Teaching Kids Programming – Introduction to Two’s Complement (Storing Negative Integers)

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Two’s complement is a mathematical method used in computers to represent positive and negative integers. It’s the most common method for representing signed integers on computers.

Two’s Complement

Here’s how it works:

Positive Numbers

For positive numbers (and zero), two’s complement representation is the same as the regular binary representation. For example, in an 8-bit system (where numbers are represented using 8 bits), the decimal number 5 is represented as 00000101.

Negative Numbers

For negative numbers, the process is a bit more complex:

• Step 1: Start with the absolute value of the number in binary. For example, for -5, start with the binary of 5, which is 00000101.
• Step 2: Invert the digits. The binary of 5 (00000101) becomes 11111010.
• Step 3: Add 1 to the inverted number. So, 11111010 becomes 11111011. This is the two’s complement representation of -5.

Why It Works: This method allows for easy arithmetic operations, especially addition and subtraction. When you add a positive number and its negative counterpart, the result is zero, due to the way binary addition works. For example, adding 5 and -5 in two’s complement (00000101 + 11111011) results in 1 00000000. In an 8-bit system, the leftmost 1 is discarded, leaving 00000000, which is zero.

Advantages

Two’s complement simplifies the design of computers and other digital systems because it allows the use of the same hardware circuits for both addition and subtraction, and it automatically handles overflow.

Special Cases

In a system with a fixed number of bits, the range of representable numbers is limited. For an 8-bit system, the range is from -128 to 127. Note that there’s one more negative number than positive, because 10000000 (the highest number in 8-bit) represents -128 in two’s complement.

Overflow

In arithmetic operations, if the result of an operation exceeds the range that can be represented in the given number of bits, an overflow occurs. This is especially important to consider in signed arithmetic.

Zero’s Two’s Complement

Understanding two’s complement is fundamental in computer science, especially when dealing with low-level programming or digital electronics, where binary arithmetic is common.

In two’s complement representation, zero has only one representation, which is all bits set to 0. This is a key feature and advantage of two’s complement over other binary representations for signed numbers.

For example, in an 8-bit system, zero is represented as 00000000. There isn’t a separate representation for “-0”. This is unlike some other number systems (like the signed magnitude or ones’ complement systems) where both positive and negative versions of zero can exist.

In two’s complement, the negation of zero (inverting all bits and adding one) still results in zero. If you invert all bits of 00000000, you get 11111111, and adding one to this results in 1 00000000 (in a 9-bit representation), but since we’re considering an 8-bit system, the leftmost bit is discarded, leaving us back with 00000000.

This unique representation of zero in two’s complement simplifies arithmetic operations and logic in computer systems, as there’s no need to handle positive and negative zero differently.

–EOF (The Ultimate Computing & Technology Blog) —