[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

[AM_20151117ISCP] Estimating a Crazy “Prosperous” surd to the Nearest Integer

Question

Introduction

     To the Chinese, the number  8  (八)  is considered to be auspicious, because it sounds like “prosper” (发) in the various Chinese languages/dialects.  But the LHS expression featured above seems too prosperous for comfort.  There is an explosion of  8s  coupled with eighth roots.  How to even handle that?

Strategy

     As usual, often one good tactic is to look for patterns or chunks.  [H04]  Can you see the sub-expressions containing the eighth roots (highlighted in green and pink)?

Do you notice any similarities between the two chunks?  Do you notice any difference(s)?

If we call the green chunk  a  and the pink chunk  b,  we can remove the roots by taking the eighth powers.  Then we get whole numbers, which are less complicated.

Solution

Remark

     The crux of the problem is the factorisation of  a⁸ – b⁸.  It is based on repeated application of the difference of squares    X² – Y² = (X + Y) (X – Y)  formula which schools expect students to know.
     For another example of using the difference of squares formula, please look at thisarticle.

Heuristics Used

H04. Look for pattern(s)

H09. Restate the problem in another way

H10. Simplify the problem

H11. Solve part of the problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels
* “IP Mathematics” so called
* challenge for students taking GCE ‘O’ Level Additional Mathematics
* other syllabuses that involve surds
* precocious kids who like to test themselves

[NumTh Expository] The Principle Behind Casting Out Nines

Introduction

Think of a number ... say 685932.  Divide by  9  and take the remainder.      685932 ¸ 9      = 76214 r 6  Add up the digits,  divide by  9  and take the remainder.      6+8+5+9+3+2 = 33 ¸ 9 = 3 r 6  What do you notice?  Try this with any other positive whole number.

Discussion

     Did you see that a number and its sum of digits always have the same remainder when divided by  9?  This is the principle behind the method of “casting out nines”, used in the past for checking arithmetical calculations.  Why does this work?  Where is its magic?

     The decimal number system that we use today is based on the number  10, which is just  1  larger than  9.  Observe that  9, 99, 999, 9 999, 99 999, ... etc are all divisible by  9.  Hence, the powers of 10, namely 10⁰ = 1,  10¹ = 10,  10² = 100,  10³ = 1 000,  10⁴ = 10 000,  10⁵ = 100 000,  etc  all leave a remainder of  1  when divided by  9.   Thus in our example,

     685932 = 6´10⁵ + 8´10⁴ + 5´10³ + 9´10² + 3´10¹ + 2´1

                  = 6´(99999+1) + 8´(9999+1) + 5´(999+1) + 9´(99+1) + 3´(9+1) + 2´1

                  = 6´99999+8´99995´999+9´993´9 + 6´1+8´1+5´1+9´1+3´1+2´1

                  = 9 ´ something + 6+8+5+9+3+2

As you can see, all the “´1” allow us to separate out the digits, and then the stuff with 9, 99, 999 etc can be lumped together as 9 ´ some whole number, but we do not need to care too much about this multiple of 9 as it would not make any difference to the remainder.  It is now obvious that  685932 and 6+8+5+9+3+2=33 will have the same number when divided by 9.

     Let us generalise the argument.   If two numbers  x  and  y  have the same remaider when divided by 9,  we say that  x  and  y  are congruent modulo 9, and we write     x  º  y  (mod 9).  Congruence is an equivalence relation and “º” behaves in many ways similar to “=”.

Theorem

For an arbitrary number  n   with digits  [dk...d3d2d1d0]                                         n º dk + ... + d3 + d2 + d1 + d0   (mod 9)

                 n = d_k ´10⁴ + ... + d₃´10³ + d₂´10² + d₁´10 + d₀.    

Since  10^k º 1 (mod 9)  for all integers  k > 0,  we have

                 n º d_k´1 + ... + d₃´1 + d₂´1 + d₁´1 + d₀

                 n º    d_k  + ... +   d₃   +    d₂  +   d₁    + d₀     (mod 9).   © (Q.E.D.)

As an example of application of this principle, please refer read thisarticle.

Suitable Levels

* Primary School Mathematics Olympiad

* syllabuses that involve congruences and Number Theory

* anybody who is interested

[AM_20150620DFFPPR] Differentiation of Powers from First Principles

Question

Introduction

     This question was posted to a Facebook forum for Secondary School and Junior College mathematics (the equivalent of grade 7 and above).  Most students, except a few enthusiastic ones or maybe some from “integrated programme” schools, will not bother with differentiation from first principles.  However, mathematical formulas are true because mathematicians took the trouble to critically analyse them and show that they always work, not because they are found in the textbook, nor because the teacher says so.

Solution

Remarks

     The answer to the original problem just falls out by letting  p = 17  and  q = 19.

     Some people suggested using Binomial Expansion for rational powers.  I feel that this is not from first principles, since Binomial Expansion relies on differentiation, which relies on first principles.

