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The binary system operates on base 2, in which all numbers are represented by two digits: 0 and 1. This system is foundational to computer science because it reflects the on (1) and off (0) states of a computer's transistors. Unlike the decimal system, which is based on ten distinct symbols (0 through 9), the binary system relies solely on 0s and 1s to convey information.

When reading a binary number, each digit represents an increasing power of 2, starting from the rightmost digit. This is known as the least significant bit (LSB), while the leftmost digit is the most significant bit (MSB). For instance, in the binary number 101101, the rightmost 1 occupies the 2⁰ (1) position, and the leftmost 1 is in the 2⁵ (32) position. Each digit's position denotes its value, contingent upon whether it's a 1 or a 0.

To convert a binary number to a decimal, or base ten, you must apply base ten conversion. This method involves multiplying each binary digit by two raised to the power of its positional value. Then, you sum these products to find the decimal equivalent.

For the binary number 101101, you calculate the products as follows:

• For the least significant bit (LSB, rightmost), 1 * 2⁰ (which is 1)
• Moving left, 0 * 2¹ (which is 0, because anything times 0 is 0)
• Continuing, 1 * 2² (which is 4)
• Then, 1 * 2³ (which is 8)
• Skipping the zero, since it won't contribute to the sum
• Lastly, 1 * 2⁵ (which is 32)

Adding these values—1, 0, 4, 8, 0, 32 — you'd arrive at the decimal number 45.

Exponential notation is a convenient way to express large numbers or small fractions without writing out long strings of zeros. It uses a base number and an exponent to show how many times the base number is multiplied by itself. In binary to decimal conversion, the base number is always 2, as binary is base 2.

The exponent corresponds to the position of the digit, starting from 0 on the far right. For example, in the binary number 101101, the second digit from the right is 0, so you'd have 0 * 2¹, and since the base is raised to the first power, it remains unchanged—still a value of 0 in the sum. The use of exponential notation simplifies the process of binary conversion and is a fundamental concept in mathematics and computer science.