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Fraction
(mathematics)
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y Fractions (from Latin: fractus, "broken") are
numbers expressed as the ratio of two numbers, and
are used primarily to express a comparison between
parts and a whole.
y The earliest fractions were reciprocals ofintegers:
ancient symbols representing one part of two, one
part of three, one part of four, and so on. A much
later development were the common or "vulgar"
fractions which are still used today (, , , etc.)
and which consist of a numerator and a
denominator, the numerator representing a number
of equal parts and the denominator telling how manyof those parts make up a whole. An example is 3/4, in
which the numerator, 3, tells us that the fraction
represents 3 equal parts, and the denominator, 4,
tells us that 4 parts make up a whole.
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yA still later development was thedecimal fraction, now called simply a
decimal, in which the denominator is apower of ten, determined by thenumber ofdigits to the right of adecimal separator, the appearance ofwhich (e.g., a period, a raised period (),a comma) depends on the locale (forexamples, see decimal separator). Thus
for 0.75 the numerator is 75 and thedenominator is 10 to the second power,viz. 100, because there are two digits tothe right of the decimal separator.
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yA third kind of
fraction still incommon use is the
percentage, in whichthe denominator is
always 100. Thus75% means 75/100.
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yOther uses for fractions
are to represent ratios,and to represent division.
Thus the fraction 3/4 isalso used to represent the
ratio 3:4 (three to four)and the division 3 4
(three divided by four).
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TerminologyyHistorically, any number that did not
represent a whole was called a
"fraction". The numbers that we nowcall "decimals" were originally called
"decimal fractions"; the numbers we
now call "fractions" were called"vulgar fractions", the word "vulgar"
meaning "commonplace".
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Writing fractionsy A common or vulgar fraction is usually
written as a pair of numbers, the top number
called the numerator and the bottom number
called the denominator. A line usuallyseparates the numerator and denominator. If
the line is slanting it is called a solidus or
forward slash, for example 3/4. If the line ishorizontal, it is called a vinculum or,
informally, a "fraction bar", thus: .
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yThe solidus may be omitted from
the slanting style (e.g. 34) where
space is short and the meaning isobvious from context, for example
in road signs in some countries.
yIn computer displays and
typography, simple fractions are
sometimes printed as a singlecharacter, e.g. (one half).
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Usage
yFractions are used most often when
the denominator is relatively small. It
is easier to multiply 32 by 316 thanto do the same calculation using the
fraction's decimal equivalent
(0.1875). It is also more accurate tomultiply 15 by , for example, than it
is to multiply 15 by a decimal
approximation of one third
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yTo change a common fraction to
a decimal, divide the numerator
by the denominator, and roundoff to the desired accuracy.
Conversely, a decimal fraction
may be converted to a common
fraction: if dealing with a finite
number of digits, it's very easy;for example, 0.1875 may be
expressed as (and later
simplified, if desired).
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Forms of fractionsy
A vulgar fraction (orcommon fraction or simplefraction) is a rational numberwritten as oneinteger(the numerator) divided by a non-zerointeger (the denominator) such as .
y A vulgar fraction is said to be a proper fraction ifthe absolute value of the numerator is less thanthe absolute value of the denominatorthat is, ifthe absolute value of the entire fraction is lessthan 1; a vulgar fraction is said to be an improper
fraction (US, British or Australian) or top-heavyfraction (British, occasionally North America) ifthe absolute value of the numerator is greaterthan or equal to the absolute value of thedenominator (e.g. ) .
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Mixed numbers
y A mixed numberis the sum of a whole
number and a proper fraction. This sum
is implied without the use of any visible
operator such as "+"; for example, in
referring to two entire cakes and three
quarters of another cake, the whole and
fractional parts of the number are writtennext to each other:
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An improper fraction can be thought of asanother way to write a mixed number. A
mixed number can be converted to animproper fraction in three steps:
y Multiply the whole part by the denominator of the
fractional part.
y Add the numerator of the fractional part to thatproduct.
y The resulting sum is the numerator of the new
(improper) fraction, with the 'new' denominator
remaining precisely the same as for the originalfractional part of the mixed number.
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Similarly, an improper fraction can beconverted to a mixed number:
yDivide the numerator by the
denominator.
yThe quotient (without remainder)becomes the whole part and the
remainder becomes the numerator
of the fractional part.yThe new denominator is the same as
that of the original improper fraction.
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Equivalent fractions
yMultiplying the numerator and denominator
of a fraction by the same (non-zero) number,
the results of the new fraction is said to be
equivalent to the original fraction. The word
equivalentmeans that the two fractions have
the same value. That is, they retain the same
integrity - the same balance or proportion.This is true because for any non-zero
numbern, .
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y Therefore, multiplying by is
equivalent to multiplying by one, and any
number multiplied by one has the samevalue as the original number. For
instance, consider the fraction :
when the numerator and denominator areboth multiplied by 2, the result is ,
which has the same value (0.5) as .
To picture this visually, imagine cutting
the example cake into four pieces; two ofthe pieces together ( ) make up half
the cake ( ).
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For example
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y Dividing the numerator and denominator ofa fraction by the same non-zero number will
also yield an equivalent fraction. This iscalled reducing or simplifying thefraction. A fraction in which the numerator
and denominator are coprime [this meansthey have no factors in common (other than1)] is said to be irreducible or in its lowestor simplest terms. For instance, is not
in lowest terms because both 3 and 9 can beexactly divided by 3. In contrast, is inlowest termsthe only number that is a
factor of both 3 and 8 is 1.
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yAny fraction can be fully reduced toits lowest terms by dividing boththe numerator and denominator bytheir greatest common divisor. For
example, the greatest commondivisor of 63 and 462 is 21,therefore, the fraction can be
fully reduced by dividing thenumerator and denominator by 21:
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Reciprocals and the "invisible
denominator"
yThe reciprocal of a
fraction is another
fraction with thenumerator and
denominator reversed.The reciprocal of , for
instance, is .
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yBecause any number divided by
one results in the same number,
it is possible to write any whole
number as a fraction by using 1
as the denominator: 17 = (1
is sometimes referred to as the"invisible denominator ").
Therefore, except for zero,
every fraction or whole numberhas a reciprocal. The reciprocal
of 17 would be .
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Complex fractions
yA complex fraction (or
compound fraction) is a fraction
in which the numerator ordenominator contains a
fraction. For example, and
are complexfractions.
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yTo simplify a
complex fraction,divide the numerator
by the denominator,as with any other
fraction (see thesection on division
for more details):
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For Examples:
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