Understanding Binary Number System Conversion

The question asks for the representation of the decimal number '22' in the binary number system. The binary number system uses base 2, meaning it only uses two digits: 0 and 1. To convert a decimal number to binary, we repeatedly divide the decimal number by 2 and record the remainders. The binary representation is obtained by reading the remainders from bottom to top.

Converting Decimal 22 to Binary Step-by-Step

Let's convert the decimal number 22 to its binary equivalent using the division method:

We stop when the quotient becomes 0. Now, we read the remainders from the last remainder to the first remainder (bottom-up). The remainders are 1, 0, 1, 1, 0.

So, the binary representation of decimal 22 is 10110.

Comparing with Options

Let's compare our result (10110) with the given options:

1. 10111
2. 11000
3. 00111
4. 10110

Our calculated binary number, 10110, matches option 4.

Verifying Binary to Decimal Conversion

We can verify our answer by converting the binary number 10110 back to decimal. In the binary system, each digit's place value is a power of 2, starting from \(2^0\) for the rightmost digit.

For 10110 binary:

\(10110_2 = (1 \times 2^4) + (0 \times 2^3) + (1 \times 2^2) + (1 \times 2^1) + (0 \times 2^0)\)

\(= (1 \times 16) + (0 \times 8) + (1 \times 4) + (1 \times 2) + (0 \times 1)\)

\(= 16 + 0 + 4 + 2 + 0\)

\(= 22\)

This confirms that the binary representation 10110 is indeed equivalent to the decimal number 22.

Revision Table: Common Number System Conversions

Additional Information: Place Value in Binary Numbers

In the binary number system (base 2), the position of each digit represents a power of 2. Starting from the rightmost digit, the place values are \(2^0, 2^1, 2^2, 2^3, \) and so on, moving left. For example, in the binary number \(d_n d_{n-1} ... d_2 d_1 d_0\), the decimal value is calculated as:

Decimal Value = \(d_n \times 2^n + d_{n-1} \times 2^{n-1} + ... + d_1 \times 2^1 + d_0 \times 2^0\)

Understanding place value is crucial for converting binary numbers to decimal and vice versa.