Understanding Decimal to Binary Conversion

This question asks for the binary equivalent of the decimal number 10. Converting a number from the decimal system (base 10) to the binary system (base 2) involves repeatedly dividing the decimal number by 2 and recording the remainders. The binary equivalent is then formed by reading these remainders from bottom to top.

Step-by-Step Conversion of Decimal 10 to Binary

Let's convert the decimal number 10 to its binary equivalent using the division-by-2 method:

1. Divide the decimal number (10) by 2.
2. Record the remainder of this division.
3. Use the quotient from the division as the new number and repeat steps 1 and 2 until the quotient becomes 0.
4. Read the remainders from the last one calculated to the first one calculated to get the binary number.

Performing the Divisions:

• Divide 10 by 2:
  • $10 \div 2 = 5$
  • Remainder: 0
• Now, use the quotient 5:
  • $5 \div 2 = 2$
  • Remainder: 1
• Now, use the quotient 2:
  • $2 \div 2 = 1$
  • Remainder: 0
• Now, use the quotient 1:
  • $1 \div 2 = 0$
  • Remainder: 1

Collecting the Remainders:

The remainders, collected from bottom to top, are 1, 0, 1, and 0. Arranging them in this order gives the binary number.

The binary equivalent of the decimal number 10 is $1010_2$.

Comparing with Options

Let's look at the provided options:

• Option 1: 1010
• Option 2: 010
• Option 3: 10
• Option 4: 0010

Our calculated binary equivalent for decimal 10 is 1010. This matches Option 1.

Options 2, 3, and 4 do not represent the correct binary equivalent of 10.

Conclusion

The binary equivalent of the decimal number 10 is 1010.

Revision Table: Number System Conversions

Additional Information: Understanding Binary Representation

Binary is a base-2 number system, meaning it only uses two digits: 0 and 1. Each position in a binary number represents a power of 2, starting from the rightmost digit which represents $2^0$ (which is 1), the next digit to the left represents $2^1$ (which is 2), then $2^2$ (4), $2^3$ (8), and so on. To convert a binary number to decimal, you multiply each digit by its corresponding power of 2 and sum the results.

For example, to convert binary 1010 to decimal:

• The rightmost digit 0 is at position $2^0$. Contribution: $0 \times 2^0 = 0 \times 1 = 0$.
• The next digit 1 is at position $2^1$. Contribution: $1 \times 2^1 = 1 \times 2 = 2$.
• The next digit 0 is at position $2^2$. Contribution: $0 \times 2^2 = 0 \times 4 = 0$.
• The leftmost digit 1 is at position $2^3$. Contribution: $1 \times 2^3 = 1 \times 8 = 8$.

Summing these contributions: $0 + 2 + 0 + 8 = 10$. This confirms that the binary 1010 is indeed the decimal number 10.

The binary system is fundamental in computing because digital electronics use two states (on/off, high/low voltage) which can be represented by 1 and 0.