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 realworld 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 livedie, yesno, onoff, leftright, and closeopen 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.
What is the difference between ? ¤ 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.

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