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Science communication is important in today's technologically advanced society. A good part of the adult community is not science savvy and lacks the background to make sense of rapidly changing technology. My blog attempts to help by publishing articles of general interest in an easy to read and understand format without using mathematics. You can contact me at ektalks@yahoo.co.uk

Wednesday 21 February 2018

Undefined/Indeterminate Mathematical Operations Involving Zero and Infinity lead to fallacies and paradoxes

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I regularly receive emails and messages that claim to prove statements like 2 is equal to 1, 0 divided by 0 is 1 etc.  Such proofs almost always contain an indeterminate mathematical step (an operation where the the result is ambiguous) leading to paradoxical/fallacious conclusions.  
Another common error is to treat infinity as a number.  While infinity is a useful concept for indicating a limiting situation of increasingly larger numbers, it must not be treated as a number for mathematical operations.

Mathematics deals with numbers; each number has a well defined value or magnitude.  Manipulation of numbers is vital for our society to function efficiently - such manipulations follow well defined rules of addition and subtraction.  Results of such operations are unique and should have no ambiguity.  If there is ambiguity - the result is indeterminate, and for this reason they are unacceptable.
First let me explain why infinity () is not a number:
a.  If we think of infinity as a number that is larger than all finite numbers, then one can always think of a real number that is larger than that.
b.  Arithmetical operations do not apply to infinity - ability to add, subtract etc. is essential to the concept of a number.  For example;  if we write
          ∞ + 1 = ∞ + 2 =  
then it implies that 1 = 2 = 0, and also that infinity is a number that is larger than itself.
Similarly, if we write   1/∞ = 0 and also  2/∞ = 0 then it seems that 1 = 2 which is absurd.
c.  Greater-than, less-than, equal-to relations do not apply in the same way to infinity as they do to finite numbers.
d.  Infinity is a useful concept to indicate the limits to which the value of an expression approaches - for example, if x decreases from positive values towards zero then the value of 1/x increases, reaching the expression 1/0 at x = 0.  While 1/0 in indeterminate, the limiting value of 1/x as x gets ever close to zero is exactly definable.
                limx0(1/x= +∞     ≥ 0

The trend is shown in the slide



















The above equation simply suggests that the limit, when x approaches zero, tends to infinity (an extremely large number) - it does not say that the value ever reaches infinity, rather that 1/x is increasing towards an extremely large positive value.
If x were changing towards zero from negative values then the limit will be written as 
                  limx0(1/x= -∞     ≤ 0

and the equation simply says that as x approaches zero, 1/x tends towards an extremely large negative value.  

Since, we can not treat infinity as a number, any mathematical operations involving infinities must be treated as 'not allowed'.  There might be special situations where one could consider infinity as a number and do maths with it, but we have to be very careful and watch out for paradoxical situations arising.  Hilbert's Infinite Hotel Paradox, Thomson's Lamp Paradox, 1 = 0.9999... are some well known examples.
Expressions (not a complete list) like 0 x ∞, ∞ + ∞, ∞ - ∞, p^∞, ∞^0 are indeterminate.

Infinite Series: Summing infinite series present some interesting situations.  If an infinite series is convergent then there is no problem, the nth term when n is very very large is going to be infinitesimally small and does not affect the sum in a material way. But what is the value of S for the infinite series:

S = 1 - 1 + 1 - 1 + 1 - 1 ...

We can organize the series in three different ways

S = (1 - 1) + (1 - 1) + (1 - 1) ...     = 0 + 0 + 0  ... = 0

S = 1 - (1 - 1) - (1 - 1) - (1 - 1) ...   = 1 - 0 - 0 - 0  ... = 1

S = 1  - S   or  2S = 1   which gives S = 1/2

Even though the first method has an infinite number of terms, in the second and third methods, we have one extra term. They are different series.

1 = 0.999...  :  This is my favourite fallacy.  Consider that 

                   x = 0.999999...            (eq.1)

three dots represent recurring nines to any large number (normally we say to infinity).  Multiply eqn. 1 by 10 on both sides

                 10 x = 9.999999...  =  9 + 0.999999...  =  9 + x          (eq.2)

From eq. 2;       10 x - x = 9 x = 9   or   x = 1

Therefore              1 = 0.999999

The problem with this type of proof is that the number of recurring nines in eq.1 is one more than in eq.2.  In eq.2, '0.999999...' is a different number from that in eq.1 and that creates the fallacious result.

