[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

[Pri20150408YYS] Yin-Yang Semicircles?

Question

Introduction

     The diagram looks a bit like the Yin and Yang symbol, doesn’t it?  This problem can be solved easily using the correct insights.  I present two solutions: the first one is by direct calculation (in terms of p), and the second solution uses the powerful concept of ratio of similar figures.

     Whichever method is used, first we must make observations.  Can you see that there are three types of semicircles (small, medium and large)?  [H02, H04]

Let  S = area of small semicircle,  M = area of medium-sized semicircle,  and  L = area of large semicircle.

Note that  area of A  = area of C  = L – M + S,  and  area of B = 2´(M – S).  [H10, H11]

Solution 1 (by direct calculation)

     L = ½p(3)² = ^9p/₂.   M = ½p(2)² = ^4p/₂.   S = ½p(1)² = ^p/₂.

     area of A  = area of C  = ^9p/₂ – ^4p/₂ + ^p/₂  =  3p

     area of B = 2´(^4p/₂ – ^p/₂) =  3p

\ area of A : area of B : area of C  =  1 : 1 : 1.  (The areas are all the same)

Many pupils feel more comfortable using concrete approximations like  p  » ²²/₇  or  p  » 3.14,  but this tends to obscure relationships between entities, and makes the calculations messier.

Solution 2 (using similar shapes)

     An powerful idea is that the ratio of areas of similar shapes is the square of the ratios of their lengths.  When a figure is enlarged by a factor of (say) 5, we get a similar figure and the area becomes  5² = 25 times as large.  So  M = (2)²S = 4S  because the radius of the medium-sized semicircle is twice that of the small semicircle.  Likewise,  L = (3)²S = 9S.
     area of A  = area of C  = 9S – 4S + S  =  6S

     area of B = 2´(4S – S) =  6S

\ area of A : area of B : area of C  =  1 : 1 : 1.  (The areas are all the same)

Commentary

     The second solution is neater because we do not need to deal with fractions or with p.  The  ratio of areas of similar shapes is the square of the ratios of their lengths.  This concept is in fact required knowledge in secondary school mathematics, including GCE ‘O’ level “Elementary” Mathematics.

Heuristics Used

H02. Use a diagram / model

H04. Look for pattern(s)

H10. Simplify the problem

H11. Solve part of the problem

Suitable Levels
* GCE ‘O’ Level “Elementary” Mathematics (“similar figures”)
* Primary School Mathematics (“areas”)
* other syllabuses that involve areas, ratios and or similar shapes.

[Maths History] #Newton's #proof that #pi is #transcendental ?

Article (from Quora, re: Alejandro Jenkins)
How do you prove that a number is a transcendental number?

Summary
Although the concept of "transcendental number" was not defined during Newton's time, Newton's Principia contained the germ of idea that seems to lead to a relatively easy proof of the idea that p  is transcendental.  Arnol'd described how Newton showed that the trajectory of planets cannot be expressed in terms of any polynomial function of time.

[#Critique] Did #Seth #Godin know what #transcendental means in #maths?

Article
Magic and irrational (by Seth Godin)


My Comments

This is an interesting piece of inspirational waffle about p to celebrate "p day".  Hopefully this special day helps to keep this important piece of knowledge that has a long cultural history fresh in our collective human consciousness, and perhaps stem the decline in mathematical literacy of some of us.

However, his article made me wonder whether he knew what transcendental means.  It is a pretty technical concept, and even if one knew what it means, it's a challenge to put it in layman's terms and say something funny with it.  If he knew the meaning, he certainly did not show it.

But I do not blame him.  Seth Godin is known for his marketing, not for his maths.  So this is not a personal attack.  I'm just following his advice, trying to be a purple cow.  Perhaps after some "moo moo" here and "moo moo" there, and transcendental meditation, this zero of a cow could break off from its roots of limited degree, eat the pi in the sky and jump over the Mooooooooon to become a hero!

;-)

[#Critique] Is #pi #infinite?

Article / Resource
The infinite life of pi - Reynaldo Lopes

My Comments
This is a well-produced video in which the team took great care in producing an interesting narrative.  The circles were drawn imperfect and shaky, so as to catch the attention of viewers and to highlight them.  A couple of points they missed out:-

1) p being irrational, does not only have an infinite number of digits in its decimal expansion, the digits are also non-repeating.  A rational number (fraction) like 1/7 also has an infinite number of digits, but they repeat: 1/7 = 0.142857 142857 142857 ...

2) near the end of the video, the animators depicted universe < p.  I find the artistic licence disturbing.  I am sure there is some way to artistically depict the known universe having less number of atoms than number of digits of p.

To answer the question in the title of this article "Is p infinite?": No, but her number of decimal digits is.