Question

Convert each of the following two's complement representations to its equivalent base 10 representation: a. 010101 b. 101010 c. 110110 d. 011011 e. 111001

Short answer

a. 21, b. -22, c. -10, d. 27, e. -7

Step-by-step solution

Understanding Two's Complement

Two's complement is a way of encoding positive and negative integers in binary. For positive numbers, the two's complement is the same as the binary number. For negative numbers, invert all bits and add 1 to the least significant bit.

Analysis of Each Binary String

We will analyze each binary number separately to determine whether it represents a positive or a negative number based on its leading bit (0 for positive, 1 for negative).

Convert 010101 (Positive Number)

010101 is a positive number (the leading bit is 0). Simply convert it to decimal using the powers of 2:\[0 \times 2^5 + 1 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 0 + 16 + 0 + 4 + 0 + 1 = 21\]

Convert 101010 (Negative Number)

101010 is a negative number (the leading bit is 1). To find its value:1. Invert all bits: 0101012. Add 1 to the least significant bit: 010101 + 1 = 0101103. Convert to decimal: \[0 \times 2^5 + 1 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 0 + 16 + 0 + 4 + 2 + 0 = 22\]Thus, the decimal value is -22.

Convert 110110 (Negative Number)

110110 is a negative number. 1. Invert all bits: 0010012. Add 1 to the least significant bit: 001001 + 1 = 0010103. Convert to decimal:\[0 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 0 + 0 + 8 + 0 + 2 + 0 = 10\]Thus, the decimal value is -10.

Convert 011011 (Positive Number)

011011 is a positive number. Convert directly to decimal:\[0 \times 2^5 + 1 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 0 + 16 + 8 + 0 + 2 + 1 = 27\]

Convert 111001 (Negative Number)

111001 is a negative number. 1. Invert all bits: 0001102. Add 1 to the least significant bit: 000110 + 1 = 0001113. Convert to decimal:\[0 \times 2^5 + 0 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 0 + 0 + 0 + 4 + 2 + 1 = 7\]Thus, the decimal value is -7.

Key concepts

Binary to Decimal Conversion

Binary to decimal conversion involves transforming a binary string, which is composed of 0s and 1s, into its decimal (base 10) equivalent. The key to this process is understanding that each position in a binary number represents a power of 2, starting from 0 on the far right.
For example, in the binary number 010101:

• The rightmost bit, 1, is in the \(2^0\) position.
• The next bit to the left, 0, is in the \(2^1\) position.
• And so on up to the leftmost bit, 0, which is in the \(2^5\) position.

To convert a binary number like 010101 to decimal:

• Multiply each bit by its corresponding power of 2.
• Add all these values together.

For 010101, the decimal conversion is calculated as:\[0 \times 2^5 + 1 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 21\]This systematic approach makes conversion from binary to decimal straightforward and clear, especially for positive representations.

Negative Number Representation

Negative number representation in computing is primarily accomplished through a method known as two's complement. It simplifies arithmetic operations and enables straightforward binary computation algorithms. Here's how it works:
To find the two's complement of a binary number:

• First, invert all the bits of the number. This means changing all 0s to 1s and all 1s to 0s, effectively flipping the binary number.
• Next, add 1 to the inverted number. This is added to the least significant bit, or the rightmost bit in the binary number.

For example, let's convert 101010, recognized by its leading bit 1 as a negative number, to decimal:

• Invert the bits: 101010 becomes 010101.
• Add 1: 010101 + 1 = 010110.
• Now convert 010110 to decimal: \(0 \times 2^5 + 1 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 22\)

Thus, the number is \(-22\). This method ensures that calculations with both positive and negatives are efficiently handled by digital systems.

Positive Number Representation

In binary systems, positive numbers are represented using the two's complement method, which is surprisingly straightforward for non-negative values. In these cases:

• The most significant bit, known as the leading bit, is set to 0 for positive numbers.
• The remaining bits are used to represent the number normally, as they would appear in standard binary form.

Take the binary number 011011 as an example:

• Since the leading bit is 0, this number is positive.
• To find the decimal equivalent, no inversion or additional steps are necessary. It's the plain sum of each bit multiplied by its respective power of 2.

Here's the conversion for 011011:\[0 \times 2^5 + 1 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 27\]This clear separation by using a leading bit helps efficiently manage and compute binary numbers in computer systems.

Bit Manipulation

Bit manipulation refers to the intentional change of bits to achieve a desired computational outcome. It is especially crucial in binary to two's complement conversions and plays a significant role in both representing numbers and performing calculations digitally.
Here's why it's important:

• **Inversion**: One of the most basic bit manipulation tasks, flipping bits, is essential in converting negative numbers in two's complement form.
• **Shifts**: Shifting bits left or right helps in efficient multiplication or division by powers of two.
• **Masking**: It isolates specific bits to toggle or read, which is useful for specific tasks like turning a set of bits on or off.

For example, in converting the binary 111001 to its decimal equivalent, bit manipulation involves:

• Inverting 111001 to get 000110.
• Adding 1 to obtain 000111.
• Then, translating 000111 to decimal \(0 \times 2^5 + 0 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 7\)

Hence, the original number is \(-7\). Understanding and mastering bit manipulation allows programmers and engineers to leverage the true power of binary computations.