Understanding Decimal to Binary Conversion

Number systems are fundamental in mathematics and computer science. The decimal system (base 10) is what we use daily, using digits from 0 to 9. The binary system (base 2) is crucial in computing, using only two digits: 0 and 1.

Converting a number from the decimal system to the binary system involves repeatedly dividing the decimal number by 2 and recording the remainders. The binary equivalent is then formed by reading the remainders from the last one obtained to the first one obtained (bottom-up).

Converting Decimal 5 to Binary Equivalent

Let's convert the decimal number 5 to its binary equivalent using the division-by-2 method:

1. Divide the decimal number 5 by 2. Note down the quotient and the remainder.
2. Take the quotient from the previous step and divide it by 2. Repeat this process until the quotient becomes 0.
3. The binary equivalent is obtained by listing the remainders in reverse order (from the last remainder to the first remainder).

Step-by-Step Conversion of 5 to Binary

Here are the steps for converting decimal 5 to binary:

Reading the remainders from bottom to top, we get 101.

Binary Equivalent Result

Based on the conversion steps, the binary equivalent of the decimal number 5 is 101.

Let's compare this result with the given options:

• Option 1: 101
• Option 2: 100
• Option 3: 001
• Option 4: 111

The calculated binary equivalent, 101, matches Option 1.

Revision Table: Common Decimal to Binary Conversions

Additional Information on Number Systems and Binary Representation

Number systems are ways to represent numbers using specific symbols or digits. The value of a digit depends on its position within the number and the base of the number system.

• Decimal System (Base 10): Uses digits 0-9. The position values are powers of 10 (e.g., \(10^0, 10^1, 10^2, \dots\)). For example, decimal \(123 = 1 \times 10^2 + 2 \times 10^1 + 3 \times 10^0\).
• Binary System (Base 2): Uses digits 0 and 1. The position values are powers of 2 (e.g., \(2^0, 2^1, 2^2, \dots\)). For example, binary \(101_2 = 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 1 \times 4 + 0 \times 2 + 1 \times 1 = 4 + 0 + 1 = 5\) in decimal.
• Binary representation is fundamental to how computers store and process information because electronic circuits can easily represent two states (on/off, high voltage/low voltage), which correspond to 1 and 0.
• Understanding decimal to binary conversion is a core concept in digital electronics and computer architecture.