Question

Assume that we use 10 bits to represent signed integers, using \(\operatorname{sign} /\) magnitude notation. What are the largest (in absolute value) positive and negative numbers that can be represented on our system?

Short answer

Largest positive: 511, largest negative: -511.

Step-by-step solution

Understanding Sign/Magnitude Notation

In sign/magnitude notation, the first bit is used as the sign bit, and the remaining bits represent the magnitude of the number. A sign bit of 0 indicates a positive number, while a sign bit of 1 indicates a negative number. Since we have 10 bits total, one bit is for the sign, and 9 bits are available for representing the magnitude.

Calculating Maximum Positive Value

To find the largest positive number, set the sign bit to 0 (indicating positive) and use all remaining 9 bits for magnitude. The maximum value for 9 bits is all ones, which is \[2^9 - 1 = 511.\] Therefore, the largest positive number is 511.

Calculating Maximum Negative Value

To find the largest in absolute value negative number, set the sign bit to 1 (indicating negative) and use all remaining 9 bits for the magnitude as all ones (same as the maximum positive magnitude). Thus, the largest negative value in absolute terms is -511.

Key concepts

Sign/Magnitude Notation

Sign/magnitude notation is one of the simplest methods to represent signed integers in binary. Imagine it like this: the first spot is reserved for the mood of the number, whether it feels positive or negative. This is known as the sign bit. A sign bit of '0' means the number is happy, or positive, while a '1' means it's quite the opposite, negative.
After the sign bit, the remaining bits are used to showcase the strength or magnitude of the number. This decision makes reading and understanding the sign of a number super easy at just a glance. However, this simplicity comes with some quirks, especially in computational terms.
For example, using 10 bits as mentioned, the first bit indicates the sign and the next 9 bits flash the digits of the number itself, determining how massive it is in value. While the sign/magnitude notation is straightforward, it is not without its catch – it can represent both positive zero and negative zero, leading to two representations of the concept "zero." It also doesn’t handle arithmetic as effectively as other notations.

Binary Representation

Binary representation is the backbone of all digital electronics, like computers and smartphones. At its simplest, binary uses only two symbols: 0 and 1. It’s a bit like a special language that computers use to interpret data and perform calculations.
In binary representation, each digit represents a power of 2. For example, as seen in our case of 10 bits, the last bit represents 2 to the power of 0, or 1, the second last bit represents 2 to the power of 1, or 2, and so on till the final bit which represents 2 to the power of 9, or 512.
Why bother with binary? It’s because it aligns so neatly with computer systems, which use electrical signals that are either on or off. So, each binary digit or 'bit' is an instruction to the computer, turning things on and off.

• Easy to process and understand for computers, helping them "think" fast.
• Ensures precision in storing and processing massive amounts of data.

Thus, when we consider 10 bits as our playground for integers in sign/magnitude notation, 9 bits show the magnitude, while 1 bit sets the sign mood.

Computer Science Problem Solving

When confronted with a problem like representing the largest positive and negative numbers using sign/magnitude notation with 10 bits, it's a puzzle-solving adventure! Here's how you can systematically approach such problems.
The first task is to understand the rules: how the sign/magnitude notation spreads its influence across bits by using the first bit to set the sign and the rest to determine magnitude. It’s just like laying down the rules before getting to a game.
Next, translate those rules into potential solutions: calculate the largest numbers possible by maximizing the available bits. When dealing with binary digits:

• Use all the right spaces (bits) to push for the "biggest" number that these bits allow.
• Remember, you'll multiply each position by 2 raised to a power starting from 0, till the last bit.

Problem-solving skills like this in computer science require knowing how to interpret signs and magnitudes, effectively splitting bits between them. This kind of understanding is foundational, as it opens doors to more complex tasks in computer architecture and design.