---
title: "Convert (53.125)₁₀ to binary"
subject: "Digital Logic"
chapter: 1
key: "semester_3/digital_logic:convert-53-125-binary"
source: https://www.bctengineer.com/q/semester_3/digital_logic/convert-53-125-binary
site: BCT Engineering
---

# Convert (53.125)₁₀ to binary

Digital Logic · Chapter 1: Introduction

- Asked in: 2075 Chaitra, 1.5 marks
- Video: [Convert  Decimal to any other Base(Binary, Octal, Hexadecimal)| Conversion from One radix to another](https://www.youtube.com/watch?v=XLrNRKsfXt4) — 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/convert-53-125-binary
- Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-1

## Question

Convert (53.125)₁₀ = (?)₂.

## Short answer

Integer 53 by repeated division gives 110101; fractional 0.125 by repeated multiplication gives 0.001. Thus (53.125)₁₀ = (110101.001)₂.

## Answer

Convert the integer part by repeated division by 2 and the fraction part by repeated multiplication by 2.

**Integer part (53):**

| Division | Quotient | Remainder |
|---|---|---|
| 53 ÷ 2 | 26 | 1 |
| 26 ÷ 2 | 13 | 0 |
| 13 ÷ 2 | 6 | 1 |
| 6 ÷ 2 | 3 | 0 |
| 3 ÷ 2 | 1 | 1 |
| 1 ÷ 2 | 0 | 1 |

Bottom to top: $53_{10} = 110101_2$.

**Fraction part (0.125):**

| Multiplication | Result | Bit |
|---|---|---|
| 0.125 × 2 | 0.25 | 0 |
| 0.25 × 2 | 0.5 | 0 |
| 0.5 × 2 | 1.0 | 1 |

Top to bottom: $0.125_{10} = 0.001_2$.

**Answer:** $(53.125)_{10} = (110101.001)_2$

## Check yourself

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

1. How is 53 converted?
   A. Repeated multiplication by 2
   B. Repeated division by 2, remainders bottom to top
   C. Division by 8
   D. Division by 10
   Answer: B. The remainders read upward give the bits. (Key point: Integer method)

2. (53)10 in binary?
   A. 101011
   B. 110101
   C. 110011
   D. 101101
   Answer: B. 32+16+4+1 = 53. (Key point: Integer result)

3. How is 0.125 converted?
   A. Repeated multiplication by 2, integer parts top to bottom
   B. Repeated division by 2
   C. Multiplication by 10
   D. Read integer parts bottom to top
   Answer: A. 0.125x2=0.25 (0), 0.5 (0), 1.0 (1). (Key point: Fraction method)

4. (53.125)10 in binary?
   A. 110101.100
   B. 101011.001
   C. 110101.01
   D. 110101.001
   Answer: D. Integer 110101 and fraction .001. (Key point: Final)

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