Data Representation

This section explores different number systems used to represent data in computers: decimal, binary, and hexadecimal. Understanding these systems is fundamental to understanding how computers store and process information.

Decimal Number System

The decimal system is the system we use in everyday life. It is a base-10 system, meaning it uses 10 digits (0-9) to represent numbers. Each position in a decimal number represents a power of 10.

For example, the number 123 can be broken down as: $1 \times 10^2 + 2 \times 10^1 + 3 \times 10^0 = 100 + 20 + 3 = 123$

• Base: 10
• Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Binary Number System

The binary system is a base-2 system, meaning it only uses two digits: 0 and 1. Computers use binary to represent all data because electronic circuits can easily represent two states (on/off, high/low voltage), corresponding to 1 and 0.

Each position in a binary number represents a power of 2.

For example, the binary number 1011 is equal to: $1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 8 + 0 + 2 + 1 = 11$ (in decimal).

Position	Value
0	$2^0 = 1$
1	$2^1 = 2$
2	$2^2 = 4$
3	$2^3 = 8$
4	$2^4 = 16$
5	$2^5 = 32$

Conversion from decimal to binary: Divide the decimal number by 2, noting the remainder. The remainders, read in reverse order, give the binary equivalent.

Conversion from binary to decimal: Multiply each binary digit by the corresponding power of 2 and sum the results.

Hexadecimal Number System

The hexadecimal system is a base-16 system. It uses 16 digits: 0-9 and A-F, where A=10, B=11, C=12, D=13, E=14, and F=15. Hexadecimal is often used to represent binary numbers in a more compact form.

Each position in a hexadecimal number represents a power of 16.

For example, the hexadecimal number 2A3 is equal to: $2 \times 16^2 + 10 \times 16^1 + 3 \times 16^0 = 2 \times 256 + 10 \times 16 + 3 \times 1 = 512 + 160 + 3 = 675$ (in decimal).

Position	Value
0	$16^0 = 1$
1	$16^1 = 16$
2	$16^2 = 256$
3	$16^3 = 4096$

Conversion from decimal to hexadecimal: Divide the decimal number by 16, noting the remainder. The remainders, read in reverse order, give the hexadecimal equivalent. Use A-F to represent values 10-15.

Conversion from hexadecimal to decimal: Multiply each hexadecimal digit by the corresponding power of 16 and sum the results.

Binary, Decimal and Hexadecimal Conversion Table

Number System	Example	Decimal Equivalent
Binary	1010	10
Decimal	255	255
Hexadecimal	FF	255