Binary Fraction Converter (2024)

Binary representation

What is a binary fraction?

Take the decimal part of a non-integer number. That is a proper fraction — its value is smaller than one. When the denominator is a power of ten ($10$10, $100$100, $1000$1000,...), we talk of decimal fractions.

$$\frac{137}{1000}=0.137$$1000137​=0.137

Representing such values in base 2 brings us the binary fractions!

How to convert fractions to binary

$$0.2912\times 2 = 0.5824$$0.2912×2=0.5824

The integer part remained $0$0: this is the first element of the binary fraction, $0.0...$0.0.... Repeat the step; we will see something different this time.

$$0.5824\times 2 =1.1648$$0.5824×2=1.1648

$$\footnotesize 0.1684\times 2=0.3368 \rightarrow \textcolor{red}{0.010}\\\footnotesize 0.3368\times 2=0.6736 \rightarrow \textcolor{red}{0.0100}\\\footnotesize 0.6736\times 2=1.3472 \rightarrow \textcolor{red}{0.01001} \\\textcolor{blue}{-1}\\\footnotesize 0.3472\times 2 = 0.6944 \rightarrow \textcolor{red}{0.010010} \\\footnotesize 0.6944\times 2 = 1.3888 \rightarrow \textcolor{red}{0.0100101} \\$$0.1684×2=0.3368→0.0100.3368×2=0.6736→0.01000.6736×2=1.3472→0.01001−10.3472×2=0.6944→0.0100100.6944×2=1.3888→0.0100101

Wait!

$$\footnotesize 0.3888\times 2=0.7776 \rightarrow \textcolor{red}{0.01001010}\\\footnotesize 0.7776\times 2=1.5552 \rightarrow \textcolor{red}{0.010010101}\\\textcolor{blue}{-1}\\\footnotesize 0.5552\times 2=1.1104 \rightarrow \textcolor{red}{0.0100101011} \\\textcolor{blue}{-1}\\\footnotesize 0.1104\times 2 = 0.2208 \rightarrow \textcolor{red}{0.01001010110} \\$$0.3888×2=0.7776→0.010010100.7776×2=1.5552→0.010010101−10.5552×2=1.1104→0.0100101011−10.1104×2=0.2208→0.01001010110

It doesn't stop! Why is the binary fraction still growing? Let's take a look at the reason in the next section.

The conversion from binary fraction to decimal fraction

$$1\rightarrow 2^0=\tfrac{1}{1}\\[0.5em]0.1\rightarrow 2^{-1}=\tfrac{1}{2^1}=\tfrac{1}{2}\\[0.5em]0.01\rightarrow 2^{-2}=\tfrac{1}{2^2}=\tfrac{1}{4}\\[0.5em]0.001\rightarrow 2^{-3}=\tfrac{1}{2^3}=\tfrac{1}{8}\\[0.5em]0.0001\rightarrow 2^{-4}=\tfrac{1}{2^4}=\tfrac{1}{16}\\[0.5em]0.00001\rightarrow 2^{-5}=\tfrac{1}{2^5}=\tfrac{1}{32}$$1→20=11​0.1→2−1=211​=21​0.01→2−2=221​=41​0.001→2−3=231​=81​0.0001→2−4=241​=161​0.00001→2−5=251​=321​

🙋 We included the 0th power of two, which equals $1$1, only to give a better understanding of this positional conversion. Remember to limit your conversions to numbers smaller than $1$1!

Take your binary fraction and rewrite it, multiplying each digit by the respective power of 2. Then sum them together.

$$0.0110101 \rightarrow\\[0.5em]\begin{align*}\footnotesize 0\!\times\! \frac{1}{2^{0}}+0\!\times\!\frac{1}{2^1}+& \footnotesize1\!\times\!\frac{1}{2^2}+1\!\times\!\frac{1}{2^3} \\[0.5em]\footnotesize +\ 0\!\times\!\frac{1}{2^4}+1\!\times\!\frac{1}{2^5}+&\footnotesize 0\!\times\!\frac{1}{2^6}+1\!\times\!\frac{1}{2^7}\end{align*}$$0.0110101→0×201​+0×211​++0×241​+1×251​+​1×221​+1×231​0×261​+1×271​​

