<s>[INST] How can $\frac{4}{3} \times 3=4$ if $ \frac{4}{3}$ is $1.3$? Ok use your closest calculator, and type $\frac{4}{3}$, which is $1.3333333333$,and then multiply it with $3$ which is $3.9999999999$ but then type $\frac{4}{3} \times 3=4$ how?. How can it be $4$ if $\frac{4}{3}$ is $1.3333333333$ and when you multiply it with $3$ is $3.9999999999$.
 [/INST] Rational numbers are numbers that can be expressed as $\frac{p}{q}$, where p and q are relatively prime integers. Notice that the definition does not mention the decimal expansion. The decimal representation of $\frac{4}{3}$ is defined as $1+\sum_{i=1}^{\infty}\frac{3}{10^i}$. Multiplying by 3, the decimal expansion for $\frac{4}{3}*3$ becomes $3+\sum_{i=1}^\infty\frac{9}{10^i}$. The value of an infinite sum is defined as the limit of the partial sums, or, in simpler terms, what number is approached when we sum up the first $n$ terms, followed by the first $n+1$ term, etc. Doing that here, we see the sequence is $.9$, $.99$, $.999$, etc. It should be obvious that we are getting closer and closer to $1$. Thus, $\sum_{i=1}^{\infty}\frac{3}{10^i}=1$. We now have $\frac{4}{3}*3=1+\sum_{i=1}^{\infty}\frac{3}{10^i}=1+3=4$.
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