Find the rational number whose decimal expansion is $0.3344444444…$
This is a homework question. I am not sure if I understand the meaning of the question. As per wiki article I think what I have to do is that I write the decimal expansion as $0 + 3\frac{1}{10}$ + $ 3\frac{1}{100}$ + $ 4\frac{1}{1000}\dots$
Please tell me if this is incorrect.
Best Answer
What you need to do is to find two integers $a,b$ such that $\frac ab=0.3344444\dots$.
You have (at least) two ways of solving the problem:
First option
call the number $0.334444444444\dots$ simply $x$, i.e.
$$x=0.3344444444\dots$$
Then, multiply this equation by $100$ to get
$$100x=33.\overline 4$$
Multiply it again by $10$ to get
$$1000 x = 334.\overline{4}$$
Now, try to subtract the above equations. What do you get?
Second option:
You can write $x=0.33\overline 4$ as $$0.3 + 0.03 + 0.004 + 0.0004 + \dots$$
which we can rewrite as $$x=0.33 + 0.004\cdot (1 + 0.1 + 0.01 + \dots)$$ or further as
$$x=\frac{33}{100} + \frac{4}{1000}\cdot \sum_{i=0}^\infty \left(\frac{1}{10}\right)^{i}$$
Now, remember how you were taught what the sum of a geometric series is?