Question

Perform each of the following additions assuming the bit strings represent values in two's complement notation. Identify each case in which the answer is incorrect because of overflow. a. \(00101+01000\) b. \(11111+00001\) c. \(01111+00001\) d. \(10111+11010\) c. \(11111+11111\) f. \(00111+01100\)

Short answer

Overflow occurs in c and f.

Step-by-step solution

Convert Bit Strings to Decimal

Before we add the bit strings, we need to interpret them as two's complement numbers. Remember, for two's complement, if the leading bit is 0, the number is positive. If it is 1, the number is negative.- For bit string \(00101\), leading bit is 0, so it's positive: 5.- For bit string \(01000\), leading bit is 0, so it's positive: 8.- For bit string \(11111\), leading bit is 1, so it's negative: -1.- For bit string \(00001\), leading bit is 0, so it's positive: 1.- For bit string \(01111\), leading bit is 0, so it's positive: 15.- For bit string \(10111\), leading bit is 1, so it's negative: -9.- For bit string \(11010\), leading bit is 1, so it's negative: -6.

Perform Binary Addition

Add the binary numbers while ignoring overflow:- a) \(00101 + 01000 = 01101\) (5 + 8 = 13)- b) \(11111 + 00001 = 00000\) (-1 + 1 = 0)- c) \(01111 + 00001 = 10000\) (15 + 1 = 16)- d) \(10111 + 11010 = 10001\) (-9 + -6 = -15)- e) \(11111 + 11111 = 11110\) (-1 + -1 = -2)- f) \(00111 + 01100 = 10011\) (7 + 12 = 19)

Check for Overflow

In two's complement arithmetic, overflow happens if: 1. The sum of two positive numbers yields a negative result. 2. The sum of two negative numbers yields a positive result. - a) No overflow (both positive, result positive) - b) No overflow - c) Overflow occurred (positive + positive = negative result) - d) No overflow (both negative, result negative) - e) No overflow - f) Overflow occurred (positive + positive = negative result)

Correct Answers Based on Overflow

Identify which additions result in incorrect answers due to overflow: - c) Overflow, result should be 16 (requiring extra bit, results in incorrect answer in provided bits) - f) Overflow, result should be 19 (requiring extra bit, results in an incorrect answer in provided bits)

Key concepts

Binary Addition

Binary addition is quite similar to decimal addition. It involves adding digits in a column, where you move from right to left. However, since we are dealing with only 1s and 0s, the rules are simpler:

• If you add 0 and 0, you get 0.
• If you add 0 and 1, or 1 and 0, the result is 1.
• If you add 1 and 1, the result is 0, but you carry over 1 to the next higher bit.

Let’s consider an example: adding the binary numbers 0101 and 0100. You start from the rightmost digits:

• 0 + 0 = 0
• 1 + 0 = 1
• 0 + 1 = 1
• 0 + 0 = 0

So, 0101 + 0100 equals 1001 in binary, which is equivalent to 9 in decimal. Binary addition is foundational in digital circuits and computer arithmetic. Being consistent with the binary system, it allows computers to perform complex calculations efficiently.

Overflow Detection

In two's complement arithmetic, detecting overflow is crucial. Overflow occurs when the arithmetic operation yields a result outside the representable range with the available number of bits. Here’s how to detect it:

• If you add two positive numbers and get a negative result, overflow has occurred.
• If you add two negative numbers and get a positive result, overflow has occurred.

Consider adding 01111 (15 in decimal) and 00001 (1 in decimal). The sum is 10000, which appears negative in a 4-bit system, suggesting overflow.
Another example is adding 00111 (7 in decimal) and 01100 (12 in decimal). The sum is 10011, which again is incorrectly negative due to overflow. It’s crucial to monitor the sign bits and check if they are flipped incorrectly due to an addition operation.
These scenarios of overflow highlight the need for cautious range checks in any computing tasks, especially while handling large-scale computations.

Bit String Conversion

Converting a bit string to its decimal equivalent, especially in two's complement notation, can be slightly tricky but manageable with practice. Here’s a simplified method:

• If the most significant bit (MSB), which is the leftmost bit, is 0 in an n-bit number, the number is positive. Calculate the decimal by straightforward addition of the powered values.
• If the MSB is 1, indicating a negative number, first flip all bits and add 1 to the least significant bit, then apply the negative sign to the result to get the decimal value.

For instance, consider the bit string 10111. It's a negative number (MSB is 1). Flip all bits to get 01000, and add 1 to it to make it 01001. This converts to 9 in decimal, hence the original was -9.
Having a clear understanding of bit string conversion serves as a foundation to advanced computational tasks, highlighting a computer’s internal representation of numbers.

Negative Number Representation

In binary systems, negative numbers are often represented using two's complement notation. It’s an elegant solution to handle both positive and negative integers using the same binary arithmetic.

• The most significant bit indicates the number's sign. A "0" means positive, and a "1" means negative.
• To find the two's complement (negative) of a number, invert all bits (flip 0s to 1s and vice versa) and add 1 to the resulting number.

For example, to represent -6 in an 8-bit number: Write 6 in binary (00000110), then flip all bits to get 11111001, and add 1 to get 11111010. So, -6 is represented as 11111010.
Representing negative numbers correctly in binary ensures arithmetic operations like addition and subtraction can be performed seamlessly, relying on the innate properties of binary systems to simplify encoding within computational processes.