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Let x be the binary number (0.001001001 . . .)2 and let y be the octal number (0.666666 . . .)8. What is x + y in decimal?

Please explain your answer.

 Feb 7, 2024

Best Answer 

 #1
avatar+394 
+2

We can convert these two seperately. 

0.001001001... when expressed in Base 2 is \(0*{2}^{-1}+0*{2}^{-2}+1*{2}^{-3}+0*{2}^{-4}+0*{2}^{-5}+1*{2}^{-6}...\)

The terms with the 0's are all 0, so this simplifies into \({2}^{-3}+{2}^{-6}+{2}^{-9}+{2}^{-12}...\)

Using the Infinite geometric series formula, (first term)/(1 - common ratio), we get \(\frac{{2}^{-3}}{1-{2}^{-3}}\)which simplifies to \(\frac{1}{7}\).

We do the second one similarly, simplifying 0.6666666666666 base 8 into \(6*{8}^{-1}+6*{8}^{-2}+6*{8}^{-3}+6*{8}^{-4}\), using the infinite geometric series formula, we get, \(6*\frac{{8}^{-1}}{1-{8}^{-1}}\)which simplifies to \(\frac{6}{7}\). Adding, 1/7 and 6/7 we get 1.

 Feb 8, 2024
 #1
avatar+394 
+2
Best Answer

We can convert these two seperately. 

0.001001001... when expressed in Base 2 is \(0*{2}^{-1}+0*{2}^{-2}+1*{2}^{-3}+0*{2}^{-4}+0*{2}^{-5}+1*{2}^{-6}...\)

The terms with the 0's are all 0, so this simplifies into \({2}^{-3}+{2}^{-6}+{2}^{-9}+{2}^{-12}...\)

Using the Infinite geometric series formula, (first term)/(1 - common ratio), we get \(\frac{{2}^{-3}}{1-{2}^{-3}}\)which simplifies to \(\frac{1}{7}\).

We do the second one similarly, simplifying 0.6666666666666 base 8 into \(6*{8}^{-1}+6*{8}^{-2}+6*{8}^{-3}+6*{8}^{-4}\), using the infinite geometric series formula, we get, \(6*\frac{{8}^{-1}}{1-{8}^{-1}}\)which simplifies to \(\frac{6}{7}\). Adding, 1/7 and 6/7 we get 1.

hairyberry Feb 8, 2024
 #2
avatar+129270 
+2

Thanks, hairyberry......very informative.....!!!

 

cool cool cool

CPhill  Feb 8, 2024
 #3
avatar+1624 
+2

Nice one Harry... I agree with CPhill! wink

proyaop  Feb 8, 2024
 #4
avatar+394 
+1

Thanks!!!

hairyberry  Feb 9, 2024

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