In decimal number system , 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are used cyclically at each place to generate number from 0 to generate natural number cyclically.

i.e.

000 , 0001, 002, 003, 004, 005, 006, 007, 008, 009

now at ones place after 9 comes zero in cyclic order , cycle get completed so at ones place comes zero where as tenth place onward number is replaced by its next number. i.e. 009 — next natural number –> (00 +1)0 = 010

In this case 01 i.e. tens place onward number indicates number of cycle completed in units place, hundred place 0 in 010 indicates zero cycle of permutation of ones and tens place completed together i.e. number of hundreds in number.

after 010 , numbers generation continues and we get

011, 012, 013, …., 099

after 099 what comes ?

in ones place after 9 comes 0 in cyclic order, cycle get completed so

09 should be increased by 1 .

increasing 09 by 1 means finding next natural number after 09.

so story goes same. after 9 of comes 0 in cyclic in 09 and cycle get completed so, we we replace 9 by 0 and 0 by next number i.e. 1 ( 10)

so ultimately , we get next natural number after 099 as 100 ..

Hope you get number generation in decimal number system.

Decimal system uses  ten symbol 0,1,…8, 9 cyclically at each place in counting / number generation so we called it as decimal number numbers. Deci means 10. The symbol 0, 1, 2, …, 9 are called as Decimal digits.

Similarly , in same manner if we use just two symbol 0, 1 cyclically in counting can you guess what it is called ?

It is called as binary. Prefix ‘Bi’ means two.

In binary number system ,  if we generate numbers by same pattern starting from zeroes we get

0 , 1, cycle at unit place completed so next number is (0+1)0 = 10,

next is 1(next of 0 in cycle order) = 11 , 100, 101, 110, 111, 01000 so on..

Thus numbers generated are

0 , 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100…

Here first is 0, then next nth number represent n .

Thus fifth number after zero in binary in above list i.e. 101 represent set of five object or simply decimal 5 . Thus binary 101 number is equivalent to decimal 5.

Hope you get correlation between binary and decimal.

In binary system , unlike decimal first ones place have cycle of 1 i.e. change after 1 symbol, where as second place left to ones changes after two symbol, third place left to unit place changes by 4 symbol so on. so we can simply multiply binary digits by 1, 2, 4, 8, .. so on to get its equivalent decimal number.

i.e. Binary (101) = Decimal 1 * 4 + 0*2 + 1*1 = 5.

1 * 4 + 0*2 + 1*1 is called as expanded form of binary number 101.

Similarly Binary (11010) = Decimal 0 *1 + 1 *2 +0*4 + 1 *8 +1 *16 = 26

Similarly , Ternary positional number system uses 3 symbol  0,1, 2 as digit for generating numbers. Its base is 3.

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This had given idea of how numbers are generated in Positional number system like decimal binary etc to you. Now following is the puzzle.

In positional number system say Bi-Decimal ones , 3rd, 5th,  so on all odd positioned places to the left and right to ‘.’ point (used for separating integer fraction part of numbers) uses 0, 1 symbol cyclically  whereas all remaining places use 0, 1, 2, 3, …9 symbols cyclically.  Then

1. In Bi-decimal number system what number will represent set ‘nineteen thousand eight hundred fifty five’ balls ?
2. What is the decimal value of Bi-decimal number 1807131 ?
3. What is the result of subtraction in Bi-decimal numbers 71100 – 131 ?

We can use same theme and define TriDecimal, QuadraDecimal, QuintDecimal, HexaDecimal, SeptaDecimal etc Number system.

This article first times in history defines such numbers. Here puzzle is set for better understanding of number system. It is the basics of our counting , whole arithmetic relies on it. Basic fundamental should be strong. If student have better understanding they can comes with better innovative knowledge idea methods.

Similary, we can define BiTernary TriOctal , Tribinary etc number system.

    – Vitthal Jadhav

      (Mathematician)