New Fraction

8/6/2019 New Fraction

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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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