Fraction (mathematics)
In a narrower sense fraction calculation refers to the calculation with common fractures (sometimes ordinary fractures ) in the " counter - fraction line - denominator notation " ( see below). Fractions thus belongs to the arithmetic, a branch of mathematics.
In a broader sense the word is used also for the computation with rational numbers, regardless of the format in which they are present.
A more important extension consists in the admission of breaking terms, which are expressions that are formally constituted as vulgar fractions, but in which the numerator and denominator terms can be that contain variables. For this fraction Terme fracture calculation rules shall apply mutatis mutandis. However, the calculation with breaking terms is part of the algebra.
The rules of fractions refer to the basic arithmetic operations, ie addition, subtraction, multiplication, division, and to the reciprocal of education. In particular, in case of breakage terms and rules for powers and roots are added.
There is also a truncation and expansion rule, which is a peculiarity of the fractions. It is based on the difference between fractions and fraction, which is shown in more detail in the following section.
The fraction notation, so the spelling with fraction line is, in general, always used in various areas of mathematics, particularly in algebra when considered in the investigated structure, the elemental fraction calculation rules, in particular the cuts and expansion rule. Here one speaks always of " fractions ", if these rules are applied.
- 3.1 Practical calculating with fractions 3.1.1 deformation of fractures 3.1.1.1 Expand and reduce
- Setting 3.1.1.2 Mixed Numbers and eliminate whole
- 3.1.1.3 make breakthroughs same name
- 3.1.2.1 Adding and Subtracting
- 3.1.2.2 Multiply
- 3.1.2.3 Divide
- 3.1.2.4 Calculating with mixed fractions
- 3.2.1 Extending and Shortening
- 3.2.2 addition
- 3.2.3 subtraction
- 3.2.4 multiplication
- 3.2.5 Division
- 3.2.6 powers
- 3.3.1 Definition area
- 3.3.2 shortening
- 3.3.3 Addition and subtraction
- 3.3.4 Multiplication and Division
- 4.1 partial fractions
- 4.2 Egyptian fractions
- 4.3 Pythagorean breaks
- 4.4 Rational numerator or denominator
Fraction and fractional
The fraction calculation based on the fact that the whole thing ( the one from the operations on natural numbers) can still divide. A cake can be divided for example into four parts. When these parts are equal, each part is one-fourth of the cake. If, as in the picture, one of the neighborhoods already missing, so three quarters cake is presented.
This is usually written in the " counter - fraction line - denominator notation ": The number below the fraction bar, the denominator indicates in how many parts the whole has been divided; the number above the fraction bar, the counter indicates how many parts of it are meant in this case. So you get a break.
If the whole ( the cake ) instead divided into eight parts and it took six, so that's another break: instead. But these two fractions are apparently for the same amount of cake: they stand for the same fraction.
For each fraction, there are many ( infinitely many ) different views, various fractures, embodying all of the same value ( the same size ), but in different ways. From a break to the other you get by extending and shortening. Thus, the value of a fraction does not change, but one gets for this number different looks: different fractions.
Definition and terms
Fractures can initially be in vulgar fractions (also called simple fractions ) and decimal fractions ( = decimal, colloquially, " point number " ) divided, and there is yet the presentation as a mixed fraction. When one speaks of a fraction, we usually mean a common fraction, the computation with decimal fractions is not usually referred to as fractions.
The following table lists common names for fractions are summarized, which are explained in this section. The standing further down the table each governed by the terms above it generic terms, for example, each certificate fracture is a common fracture, juxtaposed concepts are not mutually exclusive. It should be noted that it is writing and not number designations for the depicted figures. A certain number may have different representations, which are referred to with different terms, from the table. So you can write, for example, each improper fraction as a mixed fraction.
Other forms in which fractional values can be represented ( continued fraction, percent and per mille notation, binary fractions, etc. ) are treated in each separate articles and not listed in this table.
Common fractures
Common fractures are generally represented by a superposition position of the numerator and denominator separated by a horizontal line, :
The numerator and denominator of a fraction are integers. Here, the denominator must not be zero, since division by zero is not defined.
