[S1_Expository] Recurring Decimals and Rational Numbers

Problem

 

Introduction

     A student asked the above question on Facebook.  This article explains recurring decimals, which is part of the topic on real numbers in the Singapore Secondary 1 Mathematics syllabus. 

     By the way, I do not believe that Asians are inherently better at mathematics.   A few students are working over Christmas to prepare for the next years’ work.  Here we go!

Notation

     Personally, I prefer the horizontal bar notation which is what I learned during my time as a student, but nowadays in Singapore schools, we tend to use the dot notation.  There is no right or wrong about this, but it is just a matter of convention.  When in Rome, do as Romans do.

Coversion to a fraction

 

Concluding Remarks

     A rational number is a number that can be expressed as a ratio or  fraction  ^p/_q  where  p  and  q  are integers with  q ¹ 0.  The fraction can be proper or improper.  Since any recurring decimal can be converted to a fraction,

every (infinitely) recurring decimal is a rational number.

Now,

every finitely terminating decimal is also a rational number.

For example, 0.171 = ¹⁷¹/₁₀₀₀.  The numbers that cannot be converted to fractions are called irrational numbers.  How do these numbers look like?

The irrational numbers are exactly the numbers with non-terminating (infinite) and non-recurring decimal expansions.

Some examples of irrational numbers are
       p = 3.141 592 653 589 793 238 462 643 383 279 502 884 197 ¼
       e = 2.718 281 828 459 045 235 360 287 471 352 662 497 757 ¼

     Ö2 = 1.414 213 562 373 095 048 801 688 724 209 698 078 570 ¼

The decimal digits of irrational numbers never end, but they do not have any repeating pattern.  There are actually much “more” irrational numbers than rational numbers, but this is a fact that is technically profound, way beyond the secondary syllabus.  The interested reader can refer to this article.

Suitable Levels

* Lower Secondary Mathematics (Sec 1 ~ Grade 7)

* revision for GCE ‘O’ Level “Elementary” Mathematics (revision)

* other syllabuses that involve recurring decimals

* any independent learner who is interested

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Question

Introduction

     This is a question on percentages.  Percentages are in themselves also units.  One percent (1%) simply means ¹/₁₀₀.  And we can use units (shown circled in the diagrams below) in which each unit is  ¹/₁₀₀  or  1%  of some whole.

     It is useful to think of increases and decreases as multiplying by some percentage.  For example, a decrease by 20%  means multiplying by  100% – 20%  i.e.  80%.  After all, if you work out  100%  of something and subtract  20%  of the same thing,  you will end up with  80%  of that thing.  It is much easier to think of it that way.  Likewise, an increase of  45%  means multiplication by  100% + 45% = 145%.

Solution

     From the information given in the question, we can set up a diagram like this.  I use circles to envelop the percentage units.

We can work out the units in the “after” situation (one year later):-

40 ×  80%  = 40 ×   ⁴/₅  = 32

60 × 145% = 60 × ²⁹/₂₀ = 87

The new total is  119%  or  119 circle units.  The net increase is  19%  or  19 circle units, which we know is equivalent to  228.  Once we got this part, we can work out  1  circle unit  and then  20 circle units, which is the difference between the number of male and female members.  [Remember the check that you are answering the question that was asked.]

Ans: 240

Summary

     We have used a diagram in the form of a ratio-units model [H02].  The ratio unit used in this example happens to be the same as a percentage.  Be careful that other questions may involve different kinds of units with different bases for their percentages.  In other words, in other questions, the “100%” may stand for different things.  In the diagram, we have used the before-after concept [H06].  Increases or decreases in percentages may be re-stated as multiplications by the appropriate percentages, which, in turn, may be thought of as multiplications by fractions [H09].  It is a good idea to be able to inter-convert between fractions and percentages.  By comparison, we found the link between  19 units (or 19%)  and  228 [H11].  Having solved this part of the problem, we are able to answer the original question as asked.

HeuristicsUsed

H02. Use a diagram / model

H06. Use before-after concept

H09. Restate the problem in another way

H11. Solve part of the problem

Suitable Levels

* Primary School Mathematics

* other syllabuses that involve percentages and ratios