Question

Mark the answers true or false as follows: A. True B. False The signed-magnitude representation of numbers has two representations for zero.

Short answer

A. True

Step-by-step solution

Understanding Signed-Magnitude Representation

The signed-magnitude representation is a method for storing signed numbers in binary form. In this system, one bit is designated for the sign of the number: '0' typically represents a positive number, and '1' represents a negative number. The remaining bits represent the magnitude (absolute value) of the number.

Representing Zero in Signed-Magnitude

In signed-magnitude form, zero can be represented with both a positive sign and a negative sign. '0' for the sign bit means +0 and '1' for the sign bit means -0. Therefore, zero has two different signed representations: +0 and -0.

Conclusion and Answering

Given that there are indeed two different representations for zero in signed-magnitude form (+0 and -0), the statement that "The signed-magnitude representation of numbers has two representations for zero" is accurate.

Key concepts

Signed-Magnitude Representation

The signed-magnitude representation is a simple and intuitive method to encode signed numbers in binary format.
In this system, the leftmost bit, known as the "sign bit," indicates the sign of the number. Generally, a sign bit of '0' means the number is positive, while '1' indicates the number is negative. The rest of the bits, called the "magnitude," represent the actual absolute value of the number.
For example, in an 8-bit signed-magnitude representation:

• The number +5 would be represented as 00000101.
• The number -5 would be represented as 10000101.

An interesting aspect of signed-magnitude representation is its handling of zero. It can exist as either +0 or -0 (00000000 or 10000000, respectively). Though these two representations are semantically equivalent, they highlight a peculiar aspect of this number format.
This dual representation of zero can complicate calculations in computational systems.

Binary Numbers

Binary numbers are the foundational language of computing systems. Comprised of only two digits, 0 and 1, binary numbers represent data and perform computation in digital electronics.
Each position in a binary number reflects a power of 2, starting from the rightmost bit, which represents 2⁰ (1), moving leftward to 2¹ (2), 2² (4), and so on.
For example, the binary number 1101 represents:

• 1 × 2³ = 8
• 1 × 2² = 4
• 0 × 2¹ = 0
• 1 × 2⁰ = 1

When added together, they sum to 13. Thus, 1101 in binary equals 13 in decimal.
Binary numbers are crucial for all computer technologies, as all operations and data storage rely on this system. Understanding how to translate and perform arithmetic with binary is essential for computer science education.

Data Representation

Data representation is critical in computer systems as it defines how different types of data are captured, stored, and manipulated.
Digital systems handle various forms of data—including numbers, text, images, and sound—by converting them into binary format.
There are numerous data representation formats, each suited to different data types. For numeric data alone, possibilities include:

• Unsigned binary: Represents only positive integers.
• Signed-magnitude binary: Distinguishes positive and negative numbers using a sign bit.
• Two's complement: Simplifies arithmetic operations by maintaining a single representation for zero.

Textual data is typically encoded using standards like ASCII or Unicode, each assigning a unique binary code to every character.
By managing data representation effectively, computers balance storage efficiency with processing speed, thereby enhancing overall performance.

Number Systems

Number systems are methods for representing numbers in different bases. While humans commonly use the decimal system (base 10), computers operate on the binary system (base 2).
Apart from these, there are other important number systems used in computing:

• Octal (base 8): Groups binary digits into chunks of three, simplifying long binary numbers.
• Hexadecimal (base 16): Groups binary digits into chunks of four, often used in programming because of its concise representation.
• Decimal (base 10): Familiar number system to humans, based on ten digits (0-9).

These systems are crucial for different aspects of computing, offering efficient ways to represent, process, and interpret numbers.
Understanding how to convert between these systems or use them in different contexts helps form the backbone of computer science literacy.