Which number cannot be represented exactly on a digital computer?
A digital computer stores fractions as a finite number of bits b 1 , b 2 , b 3 , … , b n representing
2 b 1 + 2 2 b 2 + ⋯ + 2 n b n .
Which of the following numbers cannot be represented exactly in this manner?
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2 solutions
However, 0.1 cannot be written exactly as a finite sum of inverse powers of 2.
How do you know this is true?
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Hi Pi Han Goh. Thanks for your comment. I have now edited my solution to make the answer more convincing.
If we consider b = 1 , we notice that
2 b = 0 . 5 ,
2 2 b 2 = 0 . 2 5 and
2 3 b 3 = 0 . 1 2 5 ;
therefore there exists no element equals to 1 0 b = 0 . 1
You have only shown that 0.125, 0.25, and 0.5 are possible. How do you know that 0.1 is not possible as well?
I see the problem got edited but if we look at it right now, l see a geometric progression series whose ratio is 0 . 5 , therefore 0 . 1 can't be an element of this series, because every element is in the form of 2 n 1 for n = 1 , 2 , 3 , …
A computer stores decimal numbers as sums of inverse powers of 2. So, 0.5 would be stored as 0 . 1 0 0 0 0 = 1 ∗ 1 / 2 , 0.25 would be stored as 0 . 0 1 0 0 0 = 0 ∗ 1 / 2 + 1 ∗ 1 / 2 2 , and 0.125 would be stored as 0 . 0 0 1 0 0 0 = 0 ∗ 1 / 2 + 0 ∗ 1 / 2 2 + 1 ∗ 1 / 2 3 . However, 0.1 cannot be written exactly as a finite sum of inverse powers of 2. We can see that to represent 0.1 in binary we will have to select inverse powers of 2 which are less than 0.125. However, we observe that all such numbers or their sums will always end with 5, i.e, 0.0625, 0.0625+0.03125 = 0.09375. The sum of inverse powers of 2 will always have 5 in the last decimal place. Therefore 0.1 cannot be represented exactly.