Fraction (mathematics)

Usual direction of the fraction

Definition of the fraction

A fraction is a division not carried out between two relative whole numbers N and D . It is represented as follows:

n/d or or $\ frac {N} {D}$

The number the top is called the numerator ............ N
The number of bottom is called the denominator ......... D
The feature or bars fraction means that one divides the numerator by the dénominateur.

Example: 3/7 means that one divides 3 by 7; one pronounces this fraction “ three seventh ” and therefore 3 is the numerator because it indicates a number of three units (seventh) whereas 7 is the denominator because it names the unit (the seventh) with which one works. If one eats the 3/7 of a tart, numerator 3 indicates the number of shares which one eats whereas 7 indicates the full number of shares, therefore the unit considered…

One finds also sometimes the notation

N : D

N ÷ D

two points replacing the bar of fraction (this notation is to be avoided).

To draw a fraction

Fractions of which N < D

The fraction can be represented by a drawing. Very often a geometrical form which one divides into several parties.
1° the denominator D indicates the number of equal parts to draw in the form géométrique.
2° the numerator N indicates the number of equal parts utilisées.
Example:
Let us choose a rectangle like forms geometic and the fraction
The denominator is 4 thus the rectangle will be divided into 4 equal parts

The numerator is 3 thus alone 3 equal parts will be utilisées.

Other possibilté:

----

---- ==== Fractions thus N > D ==== This fraction will be equivalent to the quotient of N / D , (which will represent the number of unit) follow-up of a fraction consisted the remainder of division for numerator and D for dénominateur.

Exemple: for fraction 7/3, whole division gives 2, it remains 1.

the quotient is 2 dodsgdjsgSDHJGFBHSgdgGnc 2 units, the remainder 1 thus 2 ⅓.

It is impossible to represent this kind of fraction by a single diagram, we will consequently use several similar forms geometrical:

To take a fraction of a quantity

To take of 750, one divides 750 by 3, then one multiplies the result by 2:

750÷3 = 250; 250 × 2 = 500. Thus of 750 = 500

To take C amounts dividing C by B and multiplying the whole by A.

Equivalent fractions

If one multiplies, or divides, the numerator and the denominator of a fraction by the same number, one obtains an equivalent fraction .

Example:

In a general way, the fractions and are equivalent as soon as N × of = D ×.

Example:

\ frac {4} {6} = \ frac {6} {9}

because

6 \ times 6 = 4 \ times 9 \,

(one calls these two products the products in cross).

Certain fractions can be simplified, i.e. N and D can be divided by the same largest possible number but. This number is called the PGCD (highest common factor) of N and D . After reduction, the fraction is known as irreducible .

To carry out certain operations between fractions, all the denominators of the fractions must be equal. With this intention, it is necessary to replace each fraction by an equivalent fraction, while being arranged so that all the denominators are identical. This denominator will be more the possible small number which is divisible by each denominator. This number is called the PC (lowest common multiple) of the denominators. The operation is called to reduce to the same denominator
Example:

$\ frac {3} {4} = \ frac {3 \ times 3 \ times 3 \ times 5} {4 \ times 3 \ times 3 \ times 5} = \ frac {135} {180}$
$\ frac {1} {6} = \ frac {1 \ times 2 \ times 3 \ times 5} {6 \ times 2 \ times 3 \ times 5} = \ frac {30} {180}$
$\ frac {5} {9} = \ frac {5 \ times 2 \ times 2 \ times 5} {9 \ times 2 \ times 2 \ times 5} = \ frac {100} {180}$
$\ frac {14} {15} = \ frac {14 \ times 2 \ times 2 \ times 3} {15 \ times 2 \ times 2 \ times 3} = \ frac {168} {180}$

Comparison of fractions

For the same numerator, plus the denominator is small plus the fraction is large.

Example:

\ frac {2} {3} > \ frac {2} {5}

numerator 2 is the same one for any fraction.

the comparison of the denominators gives 3 > 5

For the same denominator, plus the numerator is large, plus the fraction is large:

Example:

\ frac {2} {7} < \ frac {5} {7}

denominator 7 is the same one for each fraction.

the comparison of the numerators gives 2 < 5

If the numerators and the denominators are different, one can always reduce the fractions to the same denominator and then compare the numerators: Comparison of 1/4 and 2/5

1/4 =5/20 and 2/5 = 8/20. However 5 < 8 thus 5/20 < 8/20 thus 1/4 < 2/5

rem: One can also use the decimal writing:

1/4 = 0,25 and 2/5 = 0,4.

