[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

[S1_20150510AESFRN] Figuring out Different Significant Figures

Question

Vishal estimated a number as 2000 when he rounded the number to 1, 2 or 3 significant figures. Ethan was shocked by Vishal's answer and he asked Vishal what is the smallest and largest possible number such that the estimated number is 2000.  Can you help Vishal answer Ethan?

Introduction

     This is taken from ShingLee’s Secondary 1 Mathematics (7th edition) textbook.  It is a good question to test if students really understand the meaning of significant figures!  If you have not done so, please read my article Significant Figures Made Easy.

     To round certain number to  n  significant figures:-

Step 1:  Identify the first non-zero digit from the left. Step 2:  From this digit, start highlighting  n  significant figures. Step 3:  Look at the digit just after the last highlighted digit.  Round up if              this is 5 and above, otherwise do nothing. Always remember to include the place-holder ‘0’s.

Solution

     This question requires one to think backwards.  You are given the number  2000.  If it is correct to  1  significant figures, then what are the possible numbers that round to it?  If it is correct to  2  significant figures, then what possible numbers round to it?  If it is correct to  3  significant figures, what are the possible numbers round to that?  The question is not specific about what sort of “number” they meant, so I shall assume that they refer to real numbers.

Technical Remark

     In case you are wondering, for example,  why in the answer for  2  significant figures i.e. the interval
1 950 < x < 2 050, the  x cannot be made equal 2 050, this is because it would be rounded to  2 100.  A number like  2 049.9999999  (seven ‘9’s after the decimal point)  will be rounded down to  2 000.  Even if there were one trillion ‘9’s after the decimal point, as long as the number of decimals terminates (i.e. is finite), it will still be rounded down to  2 000.

     Once a number touches  2 050  exactly, it will be rounded up to  2 100  for  2  significant figures.  So technically, there is no largest number, because you can keep on adding more and more  9’s  behind as long as it is a finite number.  But you cannot add an infinite number of  9’s  because that would touch  2 050. 
2 050  is not the largest number, as it is not among the possible numbers, but  1 950  is the lowest number, because it is among all the possible numbers.   Because there is actually no largest number, I used the word “largest” in quotation marks.  [ Optional: The interested reader may want to check up infimum  and supremum ]  These are some very fine details that the textbook authors and editors forgot.  Even if the editor has a PhD, the PhD could be in “mathematics education” but not “mathematics”, and may have forgotten the rigours of the real numbers system.  The answers given at the back of the book are not only wrong, one pair of answers was missing.  This caused some confusion among parents and students.  I suspect they employed a non-professional part-timer to do up the answers in order to save costs.

Suitable Levels

* Lower Secondary Mathematics (usually secondary 1 ~ equivalent to about grade 7)

* other syllabuses that involve estimation and approximation