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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 $3.9999999999$.

The Wolverine
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    Note that $.99999 \dots = 1$, a well known fact proved through the use of an infinite series. – Hyperion Feb 14 '19 at 16:36
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    This depends on how exactly the calculator represents numbers internally, which is more of a hardware engineering question than a mathematics question. I'm voting to close as off-topic for this reason. – Connor Harris Feb 14 '19 at 16:38
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    @Connor Harris I think the OP is just confused why $3.999...=4$ rather than trying to understand how their calculator works. – Lt. Commander. Data Feb 14 '19 at 16:40
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    Almost a duplicate of "Why is $0.999\cdots=1$ ?" – Peter Feb 14 '19 at 16:45
  • For this reason, modern calculators can handle fractions avoiding such rounding errors. Older calculators also do not display $0$ , when "$\sin(\pi)$" is entered. – Peter Feb 14 '19 at 16:47
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    Does $\frac33=1$ also bother you for the same reason? – saulspatz Feb 14 '19 at 16:55
  • I think you made the question ambiguous (and got irrelevant answer) by letting ellipses at the end of the last number. –  Feb 14 '19 at 17:37
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    what is $4 - 3.999999 \dots$? – John Joy Feb 14 '19 at 18:52

4 Answers4

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The fallacy is that you only consider finite many $3's$ giving only finite many $9's$ , when multiplied by $3$. But we have infinite many $3's$ , and the result is exactly $4$. The reason is exactly the same why $$0.999\cdots =1$$ which you can show with a geometric series with start value $\frac{9}{10}$ and quotient $\frac{1}{10}$. Using the formula, you get exactly $1$.

Peter
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The calculators usually keep invisible extra decimals (so-called guard digits) and round for display.

For example, computing on $10$ digits,

$$\frac43=1.333333333$$ then

$$\frac43\, 3=3.999999999$$

Now, displayed on $8$ digits with rounding,

$$4.0000000$$ or simply $$4$$

On some calculators,

$$4\div3\times3-4$$ might not return $0$, even if the intermediate result $4$ is shown.

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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 $\frac{4}{3}*3=1+\sum_{i=1}^{\infty}\frac{3}{10^i}=1+3=4$.

H Huang
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Please note that $$\frac4 3 \neq 1.\underbrace{333333333...3}_{n}, $$ no matter how large $n$ is.

Allawonder
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