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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 \(symbol*10^0\), the next (tenth) place to the left has a value \(symbol*10^1\), and the next \(symbol*10^2\). The nth place will have value \(symbol*10^n\\Thus ~~1370=1*10^3 + 3*10^2+7*10^1+ 0*10^0. ~~~~~~\\\)
. 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 \(1370_{10}\). 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 \(1370_8 =1*8^3 + 3*8^2+7*8^1+ 0*10^0 =1*512_{10} +3*64_{10}+7*8_{10}+0*1= 689_{10}\\~ in~ decimal ~system.~This~ is~ octal~,the ~8~based~system~ where 1370_8~to ~us~means~689_{10}~ \\~~~~~~\\\)
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. \(100110_2= 1*2^5+0+0+1*2^2+1*2^1+0 =64+4+2= 70_{10}.~~100110_2~~to~us~means~only~70_{10} \\~~\\ The~ hexadecimal system,~is ~16~ based ~system.~\)
The extra six symbols are (Cap or small both are OK.) \( A_{16}=10_{10},~ B_{16}=11_{10} ,~ C_{16}=12_{10},~ D_{16}=13_{10}, ~ E_{16}=14_{10},~ F_{16}=15_{10} .\\ a1370_{16} =10*16^4 + 1*16^3+3*16^2+7*16^1+0*16^0 = 660336_{10}\\a1370_{16}~a~big~number~of~660336_{10}~for~us!!\\~~~\\\)
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.

Note by Niranjan Khanderia
2 years ago

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