Question

Assume that the following 10 -bit numbers represent signed integers using sign/ magnitude notation. The sign is the leftmost bit and the remaining 9 bits represent the magnitude. What is the decimal value of each? a. 1000110001 b. 0110011000 c. 1000000001 d. 1000000000

Short answer

a: -49, b: 408, c: -1, d: 0

Step-by-step solution

Understand Sign/Magnitude Notation

In sign/magnitude notation, the leftmost bit indicates the sign of the number: 0 for positive and 1 for negative. The remaining bits represent the magnitude of the number.

Convert Binary to Decimal (a)

For the number 1000110001: The sign bit is 1, indicating a negative number. The magnitude bits are 000110001, which in decimal is 49. Hence, the decimal value is -49.

Convert Binary to Decimal (b)

For the number 0110011000: The sign bit is 0, indicating a positive number. The magnitude bits are 110011000, which in decimal is 408. Therefore, the decimal value is 408.

Convert Binary to Decimal (c)

For the number 1000000001: The sign bit is 1, indicating a negative number. The magnitude bits are 000000001, which in decimal is 1. Thus, the decimal value is -1.

Convert Binary to Decimal (d)

For the number 1000000000: The sign bit is 1, indicating a negative number. The magnitude bits are 000000000, which in decimal is 0. So, the decimal value is 0.

Key concepts

Binary to Decimal Conversion

Binary to decimal conversion is an essential skill in understanding computer representations of numbers. In binary, numbers are represented using only 0s and 1s, which are the basic units of data known as bits. Each bit represents a power of two, depending on its position.
- Starting from the rightmost bit, also known as the least significant bit, it's the power of 2 raised to 0. - The next bit to the left represents 2 raised to 1, then 2 squared, and so on.
To convert a binary number to decimal, list the powers of two corresponding to positions of 1s in the binary number. Add up these powers to get the decimal equivalent. For example, for the binary number `101`, highlight 2 squared (which is 4), 2 raised to 0 (which is 1), add them to get the decimal number 5.

Signed Integers

Signed integers are numbers that can be either positive or negative. In computer systems, these numbers are typically represented using a fixed number of bits, with one of these bits reserved as a sign bit.
- A sign bit of 0 usually denotes a positive number, while a sign bit of 1 denotes a negative number. - The rest of the bits are used to express the 'magnitude' or the absolute value of the number.
It's worth noting that there are other techniques for representing negative numbers, such as two's complement, but sign/magnitude is one of the simplest methods. It allows easy identification of the sign and magnitude, although it duplicates zero (as both positive and negative) which is a minor drawback.

Bit Representation

Bit representation is the way computers use binary digits or bits to encode information. Each bit can be one of two values, 0 or 1, which allows computers to perform complex calculations and manage data processing. All digital data, including numbers, text, images, and instructions, are converted into binary to be processed by computers.
- The bit representation is foundational to understanding computing systems, as it provides the basis for all data operations. - Combinations of bits form bytes, which are typically 8 bits long, and they are used to signify more complex data types.
For signed integers, the most significant bit often acts as the sign bit, influencing how the remaining bits are interpreted. Understanding bit representation allows us to grasp how data is stored, manipulated, and interpreted within a computer.