** Binary vs Decimal **

A number is a mathematical abstraction. We realize numbers in our real life through symbols. A certain collection of symbols associated with a set of rules is called a “Number system” or “Numeral System.” The numeric symbols manipulate almost the entire world of Mathematics. There are various number systems in the world. Number systems originate from our real-world experiences. For an example, ten fingers in our hands influenced in thinking about a number system with ten symbols. This is what is called decimal number system. Similarly, our duality in understanding as live-die, yes-no, on-off, left-right, and close-open originated the binary number system with two symbols. There are also other number systems such as octal and hexadecimal to describe the world. Computer is a marvelous machine which is governed by various number systems.

The number system used in modern mathematics is called positional number system. In this concept, each digit in a number has an associated value that depends on its position in the number. The number of distinct symbols used to define a number system is called the base. The base is an elegant way to define the concept of place value. In this sense, each place value can be represented as a power to the base.

The decimal number system comprises ten symbols (digits): 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Therefore, any number represented by this number system comprises one or more above ten symbols. For example, 452 is a number written by the decimal number system. Under positional number representation, numerals 4, 5 and 2 do not have the same importance within the number. In the decimal number system, place values are (from right to left) given by 10^{0}, 10^{1}, 10^{2}, etc. They are read as 1’s place, 10’s place and etc, from right to left.

For example, in the number 385, 5 is in 1’s place, 8 is in 10’s place, and 3 is in 100’s place. Therefore, using the concept of base we denote 385 as the summation (3×10^{2}) + (8×10^{1}) + (5×10^{0}).

The binary number system uses two symbols; 0 and 1 to represent any number. Therefore, it is a number system with base 2, and gives a set of place values as one (2^{0}), two (2^{1}), four (2^{2}), and etc. For an example, 101101_{2} is a binary number. The subscript 2 in this number representation is the base 2 of this number.

Consider the number 101101_{2}. This represents (1×2^{5}) + (0×2^{4}) + (1×2^{3}) + (1×2^{2}) + (0×2^{1}) + (1×2^{0}) = or 1×32 + 0×16 + 1×8 + 1×4 + 0×2 + 1×1 or 45.

Binary number system is widely used in the computer world. Computers use the binary number system to manipulate and store data. All mathematical operations: addition, subtraction, multiplication and division are applicable in both decimal and binary number system.

¤ Decimal number system uses 10 digits (0,1…9) to represent numbers, while the binary number system uses 2 digits (0 and 1). ¤ Number base used in decimal number system is ten, while the binary number system uses base two. |