Which binary number below represents the decimal number 20?

The binary number that represents the decimal number 20 is 10100.

Understanding Decimal to Binary Conversion

Decimal (base-10) numbers are the system we use for everyday counting, utilizing ten distinct digits (0 through 9). Binary (base-2) numbers, in contrast, form the foundational language of computers and digital electronics, employing only two digits: 0 and 1. In the binary system, the position of each digit holds a specific weight, which is a power of 2. For instance, moving from right to left, the positions represent $2^0$ (1), $2^1$ (2), $2^2$ (4), $2^3$ (8), $2^4$ (16), and so on.

To convert a decimal number to its binary equivalent, you essentially determine which combination of these powers of 2 sums up to the original decimal value.

Converting Decimal 20 to Binary

Let's meticulously break down the process of converting the decimal number 20 into its binary form:

• Step 1: Find the largest power of 2 less than or equal to 20.

  • This is $2^4 = 16$.
  • Since 16 fits into 20, the bit in the $2^4$ (or 16's) place is 1.
  • Calculate the remainder: $20 - 16 = 4$.

• Step 2: Consider the next lower power of 2, which is $2^3 = 8$.

  • Is 8 less than or equal to the current remainder (4)? No.
  • Therefore, the bit in the $2^3$ (or 8's) place is 0.
  • The remainder remains 4.

• Step 3: Move to the next power of 2, which is $2^2 = 4$.

  • Is 4 less than or equal to the current remainder (4)? Yes.
  • Since 4 fits perfectly, the bit in the $2^2$ (or 4's) place is 1.
  • Calculate the new remainder: $4 - 4 = 0$.

• Step 4: Proceed to the next power of 2, which is $2^1 = 2$.

  • Is 2 less than or equal to the current remainder (0)? No.
  • Thus, the bit in the $2^1$ (or 2's) place is 0.

• Step 5: Finally, consider the smallest power of 2, which is $2^0 = 1$.

  • Is 1 less than or equal to the current remainder (0)? No.
  • Therefore, the bit in the $2^0$ (or 1's) place is 0.

By concatenating these bits from the highest power of 2 down to $2^0$, we obtain the binary representation: 10100.

Common Decimal to Binary Equivalents

To provide further context and clarity, here is a table illustrating the binary equivalents for several decimal numbers around 20:

This table confirms that the decimal number 20 is accurately represented by the binary number 10100.