[Enrich20170101SQT] Calculating Square Roots by Hand

Introduction

          Happy New Year to our readers!  I wish this year will be a fruitful one for everybody.

          Today, I will illustrate how to calculate square roots by hand, using  54 756  as an example.  It is similar to long division, but has some modifications.

Solution

          Starting from the right, pair up the digits.

          2×2 = 4  is the nearest perfect square to  5.  Subtract and bring down the next two digits, giving  147.

          Double the digit  2  to get  4.  Think:  ? × 4?  gives  147  or nearest possible value.  We have 3×43 = 129.

          Subtracting and bringing down the next two digits gives  1856.  Replicate the digit  4  on the left and double the digit  3,  giving  46.

          Now think:  ? × 46?  gives  1856  or nearest possible value.  It turns out that  4 × 464 gives exactly  1856.  We are done!  The square root of  54 756  is  234.

How does it work?

          This relies on the algebraic identity  (10a + b)² = 100a² + 20ab + b², the right-hand expression is equal to   100a² + (20a + b)b.  For example, at stage 4, we have  a = 23,  b = 4  and  (20a + b) = 464.

          Did you learn something today?

[AM_20151119ISSD] Differences of Squares Hiding under Square Roots

Question

Introduction

     Here is another question involving surds.  As we know, surds are literally absurd, because they are irrational.  How to we do this one?

Strategy

Solution

Remark

     Reflect: What did you learn from solving this question?

     For another example of using the difference of squares formula, please look at this article.

Heuristics Used

H04. Look for pattern(s)

H09. Restate the problem in another way

H10. Simplify the problem

H11. Solve part of the problem

Suitable Levels

* GCE ‘O’ Level Additional Mathematics (indices / surds)

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

* revision for IB Mathematics HL / SL

* other syllabuses that involve indices and/or surds
* any precocious or independent learner who is interested

[S2_20151112CSSR] Change of Subject and Square Roots

Question

Introduction

     In this secondary 2 (approx. grade 8~9) algebra question, we are essentially asked to make  k  the subject.  This means that through a series of algebraic manipulation, we arrive at a final equation in which  k  appears alone on one side (conventionally the LHS) and all the other “stuff” on the other side.  This question looks challenging firstly because k  appears in more than one place and then also we need to deal with the square root.

The Square Root

     Note that the principal square root (or simply “the square root”) is by the modern definition non-negative i.e. zero or positive.  Of course, what goes under the square root must also be non-negative, otherwise it would not even make sense as a real number. 

     So observe that in the given equation, the RHS is non-negative.  Hence the LHS which is just  k,  must be non-negative.  It is tacitly understood that  3a – k² > 0  for the square root to make sense.  

     To get rid of the square root, we can square both sides of the equation.  After that we bring all the terms with  k  to the LHS.  Finally, we need to “unsquare” both sides by taking square roots.  The solution takes only about 5 steps, as shown below.

Solution

Remarks

     There is no need for  ±  in the final line because we already know that  k  is non-negative (k  is zero or positive).  Here is something that students and even teachers / tutors can get confused over.  Modern mathematics tends to take a “function” approach in which each expression can take only a single unambiguous value.  “ Ö ”  may be regarded as a function with the non-negative reals as domain and the non-negative reals as range.  Although in traditional parlance, we say things like the “square roots” of  9 are 3 and -3, once you see the  “ Ö ”  symbol (apart from the “±”), it is the result of a calculation and the result is by definition non-negative.

     Another thing that people get confused over is: What about the ± symbol ?  Note that  ±  by itself is actually meaningless!  Something like  ±3  is just a short-cut for lazy people to say “the answer is 3 or -3”  (and we tend to be lazy, don’t we?).  But this is the result of solving an equation like “x² = 9” when  x  is a real number with no other restrictions.  This equation has two roots: 3 and -3.  If  x  is known to be non-negative, then  x = 3  is the only solution.  Of course solving an equation involves calculation.

     So what is the difference between solving and mere calculation?  Solving is a process of finding values for unknowns and it usually involves a more than one step and it may include calculation.  When you see something like  Ö9   you are just calculating, and there is only one answer.  But when you see something like “Find the values of  x  such that  x² = 9” you are solving.  There is an unknown (e.g.  x)  and you are supposed to find number(s) that you can plug into  x  to satisfy that equation.  After you calculate  Ö9 = 3, you still need to write “x = 3 or x = -3” or its short form “x = ±3”.  I hope this clears the confusion.

     Remember “Ö something non-negative” Þ one non-negative answer.  “find / solve something” Þ maybe more than one answer.

Heuristics Used

H04. Look for pattern(s)

H05. Work backwards

H10. Simplify the problem

H11. Solve part of the problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* Lower Secondary Mathematics (Secondary 2 » Grade 8/9)

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

* other syllabuses that involve algebra

[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