need help with this question

[dratcliff24]

is it possible to express a decimal as a fraction without rounding

 

[Deleted member 4993]

is it possible to express a decimal as a fraction without rounding

Yes ... for example 0.5 (a decimal number) = ½ (a fraction) .... no rounding involved!

 

[HallsofIvy]

A decimal can be written as a fraction if and only if it represents a rational number. That's pretty much the definition of "rational number". One can also show that only terminating or "eventually repeating" decimals represent rational numbers.

For example, a terminating decimal can be written as $\displaystyle x= a_1a_2a_3\cdot\cdot\cdot a_m.b_1b_2b_3...b_n$ where the "a"s represent the digits before the decimal point and the "b" represent the n digits after the decimal point. Then $\displaystyle 10^nx= a_1a_2a_3\cdot\cdot\cdot a_m b_1b_2b_3\cdot\cdot\cdot b_m$ an integer. Dividing both sides by $\displaystyle 10^n$ gives the fraction $\displaystyle \frac{a_1a_2a_3\cdot\cdot\cdot a_m b_1b_2b_3\cdot\cdot\cdot b_n}{10^n}$. There may be some power of 2 or 5 that is a factor of the numerator and so will cancel with a 2 or 5 in the denominator but even so, we have it written as a fraction with only 2s and 5s in the denominator. For example, if x= 3.816, then 1000x= 3816 so that $\displaystyle x= \frac{3816}{1000}= \frac{8(477)}{8(125)}= \frac{477}{125}$

Repeating decimals are a little harder. Suppose $\displaystyle x= a_1a_2a_3\cdot\cdot\cdot a_m.b_1b_2b_3\cdot\cdot\cdot b_nc_1c_2c_3\cdot\cdot\cdot c_pc_1c_2c_3\cdot\cdot\cdot c_p \cdot\cdot\cdot$ where that last "$\displaystyle \cdot\cdot\cdot$" indicates that the "c" digits just keep repeating.

First, move the "b" digits out of the decimal part by multiplying by $\displaystyle 10^m$: $\displaystyle 10^n x= a_1a_2a_3\cdot\cdot\cdot$$\displaystyle a_mb_1b_2b_3\cdot\cdot\cdot b_n.c_1c_2c_3\cdot\dot\cdot c_pc_1c_2c_3\cdot\cdot\cdot$
Now multiply by another $\displaystyle 10^p$ to move that first group of "c"s out:
$\displaystyle 10^{m+ p}= a_1a_2a_3\cdot\cdot\cdot a_nb_1b_2b_3\cdot\cdot\cdot b_mc_1c_3c_3\cdot\cdot\cdot c_pc_1c_2c_3\cdot\cdot\cdot c_2c_2c_3\cdot\cdot\cdot c_p\cdot\cdot\cdot$

It is important to understand that because the "c" portion keeps repeating the "c" on the right never end so the decimal part of those two numbers is the same. Subtracting, we have $\displaystyle (10^{m+p}- 10^m)x= a_1a_2\cdot\cdot\cdot a_nb_1b_2\cdot\cdot\cdot b_mc_1c_2\cdot\cdot\cdot c_p- a_1a_2\cdot\cdot\cdot a_nb_1b_2\cdot\cdot\cdot b_m$. That's rather complicated but we have an integer times x equal to an integer and so x can be written as a fraction.

An easy example is $\displaystyle 0.\overline{3}$ where the overline indicates the 3 repeats- that is x= 0.3333.... Then 10x= 3.3333... and, subtracting 9x= 3 so that x= 3/9= 1/3.

A little more complicated example would be $\displaystyle x= 323.759\overline{1154}= 323.759115411541154...$. The non-repeating part decimal part, 759, has three decimal places: $\displaystyle 10^3x= 1000x= 323759.115411541154....$ The repeating part, 1154, has four decimal places: $\displaystyle 10^{3+ 4}x= 10000000x= 3237591154.11541154...$. Subtracting, $\displaystyle 9999000x= 3237267395$ and $\displaystyle x= \frac{3237267395}{9999000}$