decimal to binary numbers

Daniel Parker

3 min read

·

13-03-2025

1

0

Share

Meta Description: Learn how to convert decimal numbers to binary. This comprehensive guide explains the process step-by-step, including methods for both whole numbers and decimal fractions, with examples and helpful tips. Master binary conversion today! (158 characters)

Understanding Decimal and Binary Number Systems

Before diving into the conversion process, let's briefly review the two number systems involved: decimal and binary.

The decimal number system, also known as base-10, is the system we use every day. It uses ten digits (0-9) and each position represents a power of 10. For example, the number 123 is actually (1 x 10²) + (2 x 10¹) + (3 x 10⁰).

The binary number system, or base-2, is the foundation of digital computers. It only uses two digits: 0 and 1. Each position represents a power of 2. For instance, the binary number 1011 is (1 x 2³) + (0 x 2²) + (1 x 2¹) + (1 x 2⁰) = 8 + 0 + 2 + 1 = 11 in decimal.

Converting Whole Decimal Numbers to Binary

There are two primary methods for converting whole decimal numbers to binary:

Method 1: Repeated Division by 2

This is the most common method. Here's how it works:

1. Divide: Divide the decimal number by 2.
2. Record the Remainder: Note down the remainder (either 0 or 1).
3. Repeat: Repeat steps 1 and 2 with the quotient (the result of the division) until the quotient becomes 0.
4. Read the Remainders: Read the remainders from bottom to top. This sequence of remainders is the binary equivalent.

Example: Let's convert the decimal number 13 to binary:

Reading the remainders from bottom to top, we get 1101. Therefore, 13 in decimal is 1101 in binary.

Method 2: Using Powers of 2

This method involves finding the largest power of 2 less than or equal to the decimal number and subtracting it. Repeat this process until you reach 0.

Example: Convert 13 to binary:

1. The largest power of 2 less than or equal to 13 is 8 (2³). 13 - 8 = 5. This gives us a '1' in the 2³ position.
2. The largest power of 2 less than or equal to 5 is 4 (2²). 5 - 4 = 1. This gives us a '1' in the 2² position.
3. The largest power of 2 less than or equal to 1 is 1 (2⁰). 1 - 1 = 0. This gives us a '1' in the 2⁰ position.
4. There is no 2¹ in the result, this gives a '0' in the 2¹ position.

Putting it together, we have 1101.

Converting Decimal Fractions to Binary

Converting decimal fractions to binary requires a different approach:

1. Multiply by 2: Multiply the decimal fraction by 2.
2. Record the Integer Part: Note down the integer part of the result (either 0 or 1).
3. Repeat: Repeat steps 1 and 2 with the fractional part of the result until the fractional part becomes 0 or you reach the desired level of precision.
4. Read the Integers: Read the integer parts from top to bottom. This sequence is the binary fraction.

Example: Convert 0.625 to binary:

Reading the integer parts from top to bottom, we get .101. Therefore, 0.625 in decimal is 0.101 in binary.

Combining Whole Numbers and Fractions

To convert a decimal number with both a whole number and a fractional part, convert each part separately using the methods above and then combine the results. For example, 13.625 would be 1101.101 in binary.

Frequently Asked Questions (FAQs)

How do I check my binary conversion?

Convert your binary answer back to decimal using the powers of 2 method to verify your answer.

What is the largest decimal number I can convert?

Theoretically, there's no limit, but practically, the limit depends on your computational resources.

Are there online calculators for decimal to binary conversion?

Yes, many websites and online calculators can perform this conversion for you.

This guide provides a solid foundation for understanding and performing decimal-to-binary conversions. Practice with different numbers to build your proficiency. Remember that mastering binary is crucial for anyone interested in computer science or digital electronics.