Question

Give the 8-bit sign/magnitude representation of each of the following decimal values: a. \(+71\) b. \(-1\) c. \(-81\)

Short answer

+71: 01000111; -1: 10000001; -81: Cannot be represented in 8-bit sign/magnitude.

Step-by-step solution

Understanding Sign/Magnitude Notation

Sign/magnitude notation represents numbers using one bit for the sign and the remaining bits for the magnitude. The first bit indicates the sign (0 for positive, 1 for negative), and the remaining bits represent the magnitude (absolute value) of the number.

Converting +71 to 8-bit Sign/Magnitude

The decimal number +71 is positive, so the sign bit is 0. To convert 71 into binary, start by dividing by powers of 2: \[71_{10} = 1000111_2\] This requires 7 bits. Pad with a leading zero to make it 8 bits: \[+71 = 01000111\].

Converting -1 to 8-bit Sign/Magnitude

The decimal number -1 is negative, so the sign bit is 1. The magnitude of -1 is 1, and its binary representation is \[0000001_2\]. Thus, the 8-bit sign/magnitude representation is: \[-1 = 10000001\].

Attempting to Convert -81 to 8-bit Sign/Magnitude

The decimal number -81 is negative, so the sign bit is 1. The magnitude of -81 is 81, and its binary representation is \[1010001_2\]. However, since this already uses 7 bits, adding the sign bit results in 9 bits, which does not fit in an 8-bit format. Therefore, -81 cannot be represented in 8-bit sign/magnitude notation.

Key concepts

Binary Conversion

Binary conversion is a fundamental concept for understanding how computers process and store numbers. To convert a decimal number to binary, you must break it down into sums of powers of two. This is because the binary system is base-2, meaning it only uses the digits 0 and 1. For example, to convert the decimal number 71 into binary:

• Start with the highest power of 2 that fits into 71, which is 64, equivalent to \(2^6\), and subtract 64 from 71, leaving 7.
• Next, the largest power of 2 that fits into 7 is 4 (\(2^2\)), leaving 3.
• Then, the largest power of 2 for 3 is 2 (\(2^1\)), leaving 1.
• Finally, 1 is \(2^0\), completing the conversion.

Putting these together, 71 is represented by 1s in the positions of \(2^6\), \(2^2\), \(2^1\), and \(2^0\), resulting in the binary number \(1000111_2\). This process ensures that numbers can be expressed with the binary digits 0 and 1, essential for digital communication.

8-bit Representation

The 8-bit representation refers to using exactly 8 binary digits to represent a number. This is a common format in computing because it fits well into a byte, which is 8 bits. In 8-bit representation, the leftmost bit often serves as the sign bit in systems that use sign/magnitude or similar notations. Here's how it works:

• One bit is dedicated to indicating the sign of the number. A '0' indicates a positive number, while a '1' indicates a negative number.
• The remaining 7 bits are used for the magnitude or size of the number.

To maintain an 8-bit format:

• If the binary representation of the number requires less than 8 bits, add zeros at the beginning, known as padding, to meet the 8-bit requirement.
• This ensures a uniform number length, crucial for operations in digital systems.

For example, the positive decimal number 71 is represented as \(01000111\) to adhere to the 8-bit format, including the sign bit.

Negative Numbers in Binary

Handling negative numbers in binary can be tricky. Sign/magnitude representation is one way to express them, though it has limitations. In sign/magnitude:

• The first bit of an 8-bit number represents the sign (0 for positive, 1 for negative).
• The remaining bits reflect the absolute value of the number.

For example, the number -1 is represented as:

• The sign bit is 1 because the number is negative.
• The binary form of the number 1 is \(0000001_2\).
• Combining these gives \(10000001\) in 8-bit sign/magnitude format.

There is a drawback when converting larger negative numbers. For instance, -81 can't be represented in 8-bit sign/magnitude because its absolute value 81 needs more than the 7 bits reserved for magnitude. This brings attention to other methods like two's complement, which better handle the range limits for negative numbers in binary.