Any fraction can in fact be understood as a division problem. This is the numerator of the dividend, the denominator of the divisor:
The key to the break statement is that here each division (except by zero) is possible and has an easy- displayable result, while even in the field of integers are the divisibility rules.
Usually, natural numbers are used for the numerator and denominator used and any existing negative sign is placed before the break, so for example, instead of or. Are numerator and denominator is negative, the designated according to the rules of the division of whole numbers to the positive break:
In a variant of this notation that is often used when common fractures occur in texts, counters, fraction bar and denominator are consecutively written and uses a slash as a fraction line, for example, 1/2, 3 / 8th In the notation with slash instead of the horizontal fraction bar are (mainly ) single digit numerator and denominator sometimes reduced above and below the slash wrote: 6/7. For this purpose exist in many print character sets special characters, such as ¾ or ½.
True and improper fractions
If at a fraction of the amount of the counter is less than the denominator, then we speak of a real or actual breach (eg or ), otherwise a false or improper fraction (eg or ).
Proper fractions are therefore those whose magnitude is less than a whole.
Unit fractions and branch breaks
If the meter is in a common fraction equal to 1 (eg or ), is called a unit fraction, otherwise derived from a fracture or branch breakage.
Bill breaks
Improper fractions, in which the counter is an integral multiple of the denominator is ( for example) is referred to as slip fractures as they can be converted by cutting into integers (in the example in Figure 4). In particular, any integer can be written as a fraction bill.
Mixed fractions
Improper fractions that are not apparent fractures, can always be as mixed fractions (also: as mixed numbers, mixed case ) represent.
Initially, the integer part, ie the rounded towards zero to the number written and then right after that the remaining portion as a proper fraction. For example, instead of or in place.
A problem of mixed notation is that it can be misunderstood as a product:
So stands for and not for what would, yes.
If we write, however, it is not a break in mixed case, but (because of variable ) to a Term Here the omitted computing character must be a Malpunkt ( other arithmetic operators can not be omitted in terms ). must therefore be understood as and never as.
Calculation rules
Practical calculating with fractions
When calculating with fractions in the four basic arithmetic operations addition, subtraction, multiplication and division two fractures are linked, so that a third number is created. This should not be confused with the forming of openings, wherein a single fracture is given a new shape without its value changes.
The forming ( the strain ) is often the prerequisite that can be expected with fractions. Therefore, it is treated here first.
Change in shape of fractures
Expand and reduce
The value of the fraction represented by a fraction is not changed when the numerator and denominator of the fraction multiplied by the same number ( the fraction extended) or divided by a common divisor of the numerator and the denominator ( the fracture shortens ).
For example. The fracture was read from left to right, extended, shortened from right to left.
Setting up Mixed Numbers and eliminate whole
The value of a fraction shown in mixed case, and does not change if we as apparent fracture writes the integer part of the denominator of the fraction and the remaining fractional parts add one. Conversely, it is the fractional yield the whole split, and add the remaining as a fraction with a improper fraction.
For example. Whole were cleaved Read from left to right, right to left, the mixed number has been established.
Make breakthroughs same name
Common fractures that match in their denominator, called the same name. If fractures extended so that they then have the same denominator, it is called the same name make. In practical calculations the main denominators of the fractions should to be determined, which is the least common multiple (LCM ) of the denominators.
Example: The fractures are to be made the same name. The LCM of the denominator, ie all three fractions are extended so that their denominator is 42:
The representations of the same can now be used for example to arrange the fractions shown by size, by comparing their counters:
The basic arithmetic operations
Adding and subtracting
The fractures that are to be added or subtracted are first made the same name, then their counters are added or subtracted. The general rule for these bills in the same fractions as follows:
For example.
Multiplication
Fractions are multiplied by multiplying their numerators and denominators together. The product of the counter is the counter of the result, the product of the denominator then the denominator of the result.
For example.
Divide
Is divided by a fraction, by multiplying its inverse.
For example.
