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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: