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Binary to Decimal Tool

Binary Fraction to Decimal Converter

Convert binary fractions to decimal instantly. Enter values such as 101.101, 10.11 or 0.001 and get the decimal result together with a detailed powers-of-two breakdown and step-by-step calculation.

Exact conversion
Handles integer and fractional binary digits.

Step-by-step math
See every power of two used.

Interactive table
View each bit and its decimal contribution.

No signup
Calculate directly in your browser.

Binary Fraction Converter

● LIVE

ENTER BINARY FRACTION

BASE 2

DECIMAL RESULT BASE 10

5.625

101.101₂ = 4 + 1 + 0.5 + 0.125 = 5.625₁₀

TRY AN EXAMPLE

Input Binary digits 0 and 1

Fraction Support Binary point accepted

Output Decimal base-10 value

Method Powers of two

Example 101.101₂ = 5.625₁₀

Binary Fraction to Decimal Converter

A binary fraction is a base-2 number containing digits both before and after a binary point. While ordinary binary integers use powers such as 2⁰, 2¹, 2² and 2³, the digits after a binary point use negative powers of two such as 2⁻¹, 2⁻², 2⁻³ and 2⁻⁴.

This binary fraction to decimal converter calculates the base-10 value of a binary number automatically. You can enter a value such as 101.101₂, 0.011₂ or 1110.001₂ and immediately see its decimal equivalent.

Example: 101.101₂ equals 5.625₁₀ because 4 + 1 + 0.5 + 0.125 = 5.625.

How to Convert Binary Fractions to Decimal

Binary uses positional notation. Each position has a value equal to a power of two. The binary point separates positive powers of two from negative powers of two.

Binary place-value formula

… b₃×2³ + b₂×2² + b₁×2¹ + b₀×2⁰ + b₋₁×2⁻¹ + b₋₂×2⁻² + b₋₃×2⁻³ …

Identify the binary point Separate the whole-number portion from the fractional portion.

Assign powers to integer bits Moving left from the binary point gives powers 2⁰, 2¹, 2² and so on.

Assign negative powers to fractional bits Moving right gives 2⁻¹, 2⁻², 2⁻³ and progressively smaller values.

Add every contribution Multiply each bit by its corresponding place value and add the results.

Live Binary Fraction Calculation Breakdown

This table updates whenever you use the calculator. It shows each binary digit, the corresponding power of two, its place value and the amount contributed to the final decimal result.

Bit	Position	Power	Decimal Place Value	Contribution

Binary Fraction Place Value Chart

Every move one position to the right of the binary point divides the place value by two.

2⁻¹ 0.5

2⁻² 0.25

2⁻³ 0.125

2⁻⁴ 0.0625

2⁻⁵ 0.03125

2⁻⁶ 0.015625

2⁻⁷ 0.0078125

2⁻⁸ 0.00390625

Position	Power	Fraction	Decimal Value
1st fractional bit	2⁻¹	1/2	0.5
2nd fractional bit	2⁻²	1/4	0.25
3rd fractional bit	2⁻³	1/8	0.125
4th fractional bit	2⁻⁴	1/16	0.0625
5th fractional bit	2⁻⁵	1/32	0.03125
6th fractional bit	2⁻⁶	1/64	0.015625
7th fractional bit	2⁻⁷	1/128	0.0078125
8th fractional bit	2⁻⁸	1/256	0.00390625

Worked Example: Convert 101.101₂ to Decimal

Consider the binary fraction 101.101₂. Start by assigning a power of two to every digit.

101.101₂

= (1 × 2²)
+ (0 × 2¹)
+ (1 × 2⁰)
+ (1 × 2⁻¹)
+ (0 × 2⁻²)
+ (1 × 2⁻³)

= 4 + 0 + 1 + 0.5 + 0 + 0.125

= 5.625₁₀

The integer portion 101₂ equals 5. The fractional portion .101₂ equals 0.625. Adding both parts produces the final decimal value of 5.625.

Binary Fraction to Decimal Examples

The following examples cover some of the most common binary fractional patterns.

EXAMPLE 1 0.1₂ = 0.5₁₀ 1 × 2⁻¹ = 0.5

EXAMPLE 2 0.01₂ = 0.25₁₀ 1 × 2⁻² = 0.25

EXAMPLE 3 0.001₂ = 0.125₁₀ 1 × 2⁻³ = 0.125

EXAMPLE 4 1.01₂ = 1.25₁₀ 1 + 0.25 = 1.25

EXAMPLE 5 10.11₂ = 2.75₁₀ 2 + 0.5 + 0.25 = 2.75

EXAMPLE 6 11.001₂ = 3.125₁₀ 2 + 1 + 0.125 = 3.125

EXAMPLE 7 101.011₂ = 5.375₁₀ 4 + 1 + 0.25 + 0.125 = 5.375

EXAMPLE 8 111.111₂ = 7.875₁₀ 4 + 2 + 1 + 0.5 + 0.25 + 0.125

Binary Integers vs Binary Fractions

Binary integers and binary fractions use the same digits—0 and 1—but the position of those digits determines their value.

Binary integer positions

Digits to the left of the binary point use non-negative powers of two. Starting next to the binary point, the values are 1, 2, 4, 8, 16, 32 and so on.

