Question

Using 8 bits, what is the unsigned binary representation of each of the following values: a. 23 b. 55 c. 275 Did anything unusual happen when determining the correct answer to Part c?

Short answer

a) 23: 00010111, b) 55: 00110111, c) 275: Not representable in 8 bits.

Step-by-step solution

Converting Decimal to Binary for Part a

To convert the decimal number 23 to binary, we use the process of dividing the number by 2 and keeping track of the remainders. Doing so, we have: - 23 ÷ 2 = 11 remainder 1 - 11 ÷ 2 = 5 remainder 1 - 5 ÷ 2 = 2 remainder 1 - 2 ÷ 2 = 1 remainder 0 - 1 ÷ 2 = 0 remainder 1 Reading the remainders from bottom to top, the binary representation of 23 is 10111. Since we are using 8 bits, we add leading zeros to make it 8 bits: 00010111.

Converting Decimal to Binary for Part b

Now, let's convert the decimal number 55 to binary. Divide by 2 and note the remainders: - 55 ÷ 2 = 27 remainder 1 - 27 ÷ 2 = 13 remainder 1 - 13 ÷ 2 = 6 remainder 1 - 6 ÷ 2 = 3 remainder 0 - 3 ÷ 2 = 1 remainder 1 - 1 ÷ 2 = 0 remainder 1 The remainders from bottom to top give us the binary number 110111. Add leading zeros for an 8-bit representation: 00110111.

Converting Decimal to Binary for Part c

Now, convert the number 275 into binary by the same method: - 275 ÷ 2 = 137 remainder 1 - 137 ÷ 2 = 68 remainder 1 - 68 ÷ 2 = 34 remainder 0 - 34 ÷ 2 = 17 remainder 0 - 17 ÷ 2 = 8 remainder 1 - 8 ÷ 2 = 4 remainder 0 - 4 ÷ 2 = 2 remainder 0 - 2 ÷ 2 = 1 remainder 0 - 1 ÷ 2 = 0 remainder 1 This gives the binary number 100010011, which requires 9 bits. This is unusual because we can only represent numbers up to 255 with 8 bits. Therefore, 275 cannot be represented in an 8-bit unsigned binary format.

Key concepts

Unsigned Binary Representation

Binary numbers are the foundation of digital electronics, representing values using only two symbols, 0 and 1. An **unsigned binary representation** focuses on these numbers without considering negative values. Each binary digit (bit) represents powers of 2.

• The rightmost bit is the least significant bit (LSB), representing \(2^0\).
• As you move left, each bit represents powers increasing upwards such as \(2^1\), \(2^2\), etc.

When we use unsigned binary, all bits contribute positively, allowing us to store numbers from 0 upwards. For example, for a 4-bit binary number, the range would be from 0 (0000) to 15 (1111). In unsigned forms, there is no need for a sign bit to indicate positive or negative, offering a straightforward representation of positive integers. By using this system, digital systems can manage a predictable range of values efficiently.

Decimal to Binary Conversion

Converting from decimal to binary involves understanding the relationship between these two number systems. Decimal numbers, which are base-10, must be translated into binary, which is base-2.

• To convert a decimal number to binary, you repeatedly divide the number by 2.
• Record the remainder for each division; these represent the binary digits.
• Continue this process until the result of the division is 0.

Once complete, read the remainders in reverse order (from last to first) to obtain the binary number. For instance, converting 23 gives the binary format 10111. To ensure all bits in an 8-bit system are occupied, leading zeros are added, resulting in 00010111. This straightforward method allows us to navigate between the decimal and binary systems, providing the necessary interpretation for various computational tasks.

8-bit Binary

When dealing with computers, an **8-bit binary system** represents values using exactly 8 bits. This setup is common in computer engineering as it fits into one byte, enabling efficient data manipulation.

• The smallest value possible in an 8-bit system is 0, noted as 00000000.
• The largest is 255, displayed as 11111111.

Each bit is crucial as it represents a distinct power of 2. Therefore, in an 8-bit unsigned binary number, we can store anything from 0 through 255. When handling numbers beyond this range, such as 275 in our case, an 8-bit system cannot suffice. This illustrates the limitation, highlighting the need to potentially expand to more bits for larger numbers, indicating why understanding binary bit limitations is vital in systems design and digital computation.