Fraction Sense & Operations

See Fractions,  Ratios, Rates, Proportions  & Units for a more concrete approach to the question of what is a fraction and how to do and justify arithmetic with fractions.  The concrete approach uses line segments and measures to illustrate and develop fraction senses and operations.  The following easy bases the fraction definition and operations on the division properties of objects - properties that classify and characterize those objects of which fractions can be taken.

The division of objects into parts with the same shape, isometric or congruent geometrically, or into parts of equal value leads to the notion of unit fractions and multiples of unit fractions. When an object is divisible it, we take a half, a third, a quarter, a fifth and so on. 

The object may be a pie, a line segment, a region or a solid, or an amount of money.  

Unit Fractions

To be more precise, when N is a whole number, and when an object is divisible into N  parts, isometric or of the same value, each part provides an  N^th of the object.  We may describe that part in writing as 

The symbol 

may be read as an N^th ,and to coin a phrase, it may be called an adjective of division. An adjective of division is also an adjective of quantity if not enumeration.

Special Case:

For convenience,  an N^th of an object is the object itself when N = 1.

Multiples of Unit Fractions

Put N = 5 and M =3 on first reading.

Suppose N is a whole number, and suppose an object is divisible into N identical or equi-valued parts. Then for each whole number M less than or equal to N, we make take M of  N^ths  of the object.the object.  Symbolically, that is in writing, we have

or 

gives two more adjective of division, adjectives with the same meaning.  We write 

to indicate these adjectives of division have the same meaning. They provide simple fractions. In them the first number M is called the numerator and the second N is called the denominator. Both together may be called terms of the fraction. Mathematicians think of fractions as ordered pairs.

Proper Fractions and one Improper Fraction

When M < N above, we say 

is a proper fraction. The case M = N gives the first improper fraction

Taking N of the N-ths of an object is equivalent to taking all of the object physically and geometrically, if not biological.  

In the biological realm, an uncut apple will last longer than two halves. Two halves in this case are not a whole.   So there are exceptions to the notion that all N of the N-ths of an object equivalent to the whole. Those exceptions may be avoided may making the cuts virtual instead of real. 

More Improper Fractions

When several identical objects can be each be divided into N-ths, isometric or of equal-value, each object contribute N of the N-ths. So there more than N of the N-ths. That be said, if M is a whole number, and there sufficiently many objects divided into N-ths, we can take M of those N-ths in sequence or at random. Thus symbolically, that is in writing, we have

or 

When  M is less than N, the fraction 

is said to be proper. 

When  M is greater than or equal to N, the fraction 

is said to be improper. Improper fractions with M > N are not feasible when there is only one object.

Addition of Fractions - Like Denominators

Suppose one to  several objects are divided into N-ths. Then a group R of these N-ths added to another non-overlapping group T of these N-ths together provide  (R+T) of these N-ths. Symbolically, the foregoing may be written as 

That provides an addition rule for fractions, proper or not, with like denominators.

In the case where one to  several objects are divided into N-ths and M-ths,a group R of these N-ths added to another non-overlapping group T of the M-ths may be physically combined together to form a sum

The question of how to represent this sum as fraction follows from the concepts of equivalence for fractions and/or  mixed numbers.

Unit Fraction of Unit Fractions
Repeated Division 

If an object is divided in  R isometric (respectively equi-value) parts, we may be able to divide each of those parts into say T isometric (respectively equi-value) parts.  The product TR = RT gives the number of resulting isometric (respectively equi-value) parts. That foregoing suggests that a T-th of an R-th is an TR-th and an RT-th part.of the object.  Thus the successive division operations

yield the same result as the single division

Moreover as TR = RT due the commutative property for products of whole numbers, the successive division operations 

also has the same result. Symbolically with the times symbol * in place of the word of, we have

and 

The foregoing gives a definition and computational method for the compound adjective of division provided by a unit fraction of a unit fraction. 

Simple Fractions of  Unit Fractions

We now consider the question what is 

By definition the latter is 

Since 

the foregoing implies

If an object is divided in  R isometric (respectively equi-value) parts, we may be able to subdivide each of those parts into say T isometric (respectively equi-value) parts and then take M of the resulting parts  The product TR = RT gives the number of resulting isometric (respectively equi-value) parts. That foregoing suggests that M times T-th of an R-th is M times a TR-th or RT-th part.of the object.  

Equivalent Fractions - Fractions of equal value

A divisible  object can sometimes be divided into N isometric or equi-value parts in many different ways.  From this point on, we will consider  and asssume the case of equ-value division first and foremost.  The case of division into isometric parts will be met mainly in the form of illustration and not as a support to the unfolding theory or chains of reason.  Here the existence of a value or measure of a quantity is tacitly assumed. 