Division by zero:   In mathematics, division is opposite to multiplication.  If a divided by b is equal to c, then c multiplied by b must be equal to a.  This rule does not work when we divide by zero.  
For example, let p = q/0; but p x 0 = 0 for all values of p (I shall deal with  the case of 0/0 later). There is no number p that, when multiplied by zero gives any other number except zero, therefore, it is fallacious to say that q/0 = p.  Dividing by gives a very large number in the limiting case when x0 but the limit when x = 0 is undefined.
Another way of looking at 'division by zero' is to consider division as a subtraction process - 24 divided by 6 is a subtraction process of taking away 6 sequentially until nothing remains.  The steps are;
24 - 6 = 18
18 - 6 = 12
12 - 6 =  6
 6 - 6  =  0
Four steps - 24 divided by 6 is 4.
When we divide 24 by zero, the steps will be as follows:
24 - 0 = 24
24 - 0 = 24 ...  for ever.  
The normal rules of division do not work when we divide by zero.
  
Zero divided by zero:  From our discussion above, 0/0 is not defined.  
We can look at 0/0 as follows:

zero divided by any number is zero - so 0/0 must be 0.
Any number divided by itself is equal to one - so 0/0 must be 1.
One can not have ambiguity in mathematical manipulations and the only conclusion we can draw is that zero divided by zero is undefined/indeterminate.

Zero multiplied by infinity:  If we start with the argument that any number, however small,  multiplied by ∞ gives infinity, that is,

               a x ∞ = ∞    then for a = 0, we obtain  0 x ∞ = 

However,  if limx0(1/x= ∞  then  limx0(x/x=   limx0(1) = 1

                                                   limx0(x^2/x) =  limx0(x) = 0
                    
                                                                  limx0(x/x^2limx0(1/x) = 

Essentially, 0 x ∞ has no meaning in terms of mathematical operations.

Zero raised to the power zero (0^0): 

 What is the value of 0^0 ?  We know that any number raised to the power 0 is equal to one.  
Also we can multiply zero any number of times,  but we always get zero:

                    x^0 = 1  and  0^x = 0

these are valid mathematical operations.  However, in the limit, when x goes to 0, both expressions reduce to zero to the power zero - the first one is equal to 1 while the second one is equal to 0.
This is inconsistent with being an unambiguous result and for that reason unacceptable.  Zero to the power zero is indeterminate.

The Limit Paradox:  This is a paradox, I like very much.  Consider the equilateral triangle ABC.  All three angles of the triangle are equal to 60 degrees and the sides are the same length:

                               AB = BC = AC = a

D, F and E are midpoints of sides AB, BC and CA respectively. Therefore, triangle ADE and EFC will also be equilateral but sides of length a/2.




Now,                AB + BC = 2a = 2 x AC
also             AD +DE +EF + FC = 2 x AE + 2 x EC = 2 x AC  = 2a

We can continue to half the sides, and as shown above, the sum of the sloping sides will be equal to 2 times the base AC.

If we continue the process an infinite number of times then the sloping sides and the base coincide but according to our analysis the sum of all the sloping sides is twice the length of the base. This is a paradoxical result.

Again, the resolution is found in our concept of infinity.  The sloping sides are that way as long they are not horizontal - the height of the triangle is not zero.  then the angle of the tiny equilateral triangles formed remains at 60 degrees.  It collapses to zero as the sloping side coincides the horizontal base and in this limiting case - we do not have equilateral triangles any more - it is a different situation entirely. 

Final Word:  This publication was meant to discuss some indeterminate mathematical operations in a language accessible to non-specialists.  I have done away with formal statements as much as possible (I have not even used words like sets, axioms etc.) and for that reason, this blog piece may not be appreciated by the purist - but this is community education site.  
The main conclusions are: (a) Be very very careful when handling infinities -they are not numbers in the usual sense of the word; (b) While zero could be called a number, its position at the junction of positive and negative number lines makes it quite tricky to handle - again be very careful when doing mathematical operations with a zero.

Hope you enjoyed the excursion into mathematical paradoxes - let me know at ektalks@yahoo.co.uk

Monday 12 February 2018

Letter Frequency in Spellings of Words and Numbers in the English Language

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The subject of this blog is completely different - I think it is insane.  
But I had to write it down as it appears so fascinatingly interesting, albeit useless.

If you look at the website (http://letterfrequency.org/), you can find the order in which letters of the alphabet occur in words of the English language.  They occur in the following order - highest frequency first:


e t a o i n s r h l d c u m f p g w y b v k x j q z
The first 12 letters are found in 80% of the words. 
Actual values are (notice slight discrepancy after letter m)



















The story begins with my granddaughter writing to me to say that the spellings of numbers from zero to ninety-nine do not contain the first four letters of the alphabet, namely a, b, c and d.  I was surprised to see the letter a in the list as it is the third most frequent letter used in English language, and to be missing in the spellings of the first thousand numbers (it first appears in a thousand) would be curious.
I then got down to prepare a list of letters missing in number spellings. What use is it? - I have no idea but I think it is insanely interesting.
(The notation used here is: 10^n is 1 followed by n zeros; 10^2 is 100; 10^6 is 1,000,000 or 1 million; and so on)
       Letter                   First Appearance 

          a                 10^3  or 1,000  Thousand
          b                 10^9       or       Billion
          c                 10^27     or       Octillion
          d                 10^2       or       Hundred 
          j                 does not occur in any spellings
          k                does not occur in any spellings
          m                10^6       or       Million
          p                 10^24     or       Septillion
          q                 10^15     or       Quadrillion

I might have missed something and got one or more errors in the list - please let me know.

