Overview of Binary and Decimal Systems
The binary system, also known as base-2, uses only two digits: 0 and 1, to represent numbers through powers of 2. In contrast, the decimal system, or base-10, employs ten digits (0 through 9) and is based on powers of 10, which aligns with human counting using ten fingers. The primary difference lies in their positional notation and the symbols used to encode values.
Key Principles: Base and Digit Representation
In the decimal system, each position represents a power of 10, starting from the rightmost digit as 10^0 (units), then 10^1 (tens), and so on. For example, 123 in decimal means 1×10^2 + 2×10^1 + 3×10^0. Binary follows a similar principle but with powers of 2: the rightmost digit is 2^0, followed by 2^1, etc. Thus, 101 in binary equals 1×2^2 + 0×2^1 + 1×2^0 = 5 in decimal. This base difference affects how numbers are stored and calculated.
Practical Example: Number Conversion
Consider the decimal number 13. To convert to binary, divide by 2 repeatedly and record remainders: 13÷2=6 remainder 1, 6÷2=3 remainder 0, 3÷2=1 remainder 1, 1÷2=0 remainder 1. Reading remainders bottom-up gives 1101 in binary, which is 1×2^3 + 1×2^2 + 0×2^1 + 1×2^0 = 8+4+0+1=13. This illustrates how binary requires more digits for the same value due to its smaller base.
Importance and Applications
Decimal is the standard for everyday human use in commerce, science, and education because it matches our base-10 finger-counting system, making arithmetic intuitive. Binary, however, is essential in computing and electronics, as it directly corresponds to on/off states of digital circuits, enabling efficient data storage and processing in computers, though it can lead to longer representations of numbers.