The simple math behind decimal-binary conversion algorithms

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Looking to convert to binary floating-point? Try my floating-point converter. Looking to calculate with binary numbers? Try my binary calculator. Looking to convert numbers between arbitrary bases? Try my base converter. This is a decimal to binary and binary to decimal converter. Conversion is implemented with arbitrary-precision arithmeticwhich gives the converter its ability to convert numbers bigger than binary number to decimal conversion that can fit in standard computer word sizes like 32 or 64 bits.

Besides the converted result, the number of digits in both the original and converted numbers is displayed. For example, when converting decimal This means that the decimal input has 2 digits in its integer part and 3 digits in its fractional part, and the binary output has 6 digits in its integer part and 3 digits in binary number to decimal conversion fractional part. Fractional decimal values that are dyadic convert to finite fractional binary values and are displayed in full precision.

Fractional decimal values that are non-dyadic convert to infinite repeating binary number to decimal conversion binary values, which are truncated — not rounded — to the specified number of bits. The converter is set up so that you can explore properties of decimal to binary number to decimal conversion and binary to decimal conversion.

A decimal integer or dyadic fractional value converted to binary and then back to decimal matches the original decimal value; a non-dyadic value converts back only to an approximation of its original decimal value. Increasing the number of bits of precision will make the converted number closer to the original. You can study how the number of digits differs between the decimal and binary representations of a number. Large binary integers have about log 2 10or approximately 3.

Dyadic decimal fractions have the same number of digits as their binary equivalents. Non-dyadic decimal values, as already noted, have infinite binary equivalents. This converter also converts between bases other than binary and decimal. Skip to content Decimal to Binary Enter a decimal number e. Truncate infinite binary fractions to bits.

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In the text proper, we saw how to convert the decimal number While this worked for this particular example, we'll need a more systematic approach for less obvious cases.

In fact, there is a simple, step-by-step method for computing the binary expansion on the right-hand side of the point. We will illustrate the method by converting the decimal value. Begin with the decimal fraction and multiply by 2.

The whole number part of the result is the first binary digit to the right of the point. So far, we have. Next we disregard the whole number part of the previous result the 1 in this case and multiply by 2 once again. The whole number part of this new result is the second binary digit to the right of the point.

We will continue this process until we get a zero as our decimal part or until we recognize an infinite repeating pattern. Disregarding the whole number part of the previous result this result was. The whole number part of the result is now the next binary digit to the right of the point. So now we have. In fact, we do not need a Step 4. We are finished in Step 3, because we had 0 as the fractional part of our result there.

You should double-check our result by expanding the binary representation. The method we just explored can be used to demonstrate how some decimal fractions will produce infinite binary fraction expansions. Next we disregard the whole number part of the previous result 0 in this case and multiply by 2 once again. Disregarding the whole number part of the previous result again a 0 , we multiply by 2 once again.

We multiply by 2 once again, disregarding the whole number part of the previous result again a 0 in this case. We multiply by 2 once again, disregarding the whole number part of the previous result a 1 in this case.

We multiply by 2 once again, disregarding the whole number part of the previous result. Let's make an important observation here. Notice that this next step to be performed multiply 2. We are then bound to repeat steps , then return to Step 2 again indefinitely. In other words, we will never get a 0 as the decimal fraction part of our result. Instead we will just cycle through steps forever. This means we will obtain the sequence of digits generated in steps , namely , over and over. Hence, the final binary representation will be.

The repeating pattern is more obvious if we highlight it in color as below: