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I'm unable to get the difference between fraction and rational, we say $\frac{a}{b}$ is rational number if a and b are two integer with $b\neq 0$, and we can say also $\frac{a}{b}$ is a fraction but i don't know any reason for that, my question here is :Does fraction have the same meaning as rational in number theory ?

Note: one other thing wich is mixed me is that wolfram alpha considered rational number as fraction as shown here

Edit: I have edited the question just for specification and clarification according to the gaven answers without any change in the meaning of question

Mittens
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zeraoulia rafik
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5 Answers5

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$\frac{\sqrt3}{2}$ is a fraction, but not a rational number. You want both your denominator and numerator to be integers in order for the fraction to become a rational number

asdf
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  • The OP said that $a$ and $b$ are integers? – Dietrich Burde Jan 24 '18 at 19:11
  • Well, I thought the question was "Is fraction has the same meaning with rational in number theory", And also, he didn't specify any conditions for $a$ or $b$ – asdf Jan 24 '18 at 19:13
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    I think that's the point @asdf is making all rational numbers can be written as fractions but not all fractions are rational numbers. – Warren Hill Jan 24 '18 at 19:16
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A fraction is a certain way of writing any number. It consists of three parts: a numerator, a denominator, and a line between them. You can also write a number as a decimal expansion, and you can often translate between these two different ways of writing numbers. For instance, $2.5$ and $\frac52$ both represent the same number, but written differently. Only the second one is a fraction.

A rational number is a special type of number. One way to characterise a rational number is that it can be written as a fraction with integers as numerator and denominator. Another way to characterise them is to say that they can be written with a repeating decimal expansion. Thus the number mentioned in the previous paragraph (the one that may be represented as either $2.5$ or as $\frac52$) is rational.

So, fraction is a way of writing numbers, regardless of what number it is. Rational is a certain special type of numbers, regardless of how is written.

Arthur
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I think it's better $$\mathbb Q=\left\{\frac{a}{b}|a\in\mathbb Z,b\in\mathbb N\right\}.$$ If we say about these fractions then you are right.

Michael Rozenberg
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"Rational" has several meanings in number theory. For the field of rational numbers, it means fractions $\frac{a}{b}$ with $a,b\in\mathbb{Z}$, and $b\neq 0$ as you said, the "fraction field" of the integral domain $\mathbb{Z}$. A rational point on a curve can be a point with coordinates in $\mathbb{Q}$, but can also be a $K$-rational point, not necessarily with rational numbers. In this sense, fraction does not have the same meaning as rational in number theory.

Dietrich Burde
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Any number written in the form $\frac{a}{b}$ is a fraction provided $b \neq 0$, because division by zero is not defined.

Thus $\frac{2}{3}$, $\frac{221}{5}$ and $\frac{\pi}{2}$ are all fractions but only $\frac{2}{3}$ are $\frac{221}{5}$ rational since the decimal representation either repeats $\frac{2}{3} \approx 0.666666666666666666666$ with the 6's repeating for ever or terminates $\frac{221}{5} = 44.2$.

$\frac{\pi}{2}\approx 1.5707963...$ Is not a rational however as the decimal expansion neither ends or repeats.

If we add the requirement that $a$ and $b$ are both integers then you have the fractional representation of a rational number. I would not call $0.5$ a fraction however as it is not written in the correct format, though it is certainly a rational number. I would call $\frac{1}{2}$ a fraction though and they both have the same value.

Warren Hill
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