Real Number to Binary Calculator

Convert real numbers (both integer and fractional) to binary representation using our free online calculator. This tool helps you understand how computers store decimal numbers in binary format.

How to Convert Real Numbers to Binary

Converting a real number to binary involves two main steps: converting the integer part and converting the fractional part. Here's an overview of the process:

Separate the number into its integer and fractional parts.
Convert the integer part to binary using repeated division by 2.
Convert the fractional part to binary using repeated multiplication by 2.
Combine the results with a binary point.

Binary numbers can represent both positive and negative numbers. For negative numbers, you would typically use a sign-magnitude or two's complement representation, but our calculator focuses on the positive binary representation.

Converting Integer Parts

The integer part of a real number is converted to binary using the division-remainder method:

Divide the number by 2.
Record the integer quotient and the remainder.
Repeat the process with the quotient until you reach 0.
The binary representation is the remainders read in reverse order.

Example: Convert 13 to binary

13 ÷ 2 = 6 remainder 1
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1

The binary representation is 1101 (reading remainders in reverse).

Converting Fractional Parts

The fractional part is converted using the multiplication method:

Multiply the fractional part by 2.
Record the integer part of the result (0 or 1).
Take the fractional part of the result and repeat the process.
Continue until the fractional part becomes 0 or until you reach the desired precision.
The binary representation is the recorded integers in order.

Example: Convert 0.625 to binary

0.625 × 2 = 1.25 → record 1
0.25 × 2 = 0.5 → record 0
0.5 × 2 = 1.0 → record 1

The binary representation is 0.101.

Worked Examples

Example 1: Convert 10.375 to binary

Integer part (10):
10 ÷ 2 = 5 remainder 0
5 ÷ 2 = 2 remainder 1
2 ÷ 2 = 1 remainder 0
1 ÷ 2 = 0 remainder 1 → Binary: 1010

Fractional part (0.375):
0.375 × 2 = 0.75 → record 0
0.75 × 2 = 1.5 → record 1
0.5 × 2 = 1.0 → record 1 → Binary: 0.011

Combined: 1010.011

Example 2: Convert 7.125 to binary

Integer part (7):
7 ÷ 2 = 3 remainder 1
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1 → Binary: 111

Fractional part (0.125):
0.125 × 2 = 0.25 → record 0
0.25 × 2 = 0.5 → record 0
0.5 × 2 = 1.0 → record 1 → Binary: 0.001

Combined: 111.001

Frequently Asked Questions

Can this calculator handle negative numbers?

Our calculator focuses on the positive binary representation. For negative numbers, you would typically use a sign-magnitude or two's complement representation, which is beyond the scope of this basic converter.

How many decimal places can I convert?

The calculator can handle up to 10 decimal places for the fractional part, but you can adjust this in the settings if needed.

Is there a limit to how large a number I can convert?

The calculator can handle numbers up to 15 digits in the integer part and 10 digits in the fractional part. For larger numbers, you may need specialized software.

How accurate are the conversions?

The conversions are mathematically accurate based on the standard binary representation methods described in the guide.