It may, as shown in the example, intermediate results are truncated (in this example, the 3 and the 2 in the penultimate step).
Calculating with mixed fractions
When multiplying or dividing mixed fractions, it is usually necessary, this is first converted into simple fractions. (Except for very simple tasks, such as. )
When adding and subtracting the other hand, it is much more beneficial to look at the whole for themselves and apply fractions only at the remaining real breakthroughs. When adding an additional whole here can occur when subtracting fractions may not be sufficient, so one of the whole must be divided into a sham break:
;
.
Abstract calculation rules
The following rules apply to both at break computing in the strict sense as well as in calculating with fraction terms. When calculating with fractions: the variables are the rules for certain integers. If you rely instead for these variables other expressions, such as self back proper fractions, decimal fractions or terms, then rules for calculating with fraction terms, the fraction arithmetic is obtained in a wider sense.
When calculating with fractions abstract calculation rules always produce correct results, more often the bill with the "practical calculation rules " less burdensome.
Extending and Shortening
Helpful Mnemonics for this are:
- Factors cut, which is good; who shortens sums, which is a sheep.
- Differences and sums shorten only the stupid ones.
- What you are doing above, you're also below!
From the equivalence for arbitrary natural numbers follows that every rational number can be represented by infinitely many different breaks, because it is.
Addition
Subtraction
Multiplication
Division
So you divide by a fraction by the reciprocal of the fraction, which acts as the divisor multiplied. The division is thus fed back to the multiplication.
Potencies
Calculating with breaking terms
Breaking terms, ie arithmetic expressions in the form of common fractures play an important role in elementary algebra. In general, fracture Terme in addition numbers also variable. The rules for computing with fractions can also be applied to break Terme.
Domain
In the determination of the domain of a fraction term is to be noted that the denominator is not allowed to have the value 0. For example, the fraction of dependent term when fitting would not be defined. The domain is so if the ground set the set of real numbers is assumed. In more complicated cases in the denominator factors should be disassembled so that the domain will be visible.
For example, has the domain.
Shorten
Shorten means that you divide the numerator and denominator by the same calculation expression. It is important that only factors of products can be cut out. Sums and differences in the numerator and in the denominator may first need to be broken down into products ( factorization ).
Examples:
When shortening a fraction Terms, the domain can change! So the unabridged, left- term is only defined in the first instance, if valid, the right-wing already, if only applies. In the second example the unabridged term is only defined if and only if, the abridged is defined without restrictions.
The change in the definition domain of a fraction Terms when shortening is one of the techniques that function terms can be continued steadily.
Addition and subtraction
As with figures, it is necessary to do the breaking terms given the same name, that is, to bring to the same denominator. Determine the simplest possible common denominator ( denominator ), which is divisible by all the given denominator.
Example:
The main denominators here. The expansion factors of the three terms given fraction obtained by allowing each divides the denominator found by the previous denominator. The expansion factors are therefore, and.
Can often be the main denominator only seen when in factors divides the denominator ( factorization ). Here are accessed often resort to the method of Ausklammerns or used binomial formulas.
Example:
Multiplication and division
Multiplying by breaking terms both the numerator and denominator must be multiplied. Common factors of the numerator and denominator should be canceled out.
Example:
In more complicated tasks, you should disassemble the numerator and denominator factors in order to cut them out before the actual multiplication can.
Example:
The Division of breaking terms can be attributed to the multiplication. You divide by a fraction term by multiplying by its reciprocal.
Example:
Other forms of presentation
Partial fractions
Fractures can often be decomposed into so-called partial fractions whose denominators are all powers of primes; eg
Egyptian fractions
There are also decompositions as so-called Egyptian fractions ( unit fractions ), eg
The ancient Egyptians knew only that kind of money and have expected this.
Pythagorean breaks
The number triples is an example of a Pythagorean opening ( see also Pythagorean triplet ), for
Rational numerator or denominator
See rationalization ( fractions ).
Generalizations
The construction of the field of rational numbers as fractions of the ring of integers is generalized in abstract algebra by the concept of the quotient field of any Integrity rings.