Understanding Decimal to Binary Conversion

Numbers can be represented in different number systems. The decimal system (base 10) is what we use in everyday life, using digits 0 through 9. The binary system (base 2) is used by computers, using only digits 0 and 1. Converting a decimal number to its binary equivalent is a fundamental concept in digital electronics and computer science.

Converting Decimal 31 to Binary

To convert a decimal number to its binary representation, we use the method of repeated division by 2. The process involves dividing the decimal number by 2, noting the remainder, and then dividing the quotient by 2. We repeat this until the quotient becomes 0. The binary representation is obtained by reading the remainders from bottom to top.

Let's convert the decimal number 31 to binary:

Reading the remainders from bottom (most significant bit) to top (least significant bit), we get the binary number:

11111

Therefore, the binary expression of the decimal number 31 is 11111.

Analyzing the Options

• 1111: Let's convert this binary to decimal: \(1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 8 + 4 + 2 + 1 = 15\). This is not 31.
• 10111: Let's convert this binary to decimal: \(1 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 16 + 0 + 4 + 2 + 1 = 23\). This is not 31.
• 11011: Let's convert this binary to decimal: \(1 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 16 + 8 + 0 + 2 + 1 = 27\). This is not 31.
• 11111: As calculated earlier, this binary converts to decimal 31 (\(1 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 16 + 8 + 4 + 2 + 1 = 31\)). This matches the given decimal number.

Conclusion on Decimal 31 in Binary

Based on the conversion process, the binary representation of the decimal number 31 is 11111.

Revision Table: Number System Conversion Basics

Additional Information: Binary Numbers and Place Values

In a binary number like \(b_n b_{n-1} ... b_1 b_0\), where \(b_i\) is either 0 or 1, the decimal value is calculated as:

Decimal Value \(= b_n \times 2^n + b_{n-1} \times 2^{n-1} + ... + b_1 \times 2^1 + b_0 \times 2^0\)

For example, in the binary number 11111:

• The rightmost '1' is at position 0, corresponding to \(2^0 = 1\).
• The next '1' to the left is at position 1, corresponding to \(2^1 = 2\).
• The next '1' is at position 2, corresponding to \(2^2 = 4\).
• The next '1' is at position 3, corresponding to \(2^3 = 8\).
• The leftmost '1' is at position 4, corresponding to \(2^4 = 16\).

Summing these place values where the digit is '1' gives $16 + 8 + 4 + 2 + 1 = 31$. This confirms the conversion of 11111 binary to 31 decimal.