Question

Write binary number \[10,\overline{10111}\], as decimal fraction and reason-show it via geometric series

Answer 1

is the , meant as a decimal?

Answer 2

hmm as decimal fraction i tink something like \[\frac{x..}{x..}\]

Answer 3

\[x=\{0,1,2,3,4,5,6,7,8,9\}\]

Answer 4

no, what I meant is \[10. \overline{10111}=10, \overline{10111}\]

Answer 5

yes you are right i think its both ok

Answer 6

in question table given as with comma ","

Answer 7

so 10 = 4, yeah?

\(.\overline{10111}=23/32\)?

Answer 8

in binary i think \[10 = 2^{1}+{2}^{0} = 2 + 1 = 3 \]

Answer 9

and as such, \(.\overline{0000010111} = \Large{\frac{23} {32^2}}\)

Answer 10

.10111 = 1/2 + 0/4 + 1/8 + 1/16 + 1/32 = (16 + 0 + 4 + 2 + 1)/32 = 23/32

Answer 11

in binary, \(.1 = 1 \times 2^{-1}\text{, }.01 = 1 \times 2^{-2}\text{, }.001 = 1 \times 2^{-3}\text{, and so on}\)

Answer 12

and \[\frac{1}{2}+\frac{0}{4}+... \] is the geometric serie right ?

Answer 13

no...
\(.\overline{10111} = .101111011110111...\) what repeats is blocks of 5, which in binary is powers of 32.

so you have to break the fractional part (the part after the decimal sign) into groups of 5.
\[.10110=\frac{23}{32} \text{, }\]\[.0000010111=\frac{23}{32^2}\text{, }\]\[.000000000010111=\frac{23}{32^3}\text{, and so on...}\]

Answer 14

ok pgpilot thank you very much

Answer 15

does that make sense? do you see what the geometric part is?

Answer 16

geometric series i think \[32^{1}, 32^{2}, 32^{3}\] right ?

Answer 17

no, because the powers of 32 are in the denominator...

Answer 18

a ok like \[(2^{4})^1,(2^{4})^2 ... \] and so on right ?

Answer 19

sorry \[(2^{5}).... \]

Answer 20

\[4 + \frac{23}{32} +\frac{23}{32^2}+\frac{23}{32^3}+\frac{23}{32^4}+ \cdots\]

Answer 21

now its more clear to understand thank you very very much pgpilot