The Four Axioms

                 Conversion of a given number in one number system to its equivalent in another system has been discussed at length in the preceding sections. The methodology has been illustrated with solved examples. The complete methodology can be summarized as four axioms or principles, which, if understood properly, would make it possible to solve any problem related to conversion of a given number in one number system to its equivalent in another number system.
These principles are as follows:
1. Whenever it is desired to find the decimal equivalent of a given number in another number system,
it is given by the sum of all the digits multiplied by their weights or place values. The integer and
fractional parts should be handled separately. Starting from the radix point, the weights of different
digits are r0, r1, r2 for the integer part and r−1, r−2, r−3 for the fractional part, where r is the radix
of the number system whose decimal equivalent needs to be determined.
2. To convert a given mixed decimal number into an equivalent in another number system, the integer
part is progressively divided by r and the remainders noted until the result of division yields a
zero quotient. The remainders written in reverse order constitute the equivalent. r is the radix of
the transformed number system. The fractional part is progressively multiplied by r and the carry
recorded until the result of multiplication yields a zero or when the desired number of bits has been
obtained. The carrys written in forward order constitute the equivalent of the fractional part.
3. The octal–binary conversion and the reverse process are straightforward. For octal–binary
conversion, replace each digit in the octal number with its three-bit binary equivalent. For
hexadecimal–binary conversion, replace each hex digit with its four-bit binary equivalent. For
binary–octal conversion, split the binary number into groups of three bits, starting from the binary
point, and, if needed, complete the outside groups by adding 0s, and then write the octal equivalent
of these three-bit groups. For binary–hex conversion, split the binary number into groups of four
bits, starting from the binary point, and, if needed, complete the outside groups by adding 0s, and
then write the hex equivalent of the four-bit groups.
4. For octal–hexadecimal conversion, we can go from the given octal number to its binary equivalent
and then from the binary equivalent to its hex counterpart. For hexadecimal–octal conversion, we
can go from the hex to its binary equivalent and then from the binary number to its octal equivalent. 
Example 1.9
Assume an arbitrary number system having a radix of 5 and 0, 1, 2, L and M as its independent digits.
Determine:
(a) the decimal equivalent of (12LM.L1);
(b) the total number of possible four-digit combinations in this arbitrary number system.
Solution
(a) The decimal equivalent of (12LM) is given by
M×50 +L×51+2×52 +1×53 = 4×50 +3×51+2×52 +1×53L = 3M= 4
= 4+15+50+125 = 194
The decimal equivalent of (L1) is given by
L×5−1+1×5−2 = 3×5−1+5−2 = 064
Combining the results, (12LM.L1)5
= (194.64)10.
(b) The total number of possible four-digit combinations = 54 = 625.
Example 1.10
The 7’s complement of a certain octal number is 5264. Determine the binary and hexadecimal
equivalents of that octal number.
Solution
• The 7’s complement = 5264.
• Therefore, the octal number = (2513)8.
• The binary equivalent = (010 101 001 011)2
= (10101001011)2.
• Also, (10101001011)2
= (101 0100 1011)2
= (0101 0100 1011)2
= (54B)16.
• Therefore, the hex equivalent of (2513)8
= (54B)16 and the binary equivalent of (2513)8
=(10101001011)2.