Convert the fractions to their decimal equivalents:

$$\footnotesize 0\!\times\! 1+0\!\times\! 0.5+1\!\times\! 0.25+\\\footnotesize +1\!\times\! 0.125+0\!\times\! 0.0625+\\\footnotesize +1\!\times\! 0.03125+0\!\times\! 0.015625+\\\footnotesize +1\!\times\! 0.0078125 = 0.4140625$$0×1+0×0.5+1×0.25++1×0.125+0×0.0625++1×0.03125+0×0.015625++1×0.0078125=0.4140625

And that's it — we've converted a decimal fraction to a binary fraction!

$$0.0110101_{2}=0.4140625_{10}$$0.01101012​=0.414062510​

The limitations of binary fractions

Binary fractions can't represent every decimal fraction perfectly. It's entirely possible for decimal fractions with a finite number of decimal digits (rational) to have an infinitely long binary representation. Such a conversion results in an error, with the magnitude of the error depending on the number of digits of the representation.

Take a look at the conversion from decimal fraction to binary fraction. It is possible to build every decimal number with a sum of the negative powers of $2$2.

$$\sum_{n=1}^{\infty}\frac{1}{2^n}=1$$n=1∑∞​2n1​=1

This is a geometric series that converges to $1$1. By removing elements ad hoc, you can obtain every number between $0$0 and $1$1; however, the real issue here is to decide which contributions we must erase.

$$\left(\tfrac{1}{5}\right)_{10}=0.20_{10}$$(51​)10​=0.2010​

This is a finite fraction: the error in the 8-digit representation is zero.

Find its binary representation, you know how to do it, but we will help you this time!

$$0.2000\times 2=0.4000\rightarrow \textcolor{red}{0}\\0.4000\times 2=0.8000\rightarrow \textcolor{red}{0}\\0.8000\times 2=1.6000\rightarrow \textcolor{red}{1}\\\textcolor{blue}{-1}\\0.6000\times 2=1.2000\rightarrow \textcolor{red}{1}\\\textcolor{blue}{-1}\\0.2000\times 2 = 0.4000\rightarrow \textcolor{red}{0}$$0.2000×2=0.4000→00.4000×2=0.8000→00.8000×2=1.6000→1−10.6000×2=1.2000→1−10.2000×2=0.4000→0

The last step is the same as the first, and so if we keep going, we'll end up with a repeating sequence of $0011$0011s:

$$0.2_{10}=0.001100110011\ldots_{\ 2}$$0.210​=0.001100110011…2​

Let's truncate it at the eighth digit:

$$0.2_{10}\simeq0.00110011_{2}$$0.210​≃0.001100112​

This truncated binary fraction is not equal to $0.2$0.2, even though we started there:

$$0.00110011_{2}=0.19921875_{10}$$0.001100112​=0.1992187510​

The conversion resulted in an error of $0.2-0.19921875=0.000781$0.2−0.19921875=0.000781: small, but not negligible.

🔎 In modern computers, the truncation happens at $23$23-th or $52$52-th digit, the precision of single and double float variables. A $23$23 digits truncation in the example before has an error equal to $7.15\cdot 10^{-8}$7.15⋅10−8: extremely small, but still present.

The binary representation of a decimal fraction is exact only if $2$2 is the sole prime factor of the denominator: $1/2$1/2, $3/8$3/8, and $1013/1024$1013/1024 all have exact representations, on the other hand, $1/5$1/5 or $1/10$1/10 always have an error.

How to use our binary fraction converter?

🙋 If you need a different number of digits than the one provided by our defaults, go in advanced mode and insert the desired value. Remember to refresh the calculator if you want to use the default values again! 😀

Beyond 0s and 1s

FAQ

Can all fractions be converted to binary?

Not all fractions can be exactly converted to binary: only if the denominator is a power of 2, the binary fraction will be finite. In every other case, there will be an error in the representation. The error's magnitude depends on the number of digits used to represent it.

How to convert fractions to binary?

How do you represent 0.5 in binary?

What is 0.1101 in decimal?