Property 5
Infinive Fractions
\(a+\samefrac b{\boxed{a+\samefrac b{a+\samefrac b{\ddots}}}}=x\quad\left.\begin{array}[t]{l}\text{(is true)}\;\longleftarrow\\\longrightarrow x\;\text{(if we say x)}\end{array}\right]\)
Example
\(8+\samefrac9{8+\samefrac9{8+\samefrac9{\ddots}}}\quad\left(\text{The }8+\samefrac9{8+\samefrac9{8+\samefrac9{\ddots}}}\text{ equation}\right)\)
\(8+\samefrac9x=x\)
(\(8+\samefrac9x=x\) then the equation is solved by simplifying it)
1.
\(4+\samefrac{4+\samefrac{4+\samefrac{\mathinner{\mkern1mu\raise1pt{.}\mkern2mu\raise4pt{.}\mkern2mu\raise7pt{.}\mkern1mu}}5}5}5=?\)
2.
\(7+\samefrac8{7+\samefrac8{7+\samefrac8\ddots}}=?\)
3.
\(1+\samefrac{1+\samefrac{1+\samefrac{\mathinner{\mkern1mu\raise1pt{.}\mkern2mu\raise4pt{.}\mkern2mu\raise7pt{.}\mkern1mu}}3}3}3=?\)
4.
\(2+\samefrac8{2+\samefrac8{2+\samefrac8\ddots}}=?\)
Property 6
Decimal Numbers
\(\samefrac ab\)
Decimal numbers are another way of writing fractions and mixed numbers.
All numbers to the left of decimal point are whole numbers.
All numbers to the right of the decimal point are fractions with denominators of only powers of 10 notation.
Decimal Numbers
\(\blacksquare\quad\samefrac a{10^n}=\underbrace{0{,}0000\ldots a}_{n\text{ (times)}}\)
\(\blacksquare\quad\samefrac a{10}=0{,}a\)
\(a{,}b=\samefrac{ab}{10}=a+\samefrac b{10}\)
\(\blacksquare\quad\samefrac a{100}=0{,}0a\)
\(a{,}bc=\samefrac{abc}{100}=a+\samefrac{bc}{100}\)
\(0{,}x=0{,}x0=0{,}x00=\ldots\)
Convert the number below to the decimal number.
1.
\(\samefrac{1}{2}=?\)
2.
\(\samefrac{3}{4}=?\)
3.
\(\samefrac{4}{25}=?\)
4.
\(\samefrac{3}{5}=?\)
5.
\(\samefrac{7}{125}=?\)
6.
\(\samefrac{3}{8}=?\)
7.
\(\samefrac{13}{4}=?\)
8.
\(\samefrac{17}{8}=?\)