|
Fraction Sense & Operations
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.
Unit FractionsTo 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 Nth of the object. We may describe that part in writing as
The symbol
may be read as an Nth ,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 Nth of an object is the object itself when N = 1. Multiples of Unit Fractions
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 Nths of the object.the object. Symbolically, that is in writing, we have
or
The two symbols
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.
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 DenominatorsSuppose 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 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 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.
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.
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
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. |
www.whyslopes.com Start of Number Theory Number Theory Related Site Pages: For online automated help in senior high school maths & calculus, visit quickmath.com For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com With overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck Food for thought: Try the Twiddla Whiteboard. In principle, it allows to people to draw and chat together online on a copy of this webpage or a clean sheet. The chat may be via text or audio. Visit www.twiddla.com to set up whiteboards to work with the webpage of your choice..
|
|
Favourite
Sites: BBC News and
mathematics portion of English
National Curriculum
|