I could start looking at negative powers of 10 but I think that is taking things a bit too far.

I hope you enjoyed reading through the blog - slightly different from the usual serious stuff; this is what you get when you start talking to your grandchildren. 

Thursday 8 February 2018

Math Puzzles - How logical approach makes them much more delicious


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I love math puzzles, especially those that require some logical thinking.  
Here, I have an example of a puzzle that you can solve by brute force; 
but using some logic, the puzzle becomes deliciously beautiful.

I look at a simpler version of the puzzle to show brute-force solution: 

Joan is in a room that has five bulbs (B1, B2, B3, B4 and B5).  
Each bulb can be individually switched ON and OFF by pulling a cord attached to it. 
Initially, all bulbs are OFF.     
Step 1:  Joan pulls the cord on each bulb and they are all ON.
Step 2:  Joan now pulls cord on every second bulb
Step 3:  Joan pulls cord on every third bulb
Step 4:  Joan pulls cord on every fourth bulb
Step 5:  Joan pulls cord on every fifth bulb
Which bulbs are ON at the end of Step 5?

Brute-Force Solution:  We can follow the sequence:

Step 1:  By pulling cord on each bulb, Joan switches them all ON - all five bulbs B1, B2, B3, B4 and B5 are ON

Step 2:  Joan pulls cord on bulbs B2 and B4 and they will be switched off.  
The situation is;
B1 - ON;  B2 - OFF; B3 - ON;  B4 - OFF; B5 - ON

Step 3:  Joan pulls cord on B3 and it is switched OFF.  
Now:  B1 - ON; B2 - OFF; B3 - OFF; B4 - OFF; B5 - ON

Step 4: Joan pulls cord on B4 and it is switched ON.  
Now:  B1 - ON; B2 - OFF; B3 - OFF; B4 - ON; B5 - ON

Step 5:  Joan pulls cord on B5 and it is switched OFF.  
Now:  B1 - ON; B2 - OFF; B3 - OFF; B4 - ON; B5 - OFF

Bulbs 1 and 4 are ON after Joan finishes step 5.

I have tried brute-force method for 10 bulbs and the answer is that bulbs 1, 4 and 9 are ON.

The answer gives us a clue that bulbs that are at whole square numbers are left ON after the iteration. For 100 bulbs, bulbs 
1, 4, 9, 16, 25, 36, 49, 64, 81 and 100 will be ON.
Why is it so?

In the beginning all bulbs are OFF.  At the end of the ON/OFF/ON/OFF....sequence - 
a bulb will be OFF if its cord is pulled an even number of time  but 
it will be ON if the cord is pulled an odd number of times.

Bulbs at prime numbers will only be pulled twice (step 1 and step of prime number) so they will be OFF.
Let us look at the factors of some non-prime numbers:
12: Factors are 1,2,3,4,6,12 - 6 factors - even - bulb B12 will be OFF
20: Factors are 1,2,4,5,10,20 - 6 factors - even - bulb B20 will be OFF
44: Factors are 1,2,4,11,22,44 - 6 factors - even - bulb B44 will be OFF
72: 1,2,3,4,6,8,9,12,18,24,36,72 - 12 factors - even - bulb B72 will be OFF
In fact, all numbers that are not whole squares can be shown to have even number of factors and will therefore be OFF.

Now look at the factors of numbers that are whole squares: 
16 - 1,2,4,8,16 - five factors - odd - bulb B16 will be ON
25 - 1,5,25 - three factors - odd - bulb B25 will be ON
64 - 1,2,4,8,16,32,64 - seven factors - odd - bulb at B64 will be ON
and so on.

Why is it so? - If you look at the factors of non-whole-square numbers, you notice that factors come in pairs:  72 has (1,72 and 72,1); (2,36 and 36,2) etc.  This makes the total number of factors to always be an even number.

In whole-square numbers the square root makes its own pair and the number of factors is always an odd number - for example:
36 - has (1,36 and 36,1); (2,18 and 18,2); (3,12 and 12,3); (4,9 and 9,4); (6,6) - total factors are nine - an odd number. 

Hope you enjoyed the puzzle.  

Finally:  Do you know that 1729 is a special number such that the sum its digits (=19) multiplied by the reverse the digits in the sum (=91) is equal to 1729;  that is 19 x 91 = 1729!
Besides 1 and 81, 1458 is the only other number that I know which has this property.