Binary fractional positions

Digits to the right of the binary point use negative powers. Their values begin with 1/2, 1/4, 1/8, 1/16, 1/32 and continue dividing by two.

The binary point itself does not represent a value. It simply separates whole-number positions from fractional positions.

Where Binary Fractions Are Used

Binary fractions are not only an academic number-system concept. They are fundamental to the way digital systems represent non-integer values.

CPU

Computer Arithmetic Processors work internally with binary values, including fractional quantities.

Floating Point Binary fractions form the significand portion of floating-point representations.

FIX

Fixed-Point Systems Embedded systems frequently reserve specific bits for fractional values.

DSP

Digital Signal Processing Binary fractional values can represent coefficients, samples and scaled measurements.

MCU

Microcontrollers Low-resource devices often use binary fixed-point arithmetic instead of floating point.

EDU

Computer Science Binary fraction conversion helps students understand base-2 positional notation.

Binary Fraction Accuracy and Repeating Values

A finite binary fraction can exactly represent values whose reduced fractional denominator is a power of two. Examples include 1/2, 1/4, 3/8 and 7/16.

However, not every decimal fraction has a finite binary representation. For example, decimal 0.1 repeats indefinitely when written in binary. This is similar to how one-third repeats indefinitely as 0.3333… in decimal.

This behavior explains why computers sometimes display tiny rounding differences in calculations involving decimal fractions. The underlying binary representation may be an approximation rather than an exact finite value.

How to Convert Binary Fractions Manually

You can convert a binary fraction without a calculator by writing the place value underneath each bit and adding only the positions that contain a 1.

For example, consider 10.1011₂.

1 × 2¹ = 2
0 × 2⁰ = 0
1 × 2⁻¹ = 0.5
0 × 2⁻² = 0
1 × 2⁻³ = 0.125
1 × 2⁻⁴ = 0.0625

2 + 0.5 + 0.125 + 0.0625 = 2.6875

Therefore, 10.1011₂ = 2.6875₁₀.

Common Binary Fraction Conversion Mistakes

Using decimal tenths and hundredths. Binary positions after the point are halves, quarters, eighths and other powers of two.
Starting with 2⁰ after the binary point. The first fractional position is 2⁻¹, not 2⁰.
Ignoring zero positions. A zero contributes nothing, but its place still affects the exponent of later bits.
Reading .1₂ as 0.1 decimal. Binary .1 equals decimal 0.5.
Using digits other than 0 and 1. Standard binary numbers can contain only those two digits.
Confusing binary fractions with IEEE 754 bit patterns. An IEEE floating-point pattern must be decoded using separate sign, exponent and fraction fields.

About This Binary Fraction Calculator

The calculator above validates your binary input and calculates both the whole-number and fractional portions. It then generates an expanded equation and a detailed place-value table.

Supports binary values containing a binary point.
Also accepts binary integers without a fractional part.
Shows the decimal result immediately.
Displays active decimal contributions.
Creates a bit-by-bit conversion table.
Provides quick example values.
Works directly in modern desktop and mobile browsers.
No registration or software installation is required.

Binary Fraction to Decimal FAQs

What is a binary fraction?

A binary fraction is a base-2 number containing digits after a binary point. Fractional positions represent negative powers of two.

How do you convert binary fractions to decimal?

Multiply each binary digit by its corresponding power of two and add the resulting values. Positions after the binary point use negative powers such as 2⁻¹, 2⁻² and 2⁻³.

What is binary 0.1 in decimal?

0.1₂ equals 0.5₁₀ because the first binary fractional position represents one-half.

What is binary 0.01 in decimal?

0.01₂ equals 0.25₁₀ because the 1 occupies the 2⁻² position.

What is 101.101 binary in decimal?

101.101₂ equals 5.625₁₀. The integer portion equals 5 and the fractional portion equals 0.625.

What is the first position after a binary point?

The first position to the right of a binary point is 2⁻¹, which equals 0.5.

What is the second position after a binary point?

The second position represents 2⁻², which equals 0.25.

Is a binary point the same as a decimal point?

They serve a similar positional purpose, but the values around them depend on the number base. A decimal point uses powers of ten while a binary point uses powers of two.

Can all decimal fractions be represented exactly in binary?

No. Some decimal fractions require an infinite repeating binary representation. Decimal 0.1 is a well-known example.

Can this calculator convert binary integers too?

Yes. You can enter values such as 101, 1101 or 10000 even if no binary fractional part is present.

Which digits are valid in a binary fraction?

A standard binary fraction uses only 0 and 1, together with at most one binary point.

Why are binary fractions important in computers?

Computers represent information using bits. Non-integer numeric values therefore often need binary fractional, fixed-point or floating-point representations.

Is a binary fraction the same as IEEE 754?

No. A plain binary fraction is interpreted directly using powers of two. IEEE 754 represents a floating-point number using a structured sign, exponent and fraction field.

Is this binary fraction converter free?

Yes. The calculator runs in your browser and can be used without registration.

Convert Binary Fractions to Decimal Instantly

Use the calculator at the top of this page whenever you need to convert a binary fraction into decimal form. Enter the base-2 number, choose Convert to Decimal, and review both the final result and the detailed powers-of-two breakdown.

The additional examples, place-value chart and explanations on this page can also help you understand the conversion manually rather than simply obtaining an answer.