Binary Overflow

In binary addition, overflow occurs when the sum of two binary numbers exceeds the maximum representable value. This happens because binary numbers have a specific range of representable values based on the number of bits they contain.

When the sum exceeds this range, the result cannot be accurately represented, resulting in an overflow.

What happens in binary addition when the sum exceeds the range of representable values?

Example of Binary Overflow

Let's consider a simple 4-bit binary addition, 1101 + 0101, where we want to store the result in a 4 bit register.

The sum of these two numbers is 10010.

However, since we are working with only 4 a bit register, the most significant bit (MSB) of the sum (1) cannot be represented, resulting in overflow.

The result therefore is that we discard the most significant bit (left-most bit), which gives us an answer of 0010.

Overflow

Consequences of Binary Overflow

Incorrect Computation Results

When an overflow occurs, the result may wrap around or be truncated, leading to incorrect calculations.

Sign Errors (for Signed Numbers)

In signed representations (e.g., two’s complement), overflow can cause a positive number to become negative and vice versa.

System Crashes or Unexpected Behavior

Some systems or programming languages detect overflow and raise exceptions, which may crash programs.

In critical systems (e.g., avionics, banking), overflow can cause serious malfunctions.

Security Vulnerabilities

Overflow can lead to buffer overflows, which attackers exploit to execute malicious code (common in C/C++ programs).

Data Corruption

In databases or files, overflow may cause unintended data corruption, making stored values invalid.

What is a possible consequence of numerical overflow in calculations?

Binary Underflow

On the other hand, underflow occurs when subtraction is performed on binary numbers, and the result is smaller than the minimum representable value.

Just like overflow, underflow happens because binary numbers have limits on the representable values based on the number of bits used.

What is underflow in the context of binary numbers?

Suppose we have a 5-bit binary subtraction: 01010 - 10101.

The result of this subtraction is -10011.

However, in a 5-bit representation, the leftmost bit is used to indicate the sign (0 for positive, 1 for negative), and the remaining bits represent the magnitude.

Therefore, -10011 cannot be accurately represented in 5 bits, causing underflow. In such cases, the correct result would involve using additional bits or a different representation to accurately represent the negative value.

Binary underflow occurs when the result of a calculation is too small to be represented in the allocated .

Consequences of Binary Underflow

Rounding to Zero (Hard Underflow)

If the result of an operation is smaller than the smallest representable value, it is rounded to zero.

Loss of Significant Digits / Precision

When underflow occurs, precision is lost, leading to inaccuracies in calculations.

Errors in Programs (if not handled properly)

Some systems or languages treat underflow as an error, causing exceptions or warnings.

What is a potential consequence of underflow?

Activity Complete

Home CS Master Data Binary & Hexadecimal Binary Overflow & Underflow