The question asks us to convert the number 101 from the decimal system (base 10) to the binary system (base 2). The example shows how 25 (decimal) is converted to 11001 (binary) by expressing it as a sum of powers of 2. To convert a decimal number to binary, we use the method of repeated division by 2.

Number Conversion Method

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.

Conversion Steps for 101

Let's apply the repeated division method to the number 101:

• Divide 101 by 2: $101 \div 2 = 50$ with a remainder of $1$.
• Divide 50 by 2: $50 \div 2 = 25$ with a remainder of $0$.
• Divide 25 by 2: $25 \div 2 = 12$ with a remainder of $1$.
• Divide 12 by 2: $12 \div 2 = 6$ with a remainder of $0$.
• Divide 6 by 2: $6 \div 2 = 3$ with a remainder of $0$.
• Divide 3 by 2: $3 \div 2 = 1$ with a remainder of $1$.
• Divide 1 by 2: $1 \div 2 = 0$ with a remainder of $1$.

Now, we read the remainders from the last one obtained upwards to the first one:

The remainders are 1, 1, 0, 0, 1, 0, 1. Reading from bottom to top, we get 1100101.

So, 101 in decimal is 1100101 in binary.

Verifying Binary Result

We can verify this by converting the binary number 1100101 back to decimal, similar to the example given for 25:

$1100101_{2} = 1 \times 2^{6} + 1 \times 2^{5} + 0 \times 2^{4} + 0 \times 2^{3} + 1 \times 2^{2} + 0 \times 2^{1} + 1 \times 2^{0}$

Calculating the powers of 2:

$2^{6} = 64$

$2^{5} = 32$

$2^{4} = 16$

$2^{3} = 8$

$2^{2} = 4$

$2^{1} = 2$

$2^{0} = 1$

Substituting these values:

$1100101_{2} = 1 \times 64 + 1 \times 32 + 0 \times 16 + 0 \times 8 + 1 \times 4 + 0 \times 2 + 1 \times 1$

$1100101_{2} = 64 + 32 + 0 + 0 + 4 + 0 + 1$

$1100101_{2} = 96 + 4 + 1$

$1100101_{2} = 101_{10}$

The binary number 1100101 successfully converts back to the decimal number 101. This confirms our conversion is correct.