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I was asked this question by a kid, is -$\frac{4}{7}$ a proper fraction or not? As per my knowledge $\frac{4}{7}$ is a proper fraction. If it has a -ve number does it make any difference? Definition says A number whose numerator is smaller than denominator is called a proper fraction. Can we consider -$\frac{4}{7}$as a proper fraction? If not why not please explain. This is my first question I don't have much idea about tags of mathematics if it is tagged wrongly please edit it.

EDIT: I have got these links from comments wiki link and a link form math world and both contradicting each other.

Thank you

Dibya

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    $\begingroup$ mathworld.wolfram.com/ProperFraction.html So it's neither proper or improper. $\endgroup$ – Lazar Ljubenović Mar 10 '13 at 12:23
  • $\begingroup$ It is proper: en.wikipedia.org/wiki/… $\endgroup$ – Dennis Gulko Mar 10 '13 at 12:23
  • $\begingroup$ If I had to define a convention that makes sense in the context of where questions like this have relevance, I would say that -4/7 isn't a fraction at all; it is a numeral consisting of a negative sign - and a (proper) fraction 4/7. $\endgroup$ – Hurkyl Mar 10 '13 at 12:26
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From Wikipedia:

Common fractions can be classified as either proper or improper. When the numerator and the denominator are both positive, the fraction is called proper if the numerator is less than the denominator, and improper otherwise. In general, a common fraction is said to be a proper fraction if the absolute value of the fraction is strictly less than one—that is, if the fraction is between $-1$ and $1$.* [Italics mine]

So $\;-1 < \left(-\dfrac 47\right) < 1$ is considered a proper fraction; (alternatively $\;\;0 < \Big|-\dfrac 47 \Big| = \dfrac 47 < 1$.


I think in terms of mathworld's definition: When using the division algorithm, for example, one requires an integer quotient $\times$ an integer divisor, plus a non-negative integer remainder less than the value of the divisor. So if dividing $-4$ by $7$:

$$-4 = -1\cdot 7 + 3, \;\;q = -1;\;\;r = 3, i.e., -\dfrac 47 = -1 + \dfrac 37$$ which would be a mixed fraction.

So the fractional part would be the proper fraction $\dfrac 37$, the integer part, $-1$. Consistent with this, mathworld may require that a proper fraction occurs only when the quotient is $0$, and the remainder a positive integer less than the divisor, hence the fractional part = $\dfrac rd\; d: $divisor,$\;r: $ the remainder when dividing number by $d$.

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  • $\begingroup$ I have edited my question added a link form mathworld given by Lazar in the comment. This is confusing. Which one is correct? $\endgroup$ – NewUser Mar 10 '13 at 12:29
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    $\begingroup$ It's a matter of convention - By any other name, it'd still smell the same. $\endgroup$ – Guest 86 Mar 10 '13 at 12:33
  • $\begingroup$ The Wikipedia definition is the standard. I think mathworld left out the clause that the $0 < $ |fraction|$ < 1$ $\endgroup$ – Namaste Mar 10 '13 at 12:34

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