Question

Convert each of the following base 10 representations to its equivalent two's complement representation in which each value is represented in 8 bits: a. \(-27\) b. 3 d. 8 c. \(-18\) c. 21

Short answer

-27: 11100101, 3: 00000011, 8: 00001000, -18: 11101110, 21: 00010101.

Step-by-step solution

Converting Negative Numbers to Two's Complement

To convert a negative number to its two's complement representation in 8 bits, first find the binary representation of its positive value. For example, for -27. - Step a: Convert 27 to binary: 11011. - Step b: Pad to 8 bits: 00011011. - Step c: Invert all bits: 11100100. - Step d: Add 1 to the result: 11100100 + 1 = 11100101. Thus, -27 is represented as 11100101.

Converting Positive Numbers to Binary

A positive number can directly be converted to binary. For example, for the number 3. - Binary representation of 3: 11. - Pad this to 8 bits: 00000011. Thus, 3 is represented as 00000011.

Two's Complement for Another Negative Number

Repeat the conversion for another negative number, such as -18. - Step a: Convert 18 to binary: 10010. - Step b: Pad to 8 bits: 00010010. - Step c: Invert all bits: 11101101. - Step d: Add 1 to the result: 11101101 + 1 = 11101110. Thus, -18 is represented as 11101110.

Binary Representation for a Positive Number

Convert 8 to its binary representation. - Binary of 8: 1000. - Pad to 8 bits: 00001000. Thus, 8 is represented as 00001000.

Convert Another Positive Number

Finally, convert 21 to binary. - Binary of 21: 10101. - Pad to 8 bits: 00010101. Thus, 21 is represented as 00010101.

Key concepts

Binary Representation

Binary representation is a way of expressing numbers using only the digits 0 and 1. Computers rely on this number system since they only understand electrical signals that can be in one of two states: on (1) or off (0). To convert a decimal number to binary, one can repeatedly divide the number by 2 and note the remainder.

• Start from the least significant bit (rightmost) after initializing the number to 0.
• Divide the number by 2, noting the quotient and remainder. The remainder becomes part of the binary representation.
• Continue until the quotient reaches 0.

When converting a negative decimal number into binary, the normal process is used first to find the positive binary equivalent. This is then turned into its two's complement form, which enables the representation of negative values in binary.

Base 10 Conversion

Base 10, also known as the decimal system, is the most commonly used number system among humans. It uses ten digits, from 0 to 9. When converting from base 10 to another system, such as binary, it involves dividing the number by the base of the new system and keeping track of remainders. To convert back from binary (or any other base) to base 10, you multiply each digit by a power of the base, depending on its position. This means:

• The rightmost digit is multiplied by the base raised to the power of 0 (base⁰).
• The next digit to the left is multiplied by the base raised to the power of 1 (base¹), and so on.

The sum of these products gives you the decimal representation of the binary number.

8-bit Numbers

In computing, 8-bit refers to the number of bits available to represent a value. This means you can have up to 256 different values, ranging from binary 00000000 to 11111111. In 8-bit representation, numbers are often padded with zeros on the left to ensure they fit within the 8-bit range. This padding is essential for maintaining memory consistency and proper arithmetic calculations in digital systems. For example, a binary number like 1101 (13 in decimal) would be expressed as 00001101 in an 8-bit system. When eight bits are used for two's complement, the range of representable numbers is from -128 to 127, which allows for encoding both negative and positive values, along with zero.

Negative Numbers

Negative numbers in binary are represented using a method called two's complement. This technique enables the consistent and straightforward handling of subtraction and addition by digital electronics. Here's a quick overview of how to convert a negative decimal number to its two's complement binary form in an 8-bit system:

• Start with the positive binary equivalent of the number.
• Invert every bit, changing all 0s to 1s and all 1s to 0s.
• Add one to the inverted number to complete the two's complement representation.

This method ensures that the arithmetic operations are seamless, even with negative numbers, and it makes electronic circuits simpler by not requiring additional logic for sign handling. For instance, -27 in binary using 8 bits would be represented as 11100101 after taking the two's complement of 00011011.