Question

One of the bit patterns 01011 and 11011 represents a value stored in excess 16 notation and the other represents the same value stored in two's complement notation. a. What can be determined about this common value? b. What is the relationship between a pattern representing a value stored in two's complement notation and the pattern representing the same value stored in excess notation when both systems use the same bit pattern length?

Short answer

The common value is -5. Patterns differ due to encoding rules: two's complement/sign, excess notation/offset.

Step-by-step solution

Understand Excess 16 Notation

Excess 16 notation is a way of encoding numbers where we add 16 to the actual integer value. Therefore, if we have a bit pattern in excess 16 notation, we convert it to the true value by subtracting 16 from the bit pattern's integer equivalent. For a 5-bit pattern, 16 is the offset.

Decode Bit Patterns into Decimal

Convert the binary patterns 01011 and 11011 into decimal. For 01011: \(0 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 11\). For 11011: \(1 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 27\).

Analyze Excess 16 Conversion

Convert the excess 16 encoded value to its actual value by subtracting 16. So, if 01011 is in excess 16 notation, the represented value is \(11 - 16 = -5\). Similarly, if 11011 is in excess 16 notation, the represented value is \(27 - 16 = 11\).

Analyze Two's Complement Conversion

Interpret the bit patterns as two's complement numbers. For 01011, it's positive and equals 11 in decimal. For 11011, being a negative in two's complement since the leftmost bit is 1, convert to positive by inverting bits and adding one: \(00100 = 4\), so the original number is \(-4-1 = -5\).

Identify the Common Value

Cross-reference both interpretations: for 01011, excess 16 yields -5 and as a two's complement yields 11. For 11011, excess 16 yields 11 and as a two's complement yields -5. Therefore, the common value both represent (different encodings) is -5.

Understand Relationship Between Encoding Systems

When using the same bit pattern length (like 5 bits here), the relationship is that the conversion from two's complement to excess 16 involves interpreting the binary number according to each notation's rules—which often, as seen from our example, results in different apparent values unless further interpreted or converted properly. Two's complement is focused on signed representation, while excess notation shifts values by a constant.

Key concepts

Two's Complement Notation

Two's complement notation is a system used in computing to represent signed integers with binary numbers. It allows for the encoding of positive and negative numbers in a straightforward manner.
The most significant bit (the leftmost one) is used as a sign bit. Here's how it works:

• When the sign bit is 0, the number is positive, and its magnitude is represented by the rest of the bits.
• When the sign bit is 1, the number is negative. To find its magnitude, reverse all the other bits and then add one to the least significant bit.

This system simplifies arithmetic operations like addition and subtraction by handling them as unsigned numbers, making it integral for efficient computation.
For example, in a 5-bit system, the binary number 11011 is negative in two's complement. To find the decimal value, invert (00100) and add one (00101 = 5), resulting in a decimal value of -5.

Excess-16 Notation

Excess-16 notation (also known as biased notation) is similar but slightly different from two's complement notation.
It involves adding a constant, typically referred to as a bias, to the actual number. For a 5-bit system, this bias is 16, hence the name "excess-16".
The process is simple:

• Convert the binary number to its decimal form.
• Subtract the bias (16) from this value to find the actual integer value.

This method is frequently used in situations where non-negative integers need a simplistic encoding, like in computer graphics. For instance, for the 5-bit pattern 01011 in excess-16 notation, you convert it to a decimal (11) and then subtract 16, resulting in -5.

Bit Patterns

Bit patterns are sequences of bits (binary digits), either 0s or 1s, used in digital computing to represent various data including numbers.
Binary patterns are foundational for binary arithmetic and data encoding schemes like two's complement and excess-16 notation. Here's how they function:

• Each position in a bit pattern represents a power of 2, starting from 2^0 at the rightmost bit.
• The combination of 0s and 1s in a pattern determines the encoded value.

These bit patterns take different meanings depending on the system of representation applied, such as unsigned, two's complement, or excess notations. For example, given the bit pattern 01011, it can represent 11 in unsigned binary, -5 in two's complement, or -5 in excess-16 depending on how you interpret it.