Question

Assume that you tried to store the signed integer value \(-200\) using an 8-bit sign/ magnitude representation. What happened? What type of error does this represent?

Short answer

-200 causes an overflow error as it cannot fit in an 8-bit sign-magnitude representation.

Step-by-step solution

Understanding Sign-Magnitude Representation

In the sign-magnitude representation, one bit (usually the most significant bit) is used to indicate the sign of the number; a bit value of 0 indicates a positive number, while a bit value of 1 indicates a negative number. The remaining bits are used to represent the magnitude of the number.

Calculate the Maximum and Minimum Values

In an 8-bit sign-magnitude representation, the maximum value that can be represented is 01111111 (which is +127), and the minimum value is 11111111 (which is -127). Therefore, the range of representable numbers is from -127 to +127.

Attempt to Represent -200

To represent -200 in sign-magnitude form, we would set the first bit to 1 (indicating a negative number) and the remaining 7 bits should represent 200. However, since the 7-bit magnitude can only represent values up to 127, -200 cannot be represented.

Identify the Error Type

The inability to store -200 in an 8-bit sign-magnitude representation is a classic example of an overflow error because the magnitude exceeds the capacity of the available bits.

Key concepts

Overflow Error

Overflow error occurs when a calculation produces a number that is outside the range of representable values. In computer systems, this happens due to limited storage capacity. When using an 8-bit representation, there's a cap on the magnitude of numbers that can be stored. If a number tries to go beyond this limit, as in the case of attempting to represent

• -200
• Signs indicating direction, or positive/negative

, it results in an overflow.
In sign-magnitude representation, there are specific bounds for both positive and negative numbers:

• The maximum positive value: +127
• The maximum negative value: -127

Trying to represent any number outside of these bounds will result in an overflow error. This error is essential to consider in computations involving fixed-size storage, as it can cause unexpected results or system crashes.

8-bit Representation

An 8-bit representation refers to a format in which numbers use 8 binary digits—or bits. This is common in early computer systems and low-level computing due to its simplicity. Each bit can be 0 or 1, and the sequence of these bits determines the value they represent.
The limitation of an 8-bit system is its restricted range, which is from 0 to 255 for unsigned integers, and

• -127 to +127 for signed numbers

. In this type of representation, the first bit is often designated for indicating the sign (positive or negative), known as the sign bit. The remaining 7 bits represent the number’s magnitude. An overflow occurs when a number like -200 cannot fit within this range, indicating the need to consider alternative data representations for larger numbers.

Signed Integers

Signed integers are numbers that include both positive and negative values, distinguished by a sign bit in binary representation.

• The addition of a sign bit effectively halves the representable range of values for a given bit-size compared to unsigned integers.

In an 8-bit signed integer, the values range from -128 to +127 when using two's complement form, and from -127 to +127 for sign-magnitude representation.
When considering operations with signed integers, it is crucial to be aware of the risk of overflow. This happens when arithmetic exceeds the limits of the storage format, typically resulting in an incorrect and unexpected value. To mitigate this, careful planning and adequate bit-size allocation are necessary when designing systems expected to handle large variance in values.