Signed and Unsigned Numbers in Binary Representation

In digital systems, we need to represent both positive and negative numbers using only 0s and 1s. The distinction between signed and unsigned interpretation of binary data is fundamental to computer architecture and programming.

Unsigned Numbers

Unsigned numbers represent only non-negative values (zero and positive integers). All bits contribute to the magnitude.

Range of Unsigned Numbers

For an n-bit unsigned number:

• Minimum value: 0
• Maximum value: 2ⁿ - 1
• Total values: 2ⁿ

Conversion Example

Comparison of All Three Methods (4-bit)

Sign Extension

When converting a signed number to a larger bit width, you must extend the sign bit.

2's Complement Sign Extension

Rule: Copy the MSB (sign bit) into all new higher-order positions.

Why Sign Extension Works

Unsigned vs Signed: Same Bits, Different Values

The bit pattern doesn't change — only the interpretation matters:

This is why programming languages distinguish between int (signed) and unsigned int.

Detecting Overflow

Unsigned Overflow

Occurs when carry out of the MSB exists.

Signed Overflow

Occurs when two numbers of the same sign produce a result of opposite sign.

Overflow Detection Formula: V = Cₙ ⊕ Cₙ₋₁ (XOR of carry into MSB and carry out of MSB)

ASCII Diagram: Number Line Representation

Practical Applications

Interview Questions

Q1: What is the difference between signed and unsigned overflow?

Unsigned overflow is detected by carry out of the MSB. Signed overflow is detected when the carry into the MSB differs from the carry out (V = Cₙ ⊕ Cₙ₋₁). The same addition can overflow in one interpretation but not the other.

Q2: Why do most processors use 2's complement for signed integers?

Because: (1) unique zero representation simplifies comparisons, (2) the same addition hardware works for both signed and unsigned, (3) no end-around carry needed, and (4) one extra negative value gives slightly larger range.

Q3: How does sign extension preserve the value of a number?

By replicating the sign bit into all new upper positions. Mathematically, for 2's complement, this works because adding higher-order 1s to a negative number contributes -2ⁿ + 2ⁿ⁻¹ + ... which equals the same negative weight at the new MSB position.

Q4: What is the result of interpreting 1010 1010 as unsigned vs signed?

Unsigned: 128 + 32 + 8 + 2 = 170. Signed (2's complement): -128 + 32 + 8 + 2 = -86. Same bits, different values depending on interpretation.

Q5: In C programming, what happens when you assign -1 to an unsigned variable?

The bit pattern of -1 in 2's complement (all 1s) is reinterpreted as unsigned. For a 32-bit unsigned int, -1 becomes 4,294,967,295 (2³² - 1). The bits don't change — only the interpretation does.