# 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. ~~~~~~\$/extract_itex] . 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 6 years, 3 months ago This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science. When posting on Brilliant: • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused . • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone. • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge. • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events. MarkdownAppears as *italics* or _italics_ italics **bold** or __bold__ bold - bulleted- list • bulleted • list 1. numbered2. list 1. numbered 2. list Note: you must add a full line of space before and after lists for them to show up correctly paragraph 1paragraph 2 paragraph 1 paragraph 2 [example link](https://brilliant.org)example link > This is a quote This is a quote  # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" MathAppears as Remember to wrap math in $$ ... $$ or \[ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$