+54 1.) Find the first ten digits after the decimal point in the decimal expansion of $\frac{10}{27}=0.abcdefghij\ldots$
without a calculator.
(Express your answer as a ten-digit number.)
2.) The number $\frac{12}{13}$ can be expressed as a repeating decimal $0.\overline{abcdef}$. Find the repeating part $abcdef$ without a calculator.
3.) Convert $0.04\overline{55}$ to a fraction in simplest form.
4.) Convert $6032_8$ to decimal.
5.) Convert $999_{16}$ to decimal.
6.) Convert 35 from base 10 to base 2.
(You do not need to include the subscript 2 in this answer.)
7.) Convert $2231_4$ to base 2 without first converting to decimal.
(You do not need to include the subscript 2 in this answer.)
All help Appreciated(ASAP if possible)
+209 OK, IM guessing this is from AoPS from the dollar signs...... I don't really mind the asking and I would solve it but can you edit it so the latex is in latex???????? ok so this is my version of it...:
1.) Find the first ten digits after the decimal point in the decimal expansion of \(\frac{10}{27}=0.abcdefghij\ldots\)
without a calculator.
(Express your answer as a ten-digit number.)
2.) The number \(\frac{12}{13}\) can be expressed as a repeating decimal \(0.\overline{abcdef}\). Find the repeating part \(abcdef\) without a calculator.
3.) Convert \(0.04\overline{55}\) to a fraction in simplest form.
4.) Convert \(6032_8\) to decimal.
5.) Convert \(999_{16}\) to decimal.
6.) Convert 35 from base 10 to base 2.
(You do not need to include the subscript 2 in this answer.)
7.) Convert \(2231_4\) to base 2 without first converting to decimal.
+209 1) one is kinda easy..... just start dividing and find a pattern. I found the pattern 0.370 repeating, and so your answer for 1 would be 0.3703703703
2)So I started manually dividing and after 6 digits, I stopped... I want to show it here but have no idea how... :( but your answer is 923076
3)to convert 0.045 repeating 5 to fraction form, just do this:
1000x=45.55555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555
- x=0.045555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555
_____________________________________________________________________________________________________
subtract and get 999x=45.1
x=451/9990
x=41/90
4) \(6032_8\) to decimal?! argh! ok, so start by doing (6*6^3)+(0*6^2)+(3*6^1)+(2*6^0)=1407.... and 1407 is 6032_8 in base 10.... in other words, decimal.
5)\(999__16\)ok I cant get the base thing to work in latex... so deal with this.
(9*16^2)+(9*16)+9=2457 in decimal..
6)35/2=17r1
17/2=8r1
8/2=4r0
4/2=2r0
2/2=1r0
1/2=0r1
so i think 35 in base 2 is 100011
7)10101101...
+9673 1) \(\dfrac{10}{27} = \dfrac{370}{999} = 0.\overline{370} = 0.370\;370\;370\;3...\)
+9673 3)
\(\quad0.04\overline{55}\\ = 0.04 + 0.00\overline{5}\\ = \dfrac{4}{100} + \dfrac{0.\overline{5}}{100}\\ = \dfrac{4}{100} + \dfrac{5}{900}\\ = \dfrac{41}{900}\)
