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''arithmetical''
 
''arithmetical''
  
A number consisting of one or more equal parts of a unit. It is denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f0412001.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f0412002.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f0412003.png" /> are integers (cf. [[Integer|Integer]]). The numerator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f0412004.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f0412005.png" /> denotes the number of parts taken of the unit; this is divided by the number of parts equal to the number appearing as the denominator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f0412006.png" />. A fraction may also be considered as the ratio produced by dividing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f0412007.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f0412008.png" />.
+
A fraction is a number consisting of one or more equal parts of a unit. It is denoted by the symbol $a/b$, where $a$ and $b\ne 0$ are integers (cf.
 +
[[Integer|Integer]]). The numerator $a$ of $a/b$ denotes the number of parts taken of the unit; this is divided by the number of parts equal to the number appearing as the denominator $b$. A fraction may also be considered as the ratio produced by dividing $a$ by $b$.
  
The fraction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f0412009.png" /> remains unchanged if both the numerator and the denominator are multiplied by the same non-zero integer. Owing to this fact, any two fractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120011.png" /> may be brought to a common denominator, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120013.png" /> may be replaced by fractions equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120015.png" />, respectively, both of which have the same denominator. Moreover, fractions may be reduced by dividing their numerator and denominator by the same number; accordingly, any fraction may be represented as an irreducible fraction, i.e. a fraction the numerator and denominator of which have no common factors.
+
The fraction $a/b$ remains unchanged if both the numerator and the denominator are multiplied by the same non-zero integer. Owing to this fact, any two fractions $a/b$ and $c/d$ may be brought to a common denominator, i.e. $a/b$ and $c/d$ may be replaced by fractions equal to $a/b$ and $c/d$, respectively, both of which have the same denominator. Moreover, fractions may be reduced by dividing their numerator and denominator by the same number; accordingly, any fraction may be represented as an irreducible fraction, i.e. a fraction the numerator and denominator of which have no common factors.
  
The sum and the difference of two fractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120017.png" /> having a common denominator are given by
+
The sum and the difference of two fractions $a/b$ and $c/b$ having a common denominator are given by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120018.png" /></td> </tr></table>
+
$$\frac{a}{b} \pm \frac{c}{b} = \frac{a\pm c}{b}$$
 +
In order to add or to subtract fractions with different denominators they must first be reduced to fractions with a common denominator. As a rule, the
 +
[[Least common multiple|least common multiple]] of the numbers $b$ and $d$ is taken as the common denominator. Multiplication and division of fractions is given by the following rules:
  
In order to add or to subtract fractions with different denominators they must first be reduced to fractions with a common denominator. As a rule, the [[Least common multiple|least common multiple]] of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120020.png" /> is taken as the common denominator. Multiplication and division of fractions is given by the following rules:
+
$$\frac{a}{b}\cdot\frac{c}{d} = \frac{a\cdot c}{b\cdot d},\quad  \frac{a}{b} : \frac{c}{d} = \frac{a\cdot d}{b\cdot c},\quad (c\ne 0). $$
 
+
A fraction $a/b$ is said to be a proper fraction if its numerator is smaller than its denominator; otherwise it is an improper fraction. A fraction is said to be a decimal fraction if its denominator is a power of the number 10 (cf.
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120021.png" /></td> </tr></table>
+
[[Decimal fraction|Decimal fraction]]).
 
 
A fraction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120022.png" /> is said to be a proper fraction if its numerator is smaller than its denominator; otherwise it is an improper fraction. A fraction is said to be a decimal fraction if its denominator is a power of the number 10 (cf. [[Decimal fraction|Decimal fraction]]).
 
  
 
==Formal definition of fractions.==
 
==Formal definition of fractions.==
Fractions may be represented as ordered pairs of integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120024.png" />, for which an equivalence relation has been specified (an equality relation of fractions), namely, it is considered that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120025.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120026.png" />. The operations of addition, subtraction, multiplication, and division are defined in this set of fractions by the following rules:
+
Fractions may be represented as ordered pairs of integers $(a,b)$, $b\ne 0$, for which an equivalence relation has been specified (an equality relation of fractions), namely, it is considered that $(a,b) = (c,d)$ if $ad = bc$. The operations of addition, subtraction, multiplication, and division are defined in this set of fractions by the following rules:
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120027.png" /></td> </tr></table>
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120028.png" /></td> </tr></table>
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$$(a,b)\pm (c,d) = (ad\pm bc,bd),$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120029.png" /></td> </tr></table>
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$$(a,b)\cdot (c,d) = (ac,bd),$$
  
(thus, division is defined only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120030.png" />).
+
$$(a,b): (c,d) = (ad,bc),$$
 +
(thus, division is defined only if $c\ne 0$).
  
A similar definition of fractions is convenient in generalizations and is accepted in modern algebra (cf. [[Fractions, ring of|Fractions, ring of]]).
+
A similar definition of fractions is convenient in generalizations and is accepted in modern algebra (cf.
 +
[[Fractions, ring of|Fractions, ring of]]).
  
  
  
 
====Comments====
 
====Comments====
The set of fractions (of the integers) is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120031.png" />. With the arithmetical operations and natural order defined in the main article above it is an [[Ordered field|ordered field]]. The [[Absolute value|absolute value]] gives a metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120032.png" />. [[Completion|Completion]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120033.png" /> in this metric (e.g. by using Cauchy sequences, cf. [[Fundamental sequence|Fundamental sequence]]) leads to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120034.png" />, the ordered field of real numbers (cf. [[Real number|Real number]]). In this connection, a fraction is also called a rational number, and a number from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120035.png" /> that is not a fraction is called an irrational number, see, e.g., [[#References|[a1]]].
+
The set of fractions (of the integers) is denoted by $\Q$. With the arithmetical operations and natural order defined in the main article above it is an
 +
[[Ordered field|ordered field]]. The
 +
[[Absolute value|absolute value]] gives a metric on $\Q$.
 +
[[Completion|Completion]] of $\Q$ in this metric (e.g. by using Cauchy sequences, cf.
 +
[[Fundamental sequence|Fundamental sequence]]) leads to $\R$, the ordered field of real numbers (cf.
 +
[[Real number|Real number]]). In this connection, a fraction is also called a rational number, and a number from $\R$ that is not a fraction is called an irrational number, see, e.g.,
 +
{{Cite|HeSt}}.
  
