[AM_20150413RSD] Rationalising Denominators for #Surds

Question 

Introduction

     Expressions with surds in their denominators are cumbersome.  The good news is that we can make the denominators into rational numbers, which are nicer.  Rational numbers those that can be expressed as a ratio of integers i.e. they are (proper or improper) fractions or can be converted to fractions.  Whole numbers are also part of rational numbers because you can always put them upon a denominator of  1;  e.g. 2 = ²/₁,  so  2  is a rational number.

     The standard trick for simplifying expressions with surds in their denominators is to rationalise the denominator by mutiplying the numerator and the denominator with its conjugate surd.  For example, the conjugate surd of   Ö5 + Ö2   is   Ö5 – Ö2.   Just change the  +  to  –  or the  –  to  +.  Let us see how the magic works.

Solution

Remarks

     Note that in the first step, I pulled out 2 as the common factor of the denominator, so that I get a simpler surd to work with.  Always try to work with simpler expressions.  This not only shortens your working, it reduces your chances of making a careless mistake.
     In mathematics, “rationalising” does not mean you give some reason or excuse for something that you know you have done wrong.  It means “make it into a rational number”.  Why does rationalising the denominator work?  This is because on the bottom (denominator) we have a difference-of-squares expression of the form
                                           (a + b)(a – b)   which is equal to   a² – b².
Since squaring “gets rid” of square roots,  a²  and  b²  will give you rational numbers (whole numbers or fractions), you will end up with a nice number downstairs (on the denominator).  Pupils should make sure they have this technique in their repertoire of skills.

Suitable Levels
* GCE ‘O’ Level Additional Mathematics
* revision for GCE ‘A’ Level H2 Mathematics
* revision for IB Mathematics HL / SL
* other syllabuses that involve surds
* precocious kids who always want to learn more

[AM_20150412SQS] Perfect Squares Lurking Absurdly

Question

Introduction

     Surds are expressions involving roots like square roots.  They are usually irrational numbers.  If you try to put them into a ratio of integers, you are absurd!  Many students (and teachers?) are not sure of how to put square roots of numbers in a simple form.  The trick is to use square numbers or perfect squares.  These are squares of whole numbers.  For example:-

     1² = 1 ´ 1 = 1                      Ö1 = 1

     2² = 2 ´ 2 = 4                      Ö4 = 2

     3² = 3 ´ 3 = 9                      Ö9 = 3

     4² = 4 ´ 4 = 16                  Ö16 = 4

     5² = 5 ´ 5 = 25                  Ö25 = 5

A number like  4  can be represented by a real square whose sides have length  2  units.  Note also that if you take the square root of a perfect square, you always get a nice whole number.

     How do you deal with numbers that are not perfect squares?  You factor out as many perfect squares as possible.  This would eventually lead to surds with small numbers, which are more manageable.  Here are some examples:-

With this weapon in our hands, let us kick some butt.

Solution

Moral of the Story

     Using perfect squares reduces your square roots to surds involving square roots of prime numbers, which are easier to combine or cancel.  This gives a short and sweet solution.  In mathematics, always try to do things by the cleanest way (if you can).

Heuristics Used

H10. Simplify the problem

H11. Solve part of the problem

Suitable Levels
* GCE ‘O’ Level Additional Mathematics
* revision for GCE ‘A’ Level H2 Mathematics
* revision for IB Mathematics HL / SL
* other syllabuses that involve surds
* precocious kids who always want to learn more

[#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.