Recall a unit fraction of a unit fraction is another unit fraction. That is

when R and T are whole numbers. An R-th part of an object is also an object. Since T ocurrences of a T-th of an object equals the object, we have

That is

Now 

Therefore  

That is to say M of R-ths of an object is taken to have equal value to  MT of RT-ths of an object in cases where T times an RT-th has the same value has an R-th. That situation is assumed. 

Identification of unlike fractions with physically different meanings or operations. 

The foregiong implies two fractions representing physical different operations, here M of R-ths of an object and MT of RT-ths of the same object, are considered to be equivalent and interchangeable in our value-based calculations or consideration of adjectives of division (units fractions) and their multiplies (simple fractions). 

Assumption. The three simple fractions 

of an object give the  same value, and so can replace each other. The equal sign here is use in the sense that all three fractions (applied to an object) should give the same value.  Should more be said about equivalent (or equi-value) fractions?

Raising Terms, Lowering Terms

Here replacing the first fraction M/R by one of the other is called raising terms while replacement of one of the others by M/R is called lowering terms or simplification.  Expressing a fraction A/B in lowest terms or simpliest form means finding another fraction M/N of equal value in which no further lowering of terms is possible.  See or find a discussion of relatively prime whole numbers. 

Addition of Fraction - Unlike Denominators, First Pass  

In the case where one to  several objects are divided into N-ths and M-ths,a group R of these N-ths added to another non-overlapping group T of the M-ths may be physically combined together to form a sum

The question of how to represent this sum as fraction follows from the concepts of equivalence for fractions and/or  mixed numbers. Here we observe in terms of value, measure or or equivalent fractions

Hence 

and so

The foregoing uses the common denominator NM in order to express each fraction in the sum as an equivalent (equi-value) fraction with NM as a the common or like denominator.

Addition of Fraction - Unlike Denominators, Second Pass

Suppose N = DE and M = FE for some common factor E, a whole number.  Then 

from the distributive, commutative and associative properties of whole numbers.

Now lowering terms gives

or 

The foregoing also follows without lowering terms from the use of equivalent fractions

and the rule for addition of fractions with like denominators. Lowering terms will be possible if E is not the greatest common factor or if the addition  RF+DT results in a common factor with the product DEF. As a rule of thumb,  the most efficient way to add is to use the greatest common factor E.  A logically equivalent way of saying this, exercise show why,  is use the least common denominator. 

The choice of greatest common factor E leads to the most efficient computation in that if E is not the greatest common factor of N and M then lowering terms is possible after the fractions are combined into a single term using the rule for addition of fractions with like denominators.

Products of Simple Fractions

We now consider the question what is 

By definition the latter is 

So we need find the unit fraction of a simple fraction, that is,

and then multiply by M.

Special Case: The simplest case to visual occurs when N is a multiple of T. Say N = KT for some whole number K.  Then a T-th of KT objects (here R-ths of another object) is given by K of the R-ths.  Then

and hence multiplication by M gives

In general, we cannot assume N is a multiple of T.

General Case: Recall

So 

should be  N times one TR-th of an object. Here the division of N times one R-th of the object into T equi-valued parts is obtained by dividing each of the R-ths into T equi-valued parts, that is RT-ths, and then N of the latter. Hnece

should be N times one TR-th of the object. So in symbols

Now multiplication by M gives the product formula

for a product of simple fractions. Can a more cogent argument be given?

An example

Here 2 thirds of three fifths of a rectangle is computed graphically. 

Three fifths of rectangle are shown shaded in blue

We cut each shaded fifth into thirds

and take two thirds of the three-fifths - see the gray area.

The answer is 6 fifteenths as 15 = 3*5. 

Algebraic Properties. 

The properties of arithmetic with whole numbers, that is the associative, commutative and distributive law for whole numbers, combined with the above formulas for addition and multiplication, and raising or lowering terms implie the associative, commutative and distributive laws for addition and multiplication of fractions. Details are left to the reader.

Division and Multiplication by Reciprocal of Divisor.

The product formula 

says 

The question many multiples M/T of a fraction N/R gives another fraction K/S has the answer

Here we observe due to the associative laws for multiplication of fractions and whole numbers that

as required.   So 

works. Now if  T and Q are two fractions  with

and 

Then 

and 

So Q = T and 

is the only answer.

Conclusion

The foregoing outlines a logical development of the properties of fractions in conjunction with some assumptions about the division of objects, assumptions which characterize the objects to which fraction may be applied.  Some refinements may be possible. 

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