Data Representation: Different Number Systems

In computer science, data representation refers to the way information is stored, processed, and transmitted by computers. Since computers inherently work with electrical signals, which have two distinct states (on and off), they use a binary number system to represent data. However, in computing, other number systems are also used for ease of representation, conversion, and manipulation of data. The major number systems used in computing are binary (base-2), decimal (base-10), octal (base-8), and hexadecimal (base-16).

1. Binary Number System (Base2)

Binary Number System is the fundamental system used by computers for data storage and processing. It uses only two digits: 0 and 1, which correspond to the two states of a digital circuit (off and on). Each binary digit is referred to as a bit (short for binary digit), and a group of 8 bits is called a byte. In binary, each position represents a power of 2.

Example:

  • The binary number 1011 can be converted to decimal by evaluating its position:

1011(2) = 1 × 2^3 + 0 × 2^2 + 1 × 2^1 + 1 × 2^0 = 8 + 0 + 2 + 1 = 11(10)

The binary system is efficient for computers because it aligns with the binary nature of electrical circuits.

2. Decimal Number System (Base-10)

Decimal number system is the system most commonly used by humans in daily life, consisting of ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Each position in a decimal number represents a power of 10. The decimal system is called base-10 because it uses ten digits to represent numbers.

Example:

  • The decimal number 345 can be broken down into:

345(10) = 3 × 10^2 + 4 × 10^1 + 5 × 10^0 = 300 + 40 + 5

Even though computers primarily use binary for processing, humans often work with decimal because it is easier to understand and apply in most everyday scenarios.

3. Octal Number System (Base-8)

Octal number system is a base-8 number system that uses digits from 0 to 7. Octal numbers are often used as a shorthand for binary numbers, as each octal digit represents exactly three binary digits (bits). This makes it easier to read and write long binary numbers.

Conversion between binary and octal: To convert a binary number to octal, group the binary digits into sets of three (starting from the right), then convert each group into the corresponding octal digit.

Example:

  • The binary number 110101101 is grouped into 110 101 101, and each group is converted to an octal digit:

110(2) = 6(8), 101(2) = 5(8), 101(2) = 5(8)

Octal is used less frequently today, but it was once widely used in programming for ease of representing binary numbers, particularly in older computer systems.

4. Hexadecimal Number System (Base-16)

The hexadecimal number system (or hex) is a base-16 system that uses sixteen symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, where A through F represent the values 10 through 15, respectively. Hexadecimal is widely used in computing because it provides a more compact representation of binary numbers and is easier for humans to read than binary.

Each hexadecimal digit corresponds to four binary digits (bits), meaning that a group of two hexadecimal digits represents one byte. This makes hexadecimal a convenient shorthand for binary.

Example:

  • The binary number 110101110011 can be grouped into 1101 0111 0011, and each group is converted to hexadecimal:

1101(2) = D(16), 0111(2) = 7(16), 0011(2) = 3(16)

Hexadecimal is widely used in programming, especially in debugging, memory addressing, and low-level hardware communication, due to its ability to represent long binary numbers in a more readable form.

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