Many different base number systems are used to represent numeric values. Each number system has its own importance and usage.
Decimal number system, base 10
This is the most fundamental number system we all use in mathematics. Everybody knows there are ten unique digits 0,1,2,3,4,5,6,7,8,9; which are used to represent any known number.
Binary number system, base 2
With the advent of computer science base two number system is also proving useful and gaining popularity. It has two unique digits, 0 and 1. The machine (computer) just understands 1 and 0; the on and off state. All computer code is converted to binary code before being interpreted.The first five natural numbers in binary are represented as 001(1), 010(2), 011(3), 100(4), 101(5).
Octal number system base 8
This number system has less usage. There are eight unique digits 0, 1, 2, 3, 4, 5, 6, 7; in octal number system. Using these eight digits, one can represent any number.
Hexadecimal number system. base 16
This number system has immense usage in computer science. Large binary representation can be compacted to hexadecimal notation. Hexadecimal has sixteen unique symbols to represent numbers. Ten unique decimal digits and first six English alphabets; 0,1,2,3,4,5,6,7,8,9,a,b,c,d,e,f.
a ,b,c,d,e,f represent the numbers 10.11,12,13,14,15, respectively.
Conversion
A number given in base system can be converted to another base system.
One should have relatively good understanding of place value of each numeral in the number. Consider decimal number 746.In expanded notation it is
746 = 7 * 10^2 + 4*10^1 + 6 * 10^0 (caret ^ symbol represents, raise to the power of)
10^0 = 1, as a matter of fact any number raised to the power zero is equal to 1.We use powers of ten as it is a decimal number and each digit is given a place value starting from right hand side ones, tens, hundreds and so on.
Converting decimal number 18 to binary number
Start dividing 18 by 2 , then divide successive quotients by two, till quotient becomes less than two. Note down all the remainders as well in the process.
18/2; Q=9 R = 0
9/2; Q=4 R = 1
4/2; Q=2 R = 0
2/2; Q=1 R = 0
Answer Start from the last quotient i.e. 1 , include all remainders bottom up 10010.
18(Decimal) = 10010(Binary)
Similarly a decimal number can be converted to octal number or a hexadecimal number, the division needs to be done by eight and sixteen for the respective base number system.
Converting a binary to a decimal
First of all get well versed with powers of two like (2^1)2, (2^2)4, (2^3)8, (2^4)16, (2^5)32, (2^6)64 and so on.
Suppose you have to convert binary number 10010 into decimal. Assign each numeral from the right hand side a place value starting with zero
1*2^4 + 0*2^3 + 0*2^2 + 1*2^1 + 0*2^0
1*16 + 0*8 + 0*4 + 1*2 + 0*1
16+0+0+2+0 = 18
Hence 10010(binary) = 18 (decimal)
The conversion computation seems a bit tricky initially. After ample practice and clear understanding of the fundamentals of each number system, you will find the conversions are mere formula driven and can be quickly performed by doing just few mental calculations.
