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Understanding the place values in binary is essential for converting binary numbers, such as \(110111011_2\), into their decimal (base 10) counterparts. In the binary number system, each position to the left represents a higher power of 2. Starting from the rightmost position, which is the units or ones place, the place values increase.

For example, here's how the place values correspond to the given binary number \(110111011_2\):

• The rightmost digit (known as the least significant bit) has the place value of \(2^0\), which is 1.
• Moving left, the next digit has the place value of \(2^1\), which equals 2.
• This pattern continues, doubling the place value with each shift to the left.

This concept is analogous to the decimal system, where each position to the left is ten times that of the previous position. So, in binary, it is a matter of powers of 2 rather than powers of 10.

The binary number system is a base-2 numeration system which uses only two symbols: 0 and 1. Each digit in this system is referred to as a bit. Unlike the decimal system, which is based on base 10 and uses ten symbols (0 to 9), the binary system's simplicity makes it ideal for computers and digital electronics, where each bit can represent an on (1) or off (0) state.

In the binary system, numbers are represented by a sequence of bits and increasing powers of 2. The value of a binary number is the sum of the values of the digits in the number, each of which is a power of 2, multiplied by either 1 or 0. The absence of a digit or a '0' in any position means that the corresponding power of 2 is not present in the sum that gives the decimal value.

The conversion process from binary to base 10, also known as decimal, involves multiplying each bit of the binary number by its corresponding power of 2 and then adding up all those products.

Following the steps for converting \(110111011_2\), we multiply each bit by its power of 2 place value:

• \(1 \times 2^0 = 1\)
• \(1 \times 2^1 = 2\)
• \(0 \times 2^2 = 0\)
• \(1 \times 2^3 = 8\)
• \(1 \times 2^4 = 16\)
• \(0 \times 2^5 = 0\)
• \(1 \times 2^6 = 64\)
• \(1 \times 2^7 = 128\)
• \(0 \times 2^8 = 0\)

We then add these values together: \(1 + 2 + 0 + 8 + 16 + 0 + 64 + 128 + 0 = 219\). This sum is the decimal representation of the binary number. By performing these calculations, you gain a better understanding of the relationship between binary numbers and their decimal equivalents.