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Maybe this doesn't make sense or is too basic but if you take for example $5 \div 2 = 2 + \frac12 $ with the "remainder" being $\frac12$. If you multiply the quotient by the divisor and add the remainder you get the dividend: $2 * 2 + 1 = 5$. But here the remainder is just the numerator of $\frac12$. Is there a way to explain what happens to the denominator?

Thanks

Rolomoto
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  • Remainder gets used in two ways, because it only really means something when paired with the divisor. $\tfrac 12$, I think, is the less common way. You could also call it the "fractional part". – Alex K Feb 07 '23 at 22:38

2 Answers2

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Euclidean division requires the relationship between the dividend, the divisor, the quotient, and the remainder to be able to be written as $$p=dq+r,$$ where $p$ is the dividend, $d$ is the divisor, $q$ is the quotient, and $r$ is the remainder. It is easy to misunderstand that the remainder must take on some denominator, and this is precisely what happens when we divide [the equation] through by the divisor $d$: $$p=dq+r\implies \frac pd=q+\frac rd.$$

Using your example, we can see clearly that the equations $$5=2\times2+1$$ as well as $$\frac52=2+\frac12$$ both hold, but we have $r=1$, not $\frac12$.

Andrew Chin
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Yes there is an easy answer: $(2+\frac{1}{2})\cdot 2=2\cdot2+ 2\cdot\frac{1}{2}\text{ and }2\cdot\frac{1}{2}=1$ if you divide 5 by 3 the remainder is 1 not $\frac{1}{2}$, 2+1/2 is the exact answer without remainder.

trula
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