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The binary numeral system is the foundation of all digital systems. It operates on just two digits: 0 and 1, which we call 'bits'. This system is considered 'base 2' because it's binary, or composed of two different digits. Each position in a binary number represents a power of 2, just as in the decimal system each position represents a power of 10.

When we look at a binary number like the exercise's example, \(1101_{\text{two}}\), we start from the rightmost digit and work our way to the left. This is because, similar to base-ten counting, the significance of the bit increases with each position towards the left. Therefore, in the case of \(1101_{\text{two}}\), the digits from right to left represent the values of \(2^0\), \(2^1\), \(2^2\), and \(2^3\) respectively.

Converting from binary to decimal involves multiplying each binary digit by its corresponding power of 2 and then summing those values. This process can be visualized in a few steps to make it more understandable.

For our example, the binary number is broken down as follows:

• \(1 \times (2^0)\) which equals 1,
• \(0 \times (2^1)\) which is 0 since anything multiplied by zero is zero,
• \(1 \times (2^2)\) which equals 4, and
• \(1 \times (2^3)\) which equals 8.

After calculating these values, we add them up: \(1 + 0 + 4 + 8 = 13\). Thus, the binary number \(1101_{\text{two}}\) converts to the decimal number 13.

Base ten, also known as the decimal system, is the most commonly used numeral system and the one most familiar to us. It's called base ten because it is based on ten different digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Each position in a base-ten number has a place value ten times that of the position to its right.

For example, in the number 237, the 7 is in the units place (\(10^0\)), the 3 is in the tens place (\(10^1\)), and the 2 is in the hundreds place (\(10^2\)). Thus, they represent the values 7, 30, and 200, respectively. When we convert binary numbers to decimal, we embrace this familiar base-ten system to understand and process information in the format we use in everyday life.