0,25 < 0,4 thus <

Decimal writing, fractional writing

Any fraction has a decimal Développement finished or unlimited periodical which is obtained by posing the division of N by D.

1/4 = 0,25

2/3 = 0, 6 66… (period 6)

17/7 = 2, 428571 428571… (period 428571)

Conversely, any decimal number or having a periodic decimal development can be written in the form of fraction.

Case of the decimal number

It is enough to take as numerator the private decimal number of its comma and as denominator 10ⁿ where N is the number of figures after the comma:

0,256 = \ frac {256} {1000} = \ frac {32} {125}

15,16 = \ frac {1516} {100} = \ frac {379} {25}

Case of the unlimited decimal development

One starts by getting rid of the whole part: 3, 45 45… = 3 + 0, 45 45…

case of the simple periodic decimal development

A simple periodic number is a decimal number in which the period starts immediately after the virgule.
0,666 or 0,4545 or 0,108108
Like numerator, it is enough to use the period while the denominator will be composed of as much 9 qu ' there are figures composing the période.
Example: 0,4545
Period 45 thus numerator = 45
Period made up of two digits thus denominator = 99
Fraction = 45/99 or 5/11

consequently: 3, 45 45… = 3 + 5/11 = 38/11

Case of the mixed periodic decimal development

A mixed periodic decimal number is a decimal number in which the period does not start immediately after the virgule.
0,8333 or 0,14666
To find the numerator of the fraction, it is necessary to withdraw the mixed value of the mixed value followed by the first period. Example: 0,36981981…
mixed value: 36
Mixed value followed by the first period: 36981
Numerator = 36981 - 36 = 36945
As for the denominator, it will be composed of as much 9 qu ' there are figures composing the period, follow-ups of as many zeros as there are figures after the comma composing the mixed value.

Example 1: in value 0,36981981, period 981 consists of 3 digits thus the denominator will consist of a series of three 9 follow-ups of two zeros since the mixed value 36 is made up of two digits. Finally we will have:
0,36981981 = 36945/99900 or 821/2220

Example 2: $1,24545… = \ frac {1245-12} {990} =137/110$

Operations on the fractions

Addition and subtraction

For a common denominator

It is enough to add or withdraw the numerator of each fraction and to preserve the common denominator.

Example of a sum:

Example of a difference:

For a different denominator

Before carrying out the operation, each fraction must be transformed into an equivalent fraction whose denominator their is commun.
Example:

A = \ frac {1} {6} + \ frac {4} {9}

A = \ frac {3} {18} + \ frac {8} {18}

A = \ frac {11} {18}

Multiplication

The multiplication of two fractions is simple to carry out but it is not simple to include/understand why it functions thus.

$\ frac {2} {15} \ times \ frac {7} {11} = \ frac {2 \ times 7} {15 \ times 11} = \ frac {14} {165}$

Here is an explanation, based on an intuitive comprehension of the fractions.

One can include/understand seven eleventh as seven times one eleventh (see the charts above) are $\ frac {7} {11}$ like ${7} \ times \ frac {1} {11}$ . Thus to multiply $\ frac {2} {15}$ by $\ frac {7} {11}$ amounts carrying out $\ frac {2} {15} \ times 7 \ times \ frac {1} {11} = \ frac {2 \ times 7} {15} \ times \ frac {1} {11}$ .
But to multiply by one eleventh come down to be divided by 11, i.e. to multiply the denominator by 11 (the shares are 11 times smaller), is: $\ frac {2 \ times 7} {15 \ times 11}$ .

4 ---- 7

Minor problems

Historical problems

I found a stone but I did not weigh it. After him to have added a seventh of his weight and to have added one eleventh of the result, I weighed the whole and I found: 1 my-Na of mass. Which was in the beginning the weight of the stone  ? (Babylonian problem, shelf YBC 4652, problem 7)
an increased number of its seventh gives 19. Which is this number? (Papyrus Rhind, problem 24)
a number increased by its quarter gives 15. Which is this   number? (Papyrus Rhind, problem 26)
Supposons that one has 9 stems of yellow gold and 11 white money stems which, with the weighing, have equal very right weights. If one exchanges between them one their stems, gold becomes lighter of 13 liang of mass. One request respectively combienpèsent a gold stem and a money stem. (nine chapters on mathematical art, problem 7.17)
a lance with half and the third in water and nine palms outside. I ask you how much it is length. (medieval problem)

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