Fraction Conversion Tool

Decimal Fraction to Binary Converter

Convert decimal fractions to binary instantly. Enter a value such as 0.625, 5.625, or 13.375 and see the binary integer and fractional parts, repeated multiplication-by-2 steps, and whether the result terminates or repeats.

✓ Decimal Fractions ✓ Binary Result ✓ Exact Decimal Parsing ✓ Repeating Fraction Detection ✓ Free Unlimited Use

.10

Decimal Fraction → Binary

● Ready

Decimal Number

Fraction Precision

Enter a non-negative decimal number. Repeating binary fractions are truncated at the selected precision.

Try:

✓ Binary Result

Binary representation

Integer Part 0

Fraction Bits 0

Representation Exact

Conversion Breakdown

What Is Decimal Fraction to Binary Conversion?

Decimal fraction to binary conversion changes a base-10 number containing a fractional part into its equivalent base-2 representation.

The whole-number portion is converted using ordinary decimal-to-binary conversion. The fractional portion is converted by repeatedly multiplying the fraction by 2 and recording the integer part of each result.

Example: 0.625₁₀ Multiply repeatedly by 2: 0.625 × 2 = 1.25 → 1 0.25 × 2 = 0.5 → 0 0.5 × 2 = 1.0 → 1 Read the produced bits in order: 0.625₁₀ = 0.101₂

How to Convert a Decimal Fraction to Binary

1. Separate the integer part

For a value such as 5.625, separate 5 from 0.625.

2. Convert the integer

Convert the whole number to binary using division by 2 or positional values.

3. Multiply the fraction by 2

Multiply only the fractional portion by two.

4. Record the integer bit

Write 0 or 1 depending on whether the result is below or at least 1.

5. Repeat

Use the remaining fractional part and continue multiplying by 2.

6. Join both parts

Combine the binary integer part, binary point, and generated fractional bits.

Example: Convert 5.625 Decimal to Binary

First separate the number into its whole and fractional portions.

Integer portion: 5₁₀ = 101₂ Fractional portion: 0.625 × 2 = 1.25 → 1 0.25 × 2 = 0.5 → 0 0.5 × 2 = 1.0 → 1 Therefore: 0.625₁₀ = 0.101₂ Combine: 5.625₁₀ = 101.101₂

Example: Convert 13.375 Decimal to Binary

Integer: 13₁₀ = 1101₂ Fraction: 0.375 × 2 = 0.75 → 0 0.75 × 2 = 1.5 → 1 0.5 × 2 = 1.0 → 1 So: 0.375₁₀ = 0.011₂ Final result: 13.375₁₀ = 1101.011₂

Decimal Fraction to Binary Conversion Table

Decimal	Binary
0.5	`0.1`
0.25	`0.01`
0.125	`0.001`
0.75	`0.11`
0.625	`0.101`
1.5	`1.1`
2.25	`10.01`
5.625	`101.101`
13.375	`1101.011`

Why Do We Multiply the Fraction by 2?

Binary fractional positions represent powers of one-half. Repeated multiplication by 2 reveals whether each successive binary fractional position contains a 0 or a 1.

Binary fractional place values: 2⁻¹ = 0.5 2⁻² = 0.25 2⁻³ = 0.125 2⁻⁴ = 0.0625

Why Does Decimal 0.1 Repeat in Binary?

Not every decimal fraction can be represented by a finite number of binary fractional bits.

Decimal 0.1 is one of the best-known examples. Its binary representation continues indefinitely:

0.1₁₀ ≈ 0.000110011001100110011…₂

The pattern 0011 repeats. A computer therefore stores a finite approximation when using a fixed number of binary fraction bits.

When Does a Decimal Fraction Terminate in Binary?

A rational decimal fraction has a terminating binary representation when its reduced denominator contains no prime factors other than 2.

Examples that terminate: 1/2 1/4 3/8 5/16 These denominators are powers of 2.

Fractions whose reduced denominators contain factors such as 5 generally repeat in binary.

Why Is Precision Needed?

A terminating binary fraction needs only a finite number of bits. A repeating fraction, however, can continue indefinitely.

The precision setting tells the calculator how many fractional binary bits to generate before stopping a repeating expansion.

Important: if the fraction terminates before the selected precision, the result is exact. If a remainder still exists when the selected bit limit is reached, the result is marked as truncated.

Exact vs Truncated Binary Results

Decimal	Binary	Type
0.5	`0.1`	Exact
0.625	`0.101`	Exact
0.375	`0.011`	Exact
0.1	`0.000110011...`	Repeating
0.2	`0.00110011...`	Repeating

Why Binary Fractions Matter in Computers

Computer processors commonly represent non-integer values using binary fractions. Floating-point formats encode values using binary significands and powers of two.

This explains why some decimal calculations can display tiny rounding differences even when the decimal input appears simple.

Decimal 0.5 in Binary

0.5 × 2 = 1.0 → 1 Therefore: 0.5₁₀ = 0.1₂

Decimal 0.25 in Binary

0.25 × 2 = 0.5 → 0 0.5 × 2 = 1.0 → 1 Therefore: 0.25₁₀ = 0.01₂

Decimal 0.625 in Binary

0.625 × 2 = 1.25 → 1 0.25 × 2 = 0.5 → 0 0.5 × 2 = 1.0 → 1 Therefore: 0.625₁₀ = 0.101₂

Common Decimal Fraction to Binary Mistakes

Dividing instead of multiplying

Fractional conversion uses repeated multiplication by 2, not repeated division.

Reading bits backward

Fractional bits are recorded in the same order in which multiplication produces them.

Assuming every fraction terminates

Values such as decimal 0.1 have repeating binary expansions.

Ignoring precision

A repeating expansion must be stopped at a chosen number of fractional bits.

Related BinaryCon Tools

Decimal Fraction to Binary Converter FAQs

How do you convert a decimal fraction to binary?

Separate the integer and fractional portions. Convert the integer normally, then repeatedly multiply the fractional part by 2 and record each resulting integer bit.

What is decimal 0.5 in binary?

0.5₁₀ = 0.1₂.

What is decimal 0.25 in binary?

0.25₁₀ = 0.01₂.

What is decimal 0.125 in binary?

0.125₁₀ = 0.001₂.

What is decimal 0.625 in binary?

0.625₁₀ = 0.101₂.

What is decimal 5.625 in binary?

5.625₁₀ = 101.101₂.

What is decimal 13.375 in binary?

13.375₁₀ = 1101.011₂.

Why does decimal 0.1 repeat in binary?

The reduced fraction 1/10 contains a denominator factor of 5. A finite binary fraction can terminate only when its reduced denominator is a power of two.

What does binary precision mean?

Precision is the maximum number of fractional binary bits generated when a fraction does not terminate quickly.

Does increasing precision make a repeating result more accurate?

Yes. More fractional bits provide a closer binary approximation to a repeating value.

How can I tell whether the result is exact?

The calculator marks the representation as Exact when the fractional remainder reaches zero. Otherwise it marks the displayed expansion as Truncated.

Does this converter support whole decimal numbers?

Yes. A whole number such as 13 is valid and simply produces a binary integer without a fractional part.

Does it support negative decimal values?

This page is designed for non-negative decimal values. Signed binary representation requires additional rules and a defined encoding.

Does BinaryCon require registration?

No. The Decimal Fraction to Binary Converter can be used directly without signup.