Suitable Levels

* GCE ‘O’ Level Additional Mathematics

* college level calculus

* AP Calculus

* other syllabuses that involve differentiation from first principles, or anyone who is interested

[AM_20150616LGDFCA] Logarithms and Knowing that You are Correct

Question

Introduction

     This is a relatively straightforward question once the student has learned the rules of logarithms.  When I was first learning logarithms it took me quite some time to get used to the idea of “logs”.  Are they fallen trees?  So what are “logs”?  They are just the exponents or indices.  For example  2³ = 2 ´ 2 ´ 2 = 8  and we can write  log₂8 = 3.  Logarithm to base  2  of  8  is 3, because  3  is the index  i.e. to get  8  you need to multiply  2  by itself  3-fold.

     In general,  log_ba = x   Û   a = b^x.  Why?  Because that is exactly what logarithm means!  One way to remember this definition is to imagine: if you transport the log to the other side of the equation, the log drops off and you get the base  b  propping up the  x.  You can also do it the other way round.  If the base  b  of a power moves to the other side, it becomes a  “log”  with base  b.  [active mnemonics]

     What about the “common logarithm” lg?  It is the logarithm with base  b = 10.  In the days before pocket calculators were prevalent, students used books and slide-rules with logarithms of base  10  for multiplying and dividing large numbers.  Base 10 logarithms are still commonly used in today for the Richter Scale (in seismology, to measure earthquakes), for decibels (to compare the loudness of sounds or gain / loss in amplifiers), for pH (measurement of acidity / alkalinity in Chemistry) .... etc.  The aforementioned rule works exactly the same way, with  b = 10.

                      lg a = x   Û   a = 10^x                 (lg means log₁₀)

Note that in many calculators, their “log” button is for  lg  or logarithm of base 10.

Solution

Checking Your Answer

     The person who posted this question on Facebook got  33 333 333.3  as his answer, but did not realise that his answer is the same as the “model” answer, which is given to three significant figures in standard scientific notation.  Many students have the habit of checking their answers against the “model” answer usually given at the back of the book or worksheet, which may sometimes be wrong!  Anyway, in tests and examinations, you do not have the luxury of checking your answers like this.  In real life, if an engineer makes a calculation mistake, buildings may collapse and people die.  It is better to make it a habit to check your answers on your own and to know and be sure that you are correct.  One way to do this is to substitute the value of  x  back into the original equation to see if it works.  Nowadays, many models of calculators have a “store” function indicated by a button labelled with “STO” or an arrow “®” or something like that.  You can store the value into a variable (or memory location) like  X  and then key in something like  “log(3X) ”  and see whether you get  9  or something close.  Be aware that calculators can have rounding errors. 

Notations for “log”

     School students are taught to use “lg” to mean “log₁₀”  and  “ln”  to mean the natural logarithm “log_e”  where the special number  e = 2.7182818284 ...  discovered by the visually impaired but brilliant mathematician Euler.  Many calculators take “log” to mean “lg”  or  “log₁₀”.  For adult working professionals, “log” (without indication of the base) usually depends on what field they are in, or on the topic being discussed.  As mentioned before, base 10 is used for Richter scale, decibels and pH.  Computer scientists tend to use base  2  because of the binary system.  For rate of reaction (chemistry) or radioactive decay (chemistry / physics), the natural logarithm “ln” is often used.  In school, for the purposes of learning, we make the logarithm bases explicit.  Do not simply write “log”.  Write “lg”, “ln” or “log₂” or “log₇” or “log_b” (for whatever  b  is).  Note also that the letter “l” in all these notations is not the letter “i” or “I”, but it is the smaller case “L” (for logarithms).

Heuristics Used

H05. Work backwards

H09. Restate the problem in another way

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* GCE ‘O’ Levels Additional Mathematics

* International Baccalaureate (IB) Mathematics (revision)

* other syllabuses that involve logarithms and exponentials

[S2_20150513IXS] The Power of Repeated Chunks

Question

Introduction

     This is a question on indices (taught in secondary 2 or 3) that looks daunting.  There is a way to unravel it.  Look at the equation carefully.  Can you see anything that looks like it is repeated?  What about that small little “+1”?  How do you deal with it?

Reminders and Tips

    

Solution

Summary

     The key to solving this question is to apply the laws of indices and recognising repeated chunks.  This allows one to see the overall structure of the expression and this often leads to a simplification of the expression involved.  It is good to recognise the powers of small prime numbers.  Like atoms, prime numbers are the building blocks of all whole numbers.  Using prime bases allows us to compare indices more easily.

Heuristics Used

H04. Look for pattern(s)   e.g. repeated chunks

H09. Restate the problem in another way

H10. Simplify the problem

H11. Solve part of the problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* Lower Secondary Mathematics (usually Sec 2)

* “Elementary” Mathematics or “E Maths”

* Additional Mathematics (revision)

* other syllabuses that involve indices