You:

here is a statement in csapp (provided by the image). How could the number of bigger data type with leading 1 equals to the original number? explain it a bit.

ChatGPT:

This statement refers to the sign extension process in two’s-complement representation, which ensures that the value of a number remains the same when its bit-width is expanded.

Key Idea of Two’s-Complement Sign Extension

In two’s-complement notation, the most significant bit (MSB) is the sign bit:

• 0 indicates a non-negative number.
• 1 indicates a negative number.

When extending a two’s-complement number to a larger bit-width, the sign bit is replicated (copied) into the additional higher-order bits. This preserves the number’s original value because:

• For positive numbers (MSB = 0), adding leading 0s does not change the magnitude.
• For negative numbers (MSB = 1), adding leading 1s preserves the negative value by maintaining the two’s-complement encoding.

Why Does This Work?

The key is in how two’s-complement numbers are interpreted:

$$B2T_w(\mathbf{x}) = -x_{w-1} \cdot 2^{w-1} + \sum_{i=0}^{w-2} x_i \cdot 2^i$$

Here:

• $x_{w-1}$ is the sign bit.
• $x_i$ are the other bits of the number.

When sign-extending to a new width $w’ > w$, the replicated bits do not alter the value because:

• For positive numbers ($x_{w-1} = 0$), adding leading 0s contributes nothing to the sum.
• For negative numbers ($x_{w-1} = 1$), adding leading 1s adds a negative offset equivalent to what would be added if the value were represented at the original width $w$.

Example: 4-bit to 8-bit

Let’s see an example.

Case 1: Positive Number

Consider the 4-bit number $0110$ (6 in decimal). When expanded to 8 bits:

$$0110 \rightarrow 00000110$$

The value is unchanged, as the leading 0s contribute nothing.

Case 2: Negative Number

Consider the 4-bit number $1010$ (-6 in decimal). When expanded to 8 bits:

$$1010 \rightarrow 11111010$$

The leading 1s preserve the negative value in two’s-complement, keeping it as -6.

This process guarantees that the number’s value remains consistent regardless of its width.