Understanding Decimal to Binary Conversion

The question asks for the binary equivalent of the decimal number 10. Decimal numbers are base-10, meaning they use digits 0 through 9. Binary numbers are base-2, using only the digits 0 and 1. To convert a decimal number to its binary equivalent, we can use the method of repeated division by 2.

Steps for Decimal to Binary Conversion

Here's how to convert the decimal number 10 into its binary equivalent:

1. Divide the decimal number by 2.
2. Note the remainder (which will be either 0 or 1). This is the least significant bit (LSB) of the binary number.
3. Take the quotient from the division and repeat the process (divide by 2).
4. Continue this process until the quotient becomes 0.
5. The binary equivalent is obtained by reading the remainders from bottom to top (from the last remainder to the first remainder).

Converting Decimal 10 to Binary

Let's apply the steps to the decimal number 10:

• Divide 10 by 2:
  • \(10 \div 2 = 5\) with a remainder of \(0\). (This is the first remainder, read last)
• Now take the quotient, 5, and divide by 2:
  • \(5 \div 2 = 2\) with a remainder of \(1\).
• Take the quotient, 2, and divide by 2:
  • \(2 \div 2 = 1\) with a remainder of \(0\).
• Take the quotient, 1, and divide by 2:
  • \(1 \div 2 = 0\) with a remainder of \(1\). (This is the last remainder, read first - the most significant bit, MSB)

Now, read the remainders from bottom to top: 1, 0, 1, 0.

So, the binary equivalent of decimal 10 is 1010.

We can also represent this conversion process in a table:

Reading the remainders from bottom up (1, 0, 1, 0), we get the binary number 1010.

Analyzing the Options

Let's look at the provided options and see which one matches our result for the binary equivalent of decimal 10:

• 0010: Let's convert this binary back to decimal: \(0 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 0 + 0 + 2 + 0 = 2\). This is not 10.
• 10: This is the decimal number itself, not its binary equivalent.
• 010: This is the same as 10 in binary, which converts to \(1 \times 2^1 + 0 \times 2^0 = 2 + 0 = 2\) in decimal. This is not 10.
• 1010: Let's convert this binary back to decimal: \(1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 1 \times 8 + 0 \times 4 + 1 \times 2 + 0 \times 1 = 8 + 0 + 2 + 0 = 10\). This matches the original decimal number.

Therefore, the binary equivalent of the decimal number 10 is indeed 1010.

Revision Table: Number Systems

Additional Information: Place Values in Binary

In the binary number system, the position of each digit (bit) has a specific place value, which is a power of 2. Starting from the rightmost digit (LSB) and moving left, the place values are \(2^0, 2^1, 2^2, 2^3,\) and so on.

• Rightmost digit: represents \(2^0 = 1\)
• Second digit from the right: represents \(2^1 = 2\)
• Third digit from the right: represents \(2^2 = 4\)
• Fourth digit from the right: represents \(2^3 = 8\)
• And so on...

To convert a binary number to decimal, you multiply each digit by its corresponding place value and sum the results. For binary 1010:

• \(1010_2 = (1 \times 2^3) + (0 \times 2^2) + (1 \times 2^1) + (0 \times 2^0)\)
• \(= (1 \times 8) + (0 \times 4) + (1 \times 2) + (0 \times 1)\)
• \(= 8 + 0 + 2 + 0\)
• \(= 10_{10}\)

This confirms our conversion result and the method used to find the binary equivalent of decimal 10.