Proper Fractions and Subdivision of an Object

In primary school, we learn about dividing a group or object into a number of parts of equal size or value.

• Division into three parts gives thirds, three of them and no more.
• Division into four parts gives quarters, four of them and no more.
• Division into five parts gives fifths, five of them and no more.
• Division into six parts gives sixths, six of them and no more.

• Division into N parts gives N-ths, N of them and no more.

In the case of quarters, we may take zero quarters, one quarter, two quarters, three quarter or four quarters. Here one, two and three quarters are known as proper fractions. When there is only one object to divide, we can not have five quarters of it.

When four people are equal, dividing a pie equally among them gives one quarter for each. Likewise dividing a pie between two equally will give each a half.

When there is only one person present, he or she gets the whole pie. He or she may decide to eat one or more quarters of a pie, and leave the rest for later.

Arithmetic can be done with equal parts of an object: \begin{eqnarray*} \frac 2{10} + \frac5{10} &=& \frac7{10} \\ \frac 8{10} - \frac6{10} &=& \frac2{10} \\ 2 \times \frac3{10} &=& \frac6{10} \end{eqnarray*}

Improper Fractions and Whole or fractional Units of Measure

In the division of a single object, we cannot have five quarters of it. But in the case of many objects of the same type, we divide each object into four equal parts. That will give four quarters for each object. In particular, two objects will be the same as 2 times four quarters, that is 8 quarters.

For example, we may measure length in meters. Counting each meter in a two meter length as four quarter leads to eight quarter meters in the two meter length. In symbols. \[2 = 2 \times \frac44 = \frac84 \] Here we take 5 of quart to get a single length of length five quarters of a meter. In symbols. \[\frac54 = 5 \times \frac14 = 1+\frac14 \] In the first instance, measures of length, time, money and further amounts may be done by counting whole units. But each whole unit consists of 2 half-units, 3 one-third units, 4 one-quarter units, 5 one-fifth units, 10 one-tenth units, 100 one-hundreth units. So measures may be also be done by counting half-units, one-third units, one-quarter units, one-fifth units, one-tenth units, one-hundreth units and so on. For example, a measure might be 25 one tenth units. In symbols that would be \[ 25 \times \frac1{10} \mbox{ units} = \frac{25}{10} \mbox{ units} \] Now 25 = 2 × 10 + 5. So 25 one-tenths can be grouped into 2 groups of 10 and one group of five. With the units understood and not written, \[ \frac{25}{10} \mbox{ units} = 2 \times \frac{10}{10}\mbox{ units} + \frac 5{10}\mbox{ units}\] Here the two groups of 10 tenths equal a single unit. So the count of one-tenths units \[ \frac{25}{10} \mbox{ units} = 2 \mbox{ units} + \frac 5{10}\mbox{ units} = \left(2 \frac 5{10}\right)\mbox{ units} = 2.5 \mbox{ units} \] That gives the measurement as count of whole units plus a count of tenth units. That provide what we will call a mixed measure of length, time, money or another quantity, depending on what is measured. If we are just counting whole or ones, the mixed measure becomes a mixed number.

Raising and Lowering Terms

Numerals, Number and Measuring Systems

In writing whole numbers, we write them in as Roman Numeral I, II, III, IV, V, VI, VII and so on, or we may write them as decimal numerals 1, 2, 3, 4, 5, 6, 7 and so on. Counting and comparing how many with decimal and Roman are both simple matters, with decimal ways of counting being the most familair for most. But for arithmetic with whole numbers, written decimal place value methods for addition, subtraction, mutliiplication and long division are easier than any corresponding method for Roman Numerals. And in days gone by, abacuses were not common outside of big merchants or trading houses - I presume.

Roman numeral, decimal numerals and tally marks all provide different ways to write and record counts. The decimal count 7 = 7 ones.
463 = 4 hundreds + 6 tens + 3 ones
represents a count as number of hundreds plus a number of tens plus a number of ones. Here hundreds and tens represent possible units of counting. So decimal like 463 represent mixed counts.

As said above, measures can be written in different forms and by grouping may be expressed as mixed measures with the aid of mixed numbers - proper fractions added to a whole number. Measures are effectively counts of units and subunits. Grouping and regrouping units and subunits leads to different forms for a measure. So the numerals \[ \frac{25}{10} \mbox{ units} = \left(2 \frac 5{10}\right)\mbox{ units} = 2.5 \mbox{ units} \] are equivalent.

In practice, all are taken to be equivalent - interchange. In other words, in the practice of counting, measuring and figuring - figuring being arithmetic, we assume or gamble that the value of measures do not depend on how they are expressed nor on the method by which they are calculated from other counts and measures, nor on the form or representation of the latter counts and measures. The foregoing practice is normally learnt and done in silence. Mention here provides an option for course design and delivery. In learning and teaching mathematics, which nuances have to be discussed in the development of skills and concepts with take home value, or with value for college programs.

Comparing Measures and Fractions

Given two mixed numbers or fractions, the question arises which one is greater. For example consider the two fractions \[ \frac{78}4 = 78 \times \frac14 \mbox{ and } \frac{84}5 = 84\times \frac15 \] One counts quarters and the other counts fifths. Direct comparison is not possible. But 20 is a common multiple of both 4 and 5. Moreover, \[ \frac14 = 5 \times \frac1{20} \mbox{ and } \frac15 = 4 \times \frac1{20} \] Therefore, \[ \frac{78}4 = \frac{78 \times 5}{20} \mbox{ and } \frac{84}5 = = \frac{84 \times 4}{20} \] We may compare the two fractions by expressing quarters and fifths as multiples of 20. Here 78 × 5 = 390 while 84 × 4 = 336. So the first fraction is more than the second.

The foregoing comparison may be done by rote with the aid of cross-multiplication, but the raising terms method above provides a thought-based perspective which at least the college bound should see. ,p>

More Arithmetic

Other site sections describe the addition, subtraction, multiplication and division of fractions.

Half-Line Coordinates

Unsigned fractions, whole numbers and mixed numbers may be used as coordinate on a half-line or ray.

Sets of Fractions

$ G = \left\{ \frac 12, \frac 22, \frac13, \frac23, \frac33,\frac14,\frac24, \frac34, \frac45, \frac55, \frac16,\frac26,\frac36, \ldots \right\} $

Symbolically, the foregoing set of fractions \[ G = \left\{ \frac pq \big| p \in N \mbox{ and } q \in W \right\} \] Read the latter as the collection of all fractions $\frac pq$ with a natural number p in the numerator and whole number q as a denominator - whole numbers are nonzero. This set also consists of all natural number multiples of the one numerator fractions \[ G_{one} = \left\{ \frac 12, \frac13, \frac14, \frac15, \frac17, \ldots \right\} \] minus those fractions that are reducible

Later on, we change notation and put $Q_+=G.$

The multiple is a proper fraction when the numerator is less than the denominator.

Alternative Approaches. Fractions and sets of Fractions can be introduced in different ways. The Cantor Diagonal Enumaration method for fractions suggested the representations

$ G = \left\{ \frac 12, \frac 22, \frac13, \frac23, \frac33,\frac14,\frac24, \frac34, \frac45, \frac55, \frac16,\frac26,\frac36, \ldots \right\} $

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science