---
title: "(10.001)₂ − (11.101)₂ by 2's complement"
subject: "Digital Logic"
chapter: 5
key: "semester_3/digital_logic:twos-complement-binary-subtraction"
source: https://www.bctengineer.com/q/semester_3/digital_logic/twos-complement-binary-subtraction
site: BCT Engineering
---

# (10.001)₂ − (11.101)₂ by 2's complement

Digital Logic · Chapter 5: Arithmetic Circuits

- Asked in: 2072 Chaitra, 2 marks
- Video: [Binary Subtraction | 1's | 2's | Complement | Digital logic design | Digital Electronics | STLD | CO](https://www.youtube.com/watch?v=TCGuxRbXU2Q) — Sudhakar Atchala
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/digital_logic/twos-complement-binary-subtraction
- Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-5

## Question

Perform (10.001)₂ - (11.101)₂ using 2's complement method.

## Short answer

Make both numbers equal length. 2's complement of 11.101 is 00.011. Add to 10.001: 10.001+00.011=10.100. No end carry, result is negative. Take 2's complement of sum: invert 10.100 to 01.011, add 0.001 yields 01.100, so answer is -(1.1)₂ = -1.5.

## Answer

To find $A - B$ by 2's complement: add the 2's complement of $B$ to $A$. If an end carry occurs, drop it and the result is positive. If no end carry occurs, the result is negative: take the 2's complement of the sum and put a minus sign.

Make both numbers the same length: $A = 10.001$, $B = 11.101$.

**Step 1: 2's complement of B**

$$
\begin{aligned} \text{1's complement of } 11.101 &= 00.010 \\ \text{add 1 at LSB} &= 00.010 + 00.001 = 00.011 \end{aligned}
$$

**Step 2: Add to A**

```text
   10.001
 + 00.011
 --------
   10.100    (no end carry)
```

**Step 3: No end carry, so the answer is negative.** Take the 2's complement of 10.100:

$$
\begin{aligned} \text{1's complement} &= 01.011 \\ +\,0.001 &= 01.100 \end{aligned}
$$

**Answer:** $(10.001)_2 - (11.101)_2 = -(1.1)_2$

Check in decimal: $2.125 - 3.625 = -1.5 = -(1.1)_2$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How is A - B performed by 2's complement?
   A. Add the 2's complement of B to A
   B. Add the 1's complement of A to B
   C. Subtract bit by bit with borrow only
   D. Add B to A and invert
   Answer: A. Subtraction becomes addition of the 2's complement. (Key point: Method)

2. What does the absence of an end carry indicate?
   A. The result is positive
   B. The result is negative, so take its 2's complement
   C. An overflow error
   D. The result is zero
   Answer: B. No end carry means B > A, so the answer is the negative of the sum's 2's complement. (Key point: End carry rule)

3. What is (10.001)2 - (11.101)2?
   A. +(1.1)2
   B. -(10.100)2
   C. -(1.1)2
   D. -(0.11)2
   Answer: C. 2.125 - 3.625 = -1.5 = -(1.1)2. (Key point: Result)

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