G
Guest
Guest
I can convert integers and fixed decimals, but I need to convert 5/7 to binary.
How do you convert an irrational number to binary?
Thanks!
How do you convert an irrational number to binary?
Thanks!
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G
Guest
Guest
oops! an assumption I should have checked first. Thanks!
Hello, everyone!
Do you want to see a truly primitive way to convert decimals?
We know that: \(\displaystyle \L\,\frac{5}{7}\:=\:0.714285...\)
Assume that it equals a binary "decimal": .\(\displaystyle 0.7142857\,\cdots\;=\;0.abcdefgh\,\cdots\)
Multiply both sides by 2.
. . On the left, multiply as usual.
. . On the right, the "decimal point" moves one place to the right.
We have: \(\displaystyle \:1.42857\,\cdots\;=\;a.bcdefgh\,\cdots\)
. . Hence: .\(\displaystyle \fbox{a\,=\,1}\)
Remove the integers from the equation: \(\displaystyle \:0.428571\,\cdots\;=\;0.bcdefgh\,\cdots\)
Multiply both sides by 2: \(\displaystyle \:0.857142\,\cdots \;=\;b.cdefgh\,\cdots\)
. . Hence: .\(\displaystyle \fbox{b\,=\,0}\)
Remove the integers from the equation: \(\displaystyle \:0.857142\,\cdots \;=\;0.cdefgh\,\cdots\)
Multiply both sides by 2: \(\displaystyle \:1.714285\,\cdots\;=\;c.defgh\,\cdots\)
. . Hence: .\(\displaystyle \fbox{c\,=\,1}\)
Remove the integers from the equation: \(\displaystyle \:0.714285\,\cdots\;=\;0.defgh\,\cdots\)
But this decimal is identical to the one we started with.
. . Hence, the pattern repeats itself.
Therefore: \(\displaystyle \L\:\frac{5}{7}\;=\;0.\overline{101}_{_2}\)
Do you want to see a truly primitive way to convert decimals?
We know that: \(\displaystyle \L\,\frac{5}{7}\:=\:0.714285...\)
Assume that it equals a binary "decimal": .\(\displaystyle 0.7142857\,\cdots\;=\;0.abcdefgh\,\cdots\)
Multiply both sides by 2.
. . On the left, multiply as usual.
. . On the right, the "decimal point" moves one place to the right.
We have: \(\displaystyle \:1.42857\,\cdots\;=\;a.bcdefgh\,\cdots\)
. . Hence: .\(\displaystyle \fbox{a\,=\,1}\)
Remove the integers from the equation: \(\displaystyle \:0.428571\,\cdots\;=\;0.bcdefgh\,\cdots\)
Multiply both sides by 2: \(\displaystyle \:0.857142\,\cdots \;=\;b.cdefgh\,\cdots\)
. . Hence: .\(\displaystyle \fbox{b\,=\,0}\)
Remove the integers from the equation: \(\displaystyle \:0.857142\,\cdots \;=\;0.cdefgh\,\cdots\)
Multiply both sides by 2: \(\displaystyle \:1.714285\,\cdots\;=\;c.defgh\,\cdots\)
. . Hence: .\(\displaystyle \fbox{c\,=\,1}\)
Remove the integers from the equation: \(\displaystyle \:0.714285\,\cdots\;=\;0.defgh\,\cdots\)
But this decimal is identical to the one we started with.
. . Hence, the pattern repeats itself.
Therefore: \(\displaystyle \L\:\frac{5}{7}\;=\;0.\overline{101}_{_2}\)