+0

+1
20
4
+0

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?

Feb 7, 2024

#1
+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
+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.

hairyberry Feb 8, 2024
#2
+129270
+2

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

CPhill  Feb 8, 2024
#3
+1624
+2

Nice one Harry... I agree with CPhill!

proyaop  Feb 8, 2024
#4
+394
+1

Thanks!!!

hairyberry  Feb 9, 2024