For a construction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120036.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120037.png" /> using Dedekind cuts (cf. also [[Dedekind cut|Dedekind cut]]) see, e.g., [[#References|[a2]]].
+
For a construction of $\R$ from $\Q$ using Dedekind cuts (cf. also
 +
[[Dedekind cut|Dedekind cut]]) see, e.g.,
 +
{{Cite|Ru}}.
  
For aliquot fractions (i.e. numbers of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041200/f04120039.png" /> a positive [[Integer|integer]]) see [[Aliquot ratio|Aliquot ratio]].
+
For aliquot fractions (i.e. numbers of the form $1/n$, $n$ a positive
 +
[[Integer|integer]]) see
 +
[[Aliquot ratio|Aliquot ratio]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Hewitt,  K.R. Stromberg,  "Real and abstract analysis" , Springer  (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W. Rudin,  "Principles of mathematical analysis" , McGraw-Hill  (1953)</TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|HeSt}}||valign="top"| E. Hewitt,  K.R. Stromberg,  "Real and abstract analysis", Springer  (1965) {{MR|0188387}}  {{ZBL|0137.03202}}
 +
|-
 +
|valign="top"|{{Ref|Ru}}||valign="top"| W. Rudin,  "Principles of mathematical analysis", McGraw-Hill  (1953) {{MR|0055409}}  {{ZBL|0052.05301}}
 +
|-
 +
|}

Revision as of 20:59, 6 April 2012

arithmetical

A fraction is a number consisting of one or more equal parts of a unit. It is denoted by the symbol $a/b$, where $a$ and $b\ne 0$ are integers (cf. Integer). The numerator $a$ of $a/b$ denotes the number of parts taken of the unit; this is divided by the number of parts equal to the number appearing as the denominator $b$. A fraction may also be considered as the ratio produced by dividing $a$ by $b$.

The fraction $a/b$ remains unchanged if both the numerator and the denominator are multiplied by the same non-zero integer. Owing to this fact, any two fractions $a/b$ and $c/d$ may be brought to a common denominator, i.e. $a/b$ and $c/d$ may be replaced by fractions equal to $a/b$ and $c/d$, respectively, both of which have the same denominator. Moreover, fractions may be reduced by dividing their numerator and denominator by the same number; accordingly, any fraction may be represented as an irreducible fraction, i.e. a fraction the numerator and denominator of which have no common factors.

The sum and the difference of two fractions $a/b$ and $c/b$ having a common denominator are given by

$$\frac{a}{b} \pm \frac{c}{b} = \frac{a\pm c}{b}$$ In order to add or to subtract fractions with different denominators they must first be reduced to fractions with a common denominator. As a rule, the least common multiple of the numbers $b$ and $d$ is taken as the common denominator. Multiplication and division of fractions is given by the following rules:

$$\frac{a}{b}\cdot\frac{c}{d} = \frac{a\cdot c}{b\cdot d},\quad \frac{a}{b} : \frac{c}{d} = \frac{a\cdot d}{b\cdot c},\quad (c\ne 0). $$ A fraction $a/b$ is said to be a proper fraction if its numerator is smaller than its denominator; otherwise it is an improper fraction. A fraction is said to be a decimal fraction if its denominator is a power of the number 10 (cf. Decimal fraction).

Formal definition of fractions.

Fractions may be represented as ordered pairs of integers $(a,b)$, $b\ne 0$, for which an equivalence relation has been specified (an equality relation of fractions), namely, it is considered that $(a,b) = (c,d)$ if $ad = bc$. The operations of addition, subtraction, multiplication, and division are defined in this set of fractions by the following rules:

$$(a,b)\pm (c,d) = (ad\pm bc,bd),$$

$$(a,b)\cdot (c,d) = (ac,bd),$$

$$(a,b): (c,d) = (ad,bc),$$ (thus, division is defined only if $c\ne 0$).

A similar definition of fractions is convenient in generalizations and is accepted in modern algebra (cf. Fractions, ring of).


Comments

The set of fractions (of the integers) is denoted by $\Q$. With the arithmetical operations and natural order defined in the main article above it is an ordered field. The absolute value gives a metric on $\Q$. Completion of $\Q$ in this metric (e.g. by using Cauchy sequences, cf. Fundamental sequence) leads to $\R$, the ordered field of real numbers (cf. Real number). In this connection, a fraction is also called a rational number, and a number from $\R$ that is not a fraction is called an irrational number, see, e.g., [HeSt].

For a construction of $\R$ from $\Q$ using Dedekind cuts (cf. also Dedekind cut) see, e.g., [Ru].

For aliquot fractions (i.e. numbers of the form $1/n$, $n$ a positive integer) see Aliquot ratio.

References

[HeSt] E. Hewitt, K.R. Stromberg, "Real and abstract analysis", Springer (1965) MR0188387 Zbl 0137.03202
[Ru] W. Rudin, "Principles of mathematical analysis", McGraw-Hill (1953) MR0055409 Zbl 0052.05301
How to Cite This Entry:
Fraction. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fraction&oldid=16993
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article