how does a byte show a negative number

Yulia Chive

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07-12-2024

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How Does a Byte Show a Negative Number? The Magic of Two's Complement

Computers store information as sequences of bits, the smallest units of data. Eight bits grouped together form a byte, capable of representing numbers, letters, or other data. But how does a seemingly simple byte, only capable of representing numbers from 0 to 255 (2⁸ - 1) in unsigned representation, manage to also represent negative numbers? The answer lies in a clever method called two's complement.

Understanding Binary Representation

Before diving into two's complement, let's refresh our understanding of how positive numbers are represented in binary. Each bit position represents a power of 2, starting from 2⁰ (the rightmost bit) and increasing to the left. For example:

• 0101 (binary) = 0 * 2³ + 1 * 2² + 0 * 2¹ + 1 * 2⁰ = 5 (decimal)

Introducing Two's Complement

To represent negative numbers, we modify the most significant bit (MSB), the leftmost bit, to act as a sign bit. A 0 in the MSB indicates a positive number, and a 1 indicates a negative number. However, simply using the MSB as a sign bit doesn't work efficiently; it leads to ambiguity and complicates arithmetic. This is where two's complement comes in.

Here's how it works:

1. Find the one's complement: Invert all the bits of the positive number. A 0 becomes a 1, and a 1 becomes a 0.

2. Add 1: Add 1 to the result of step 1. This is the two's complement representation of the negative number.

Let's illustrate this with an example. Let's represent -5 using an 8-bit byte:

1. Positive 5: 00000101 (binary)

2. One's complement: 11111010 (invert all bits)

3. Two's complement: 11111011 (add 1)

Therefore, 11111011 is the two's complement representation of -5 in an 8-bit byte.

Why Two's Complement?

Two's complement is superior to other methods for representing signed numbers because:

• Simplified Arithmetic: Addition and subtraction work the same way for both positive and negative numbers, simplifying hardware design. You don't need separate circuits for positive and negative operations.

• Unique Representation: Each number has a unique representation, eliminating ambiguity. There's no "+0" and "-0".

• Efficient Range: With an 8-bit byte, we can represent numbers from -128 ( 10000000) to +127 (01111111). This provides a balanced range around zero.

Converting Back to Decimal

To convert a two's complement number back to decimal:

1. Check the MSB: If the MSB is 1, the number is negative.

2. Find the two's complement: If negative, find the two's complement (invert the bits and add 1).

3. Convert to decimal: Convert the resulting binary number to decimal. If the original number was negative, negate the decimal value.

For example, let's convert 11111011 back to decimal:

1. MSB is 1: The number is negative.

2. Two's complement: Inverting the bits gives 00000100, adding 1 gives 00000101 (which is 5).

3. Decimal value: Since the original number was negative, the decimal value is -5.

Conclusion

Two's complement is a brilliant and efficient method for representing negative numbers in binary. It simplifies arithmetic operations and ensures a unique representation for each number within a given byte size, making it the standard approach in almost all modern computer systems. Understanding this technique is fundamental to comprehending how computers handle numerical data at a low level.