Understanding the Binary Number Representation

The question provides a binary number in a symbolic form: (cdccddcccddd)₂. It also gives a condition that 'c > d'. Since this is a binary number, the only allowed digits are 0 and 1.

We need to figure out what digits 'c' and 'd' represent based on the given condition and the rules of binary numbers.

Determining the Values of c and d

In the binary system, the digits can only be 0 or 1. The condition given is c > d.

• If we assume c = 0, there is no binary digit 'd' that is smaller than 0.
• If we assume c = 1, the only binary digit 'd' that is smaller than 1 is 0.

Therefore, to satisfy the condition c > d within the binary system, 'c' must be 1 and 'd' must be 0.

Reconstructing the Binary Number

Now that we know c = 1 and d = 0, we can replace the symbols in the given binary number (cdccddcccddd)₂:

• Replace 'c' with '1'.
• Replace 'd' with '0'.

The sequence of symbols is cdccddcccddd. Replacing them gives us:

1 0 1 1 0 0 1 1 1 0 0 0

So, the actual binary number is (101100111000)₂.

Converting Binary to Decimal Explanation

To convert a binary number to its decimal equivalent, we use the positional value of each digit. In a binary number, each position represents a power of 2, starting from \(2^0\) for the rightmost digit, \(2^1\) for the next digit to the left, \(2^2\) for the next, and so on.

The decimal value is the sum of each digit multiplied by its corresponding power of 2.

For a binary number \((b_{n-1} b_{n-2} \dots b_1 b_0)_2\), the decimal equivalent is calculated as:

\(\text{Decimal} = b_{n-1} \times 2^{n-1} + b_{n-2} \times 2^{n-2} + \dots + b_1 \times 2^1 + b_0 \times 2^0\)

Step-by-Step Conversion of (101100111000)₂ to Decimal

Our binary number is (101100111000)₂. It has 12 digits. We start assigning powers of 2 from the rightmost digit, which is at position 0 (for \(2^0\)). The leftmost digit is at position 11 (for \(2^{11}\)).

Now, we sum up the products obtained from each position:

Decimal Equivalent = 2048 + 0 + 512 + 256 + 0 + 0 + 32 + 16 + 8 + 0 + 0 + 0

Decimal Equivalent = 2048 + 512 + 256 + 32 + 16 + 8

Let's perform the addition:

• 2048 + 512 = 2560
• 2560 + 256 = 2816
• 2816 + 32 = 2848
• 2848 + 16 = 2864
• 2864 + 8 = 2872

The decimal equivalent of the binary number (101100111000)₂ is 2872.

Conclusion

By understanding the constraints of the binary system and the given condition c > d, we determined the specific binary number. Converting this binary number (101100111000)₂ to base 10 using positional notation gives us the decimal equivalent of 2872.

Revision Table: Binary to Decimal Conversion Key Points

Additional Information: Understanding Number Bases

Number bases, or radixes, define the number of unique digits used in a number system and the value each digit position represents.

• Base 10 (Decimal): Uses ten digits (0-9). Position values are powers of 10 (\(10^0, 10^1, 10^2, \dots\)).
• Base 2 (Binary): Uses two digits (0-1). Position values are powers of 2 (\(2^0, 2^1, 2^2, \dots\)). This is the native language of computers.
• Base 8 (Octal): Uses eight digits (0-7). Position values are powers of 8. Useful because 8 is \(2^3\), so one octal digit represents three binary digits.
• Base 16 (Hexadecimal): Uses sixteen "digits" (0-9 and A-F). Position values are powers of 16. Useful because 16 is \(2^4\), so one hex digit represents four binary digits. Often used in programming and computer memory representation.

Converting between these bases is a common task in computer science and digital electronics.