_{16}=14_{10},~ F_{16}=15_{10} .\\ a1370_{16} =10*16^4 + 1*16^3+3*16^2+7*16^1+0*16^0 = 660336_{10}\\a1370_{16}~a~big~number~of~660336_{10}~for~us!!\\~~~\\
Binary, Octal and Hexadecimal systems are ued in computers. Incidentally, before switching to matric system India was using Hexadecimal system for there currency and measurements.

#NumberTheory #NumberBases

Note by Niranjan Khanderia
6 years, 5 months ago

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