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I was shown the following:

$$ 0.110 < 0.1101 < 0.111$$

and told that the middle number is halfway in between those two numbers. Is this correct? How can I see that?

Update:

If I add a zero to the first and the last number, I get:

$$ 0.1100 < 0.1101 < 0.1110$$

Now, let write out the last two digits and we can see the relationship:

$$ \underbrace{00 < 01 < 10}_{base-2} = \underbrace{0 < 1 < 2}_{base-10} $$

Max Koretskyi
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2 Answers2

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Calculate the average of $0.110$ and $0.111$, by adding them and dividing the result by $2$ (keep in mind that it is binary), and see that you get $0.1101$.

Arthur
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  • I see, thanks. Is there an easier way to see this relationship, like the one suggested by Ahmed – Max Koretskyi May 21 '16 at 14:01
  • @Maximus Mathematically equivalent, but perhaps more intuitive is to calculate the differences $0.1101 - 0.110$ and $0.111 - 0.1101$ and see that you get the same number. – Arthur May 21 '16 at 14:04
  • thanks, can you please tell me whether the logic presented in my update is valid? – Max Koretskyi May 21 '16 at 14:15
  • @Maximus It seems valid, yes. But I wouldn't phrase it like that. What you have done is the following: 1) Multiply all the numbers by $1000$ 2) Subtract $1100$ 3) Convert to base ten to inspect the values. Perhaps you should convince yourself that none of these three steps ruins the "halfwayness" that you're checking for. – Arthur May 21 '16 at 14:39
  • thanks. Here is the other explanation. Can I say that the number is always two times more that the number following it? For example, in the fraction 0.111=0+1/2+1/4+1/8 the second number is two times larger than the third, so any number 0.111...(1*X) will be two times larger, than 0.111...(0*X)1? – Max Koretskyi May 21 '16 at 14:45
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\begin{align} 0.1100 &= 2^{-1} + 2^{-2} \\ 0.1101 &= 2^{-1} + 2^{-2} + 2^{-4} \\ 0.1110 &= 2^{-1} + 2^{-2} + 2^{-3} = 2^{-1} + 2^{-2} + 2\cdot2^{-4} \end{align}

Therefore, the difference between $0.1100$ and $0.1101$ as well as between $0.1101$ and $0.1110$ is $2^{-4}$, and hence $0.1101$ does indeed lie halfway between the two other numbers.

sbares
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