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Science communication is important in today's technologically advanced society. A good part of the adult community is not science saavy 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. I also give free lectures in community events - you can arrange these by writing to me.

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

Sunday, 18 February 2018

Physics of Humidity, Relative Humidity, Health Implications of Low or High Humidity

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For the last two months, I have struggled to explain to my family the amount of water vapour present in the house.  We would have liked to kill dust mites in the house and for this, the level of relative humidity (RH) should be maintained at less than 45%.  But what does relative humidity of less than 45% mean? Searching many online resources, one could eventually find that it refers to a room temperature of 20°C.  And that is where the subject becomes confusing for the general public - for the same amount of water vapour in the air, RH is different at different temperatures.
In this blog, I shall go over the physics of humidity and discuss the importance of maintaining RH in the correct range (45 - 55%).  The physics is thankfully quite straightforward.  
The main constituents of air in a house are nitrogen, oxygen and water vapour.  The pressure of air is measured in millimeters of mercury or torr and for our purpose we can take it be fixed at 760 torr. Most of the 760 torr is due to the amount of nitrogen and oxygen present in the air.  At normal room temperatures, water vapour contributes only a small amount - called the partial pressure due to water vapour.  If we boil a kettle full of water then more water vapour will enter the surroundings; partial pressure of water and hence the humidity will increase. 
However, at any given temperature the air can only hold so much water vapour - it gets saturated - and any extra input of water (say boiling off from a kettle) will simply precipitate out and condense on the coldest surface in the room.  
So - at any temperature, it is only possible to have a maximum value of the vapour pressure of water - maximum humidity  - or as we say - the relative humidity is 100% at that temperature.  
The slide shows how the saturated vapour pressure of water (to create 100% RH) changes with room temperature:
It might be useful to list some saturated water vapour pressures (WVP) for 100% RH for temperatures (T) of interest to us:

              T (°C)      WVP (Torr)

          0              4.6
                4              6.1
                8              8.1
               12             10.5
               16             13.6
               20             17.5
               24             22.4
               32             35.9
               36             44.6

Notice that these values are much smaller than 760 torr.  At normal room temperatures the maximum amount of water that air can hold is not that much.

At T = 20°C , WVP is 17.5 torr - this is the maximum amount of water that the air at 20°C can hold and relative humidity (RH) is 100%.  
If the WVP is reduced to half this value, then the amount of water in the air is also halved and we say that RH is 50% at 20°C.

Consider the situation:  Outside temperature is 8°C and the air is quite damp - WVP or RH outside is 100% or 8.1 torr (see table).  If we open the windows wide and let the air in the room be replaced by the colder outside air, then RH in the room (assuming it stays at 20°C) will be reduced to 8.1/17.5 = 46.3%.  
If the room temperature has dropped to 16°C then RH = 8.1/13.6 or 59.6%.
The value of RH depends on both the amount of water in the air and the temperature.

Mass of water in the room:   Consider the contents of air at 760 torr.  The WVP is 17.5 torr -  remaining 760 - 17.5 = 742.5 torr is due to other constituents of air (mostly nitrogen and oxygen).
There is a law in physics that states that the weight of 22.4 litres of a gas is equal to its gram molecular weight at 760 torr and 0°C.  We shall use this law at other temperatures too - it is accurate enough for our purpose.  
The gram molecular weight of water is 18 gram while for air it is approximately 29 gram.
Therefore the weight of water vapour in 22.4 litres of air is 18*17.5/760 = 0.414 gram
Weight of 22.4 litres of air is = 29*742.5/760 = 28.3 gram
Amount of water = 0.414/28.3*1000 = 14.6 gram/kg for saturated vapour pressure. This is about 15 cc of water in 22.4 litres of air (1 litre = 1000 cc)  
For RH of 45%; Amount of water is 6.6 g/kg.  A level lower than 6.6 g/kg is the recommended level of water content at 20°C to kill dust mites.

We can now extend our table of WVP as follows:

   T     WVP    Water content in gram per kg of air
 (°C)    (torr)        Relative Humidity (RH)
                       100%   60%   45%    30%       

    0     4.6        3.9        2.3     1.7       1.2      
   4      6.1        5.1     3.1     2.3       1.5    
   8      8.1        6.8     4.1     3.1       2.1        
  12   10.5        8.8     5.3      4.0       2.7        
  16   13.6       11.4    6.8      5.1       3.4  
  20   17.5       14.6    8.8      6.6       4.4         
  24   22.4       18.8   11.3     8.5       5.7        
  32   35.9       30.0   18.0    13.5      9.0  
  36   44.6       37.3   22.4    16.8     11.2          

I have highlighted in red the 'recommended' value of RH at different temperatures. 
Water content of 6.6 g/kg is easy to achieve for room temperatures of 20°C or less.  At higher room temperatures, common in the summer and in tropical climates, RH will have to be less than 30% to kill dust mites.  Interestingly, water content levels of 6 to 8 g/kg are recommended for healthy living.  These levels are also suitable for dust mites, many fungii and molds to reproduce and grow. 

To complete our discussion, let us calculate the amount of water in a normal size room - say 5m x 4m x 2.5m  or 50 m^3.  
The density of air is 1.225 kg/m^3 
The mass of air in the room is 50 x 1.225 = 61.25 kg
Mass of water at 20°C and 45% RH = 6.6 gram/kg x 61.25 kg = 404 gram
An average size of room contains, at 20°C,  about 400 gram or 0.4 litres of water at RH of 45% or about 8 cc water per m^3.  This is not a lot of water.  We can increase RH in a room by boiling some water in a pan or electric kettle. As a rough guide, 0.1 litre water, when boiled off, will increase humidity by 10% in a 50 m^3 space at 20°C.

How is humidity measured:  It is common to use a digital hygrometer which displays the temperature and relative humidity of the surrounding air.  A hygrometer calculates the humidity by measuring the capacitance or resistance of the element.
A capacitor has two metal plates with air in between them,  It is used to store electric charge - its capacity to store charge is affected by the amount of water vapour between the metal plates. Measurement of the capacity provides an accurate value of the humidity.
A resistive sensor is generally a piece of ceramic that is exposed to surrounding air.  The humidity of the air in the ceramic resistor affects its resistance and hence the current flowing in it when connected to a battery.  

Adverse effects of too high or too low humidity: 

Low Humidity:  In winter months, it is quite likely that the water content in your house will be less than 7 g/kg; with good ventillation, it could quite easily drop to 4 or 5 g/kg.  Remember, it is the water content that is meaningful, relative humidity numbers depend on room temperature and are less useful.

The low humidity air can lead to dry skin, itchy/dry eyes, irritated sinuses and throat.  A hygrometer is the best way of monitoring humidity in the house, but tell-tale sign of houseplants drying out, wallpaper peeling at the edges or static electricity point to low humidity conditions. 
Exposure to low humidity can dry out and inflame the mucous membrane lining of the respiratory tract increasing risk of infections like cold and flu. In low humidity environment some viruses may be able to survive longer, further increasing the risk of infection.

High HUMIDITY interferes with THE BODY'S Cooling Mechanism:  Human body works best when the core temperature is 37°C.  When outside temperatures approach 37°C, the body’s thermal regulation system attempts to cool it by transporting heat from the core organs by increasing blood circulation to the skin and sweating. Sweating, one of the main cooling mechanisms of the body, works by evaporating water that is excreted through the skin. This is where humidity becomes important. The concentration of water in the air (humidity) determines the rate at which water can evaporate from the skin. When the humidity in the air is high, it is not able to absorb the extra moisture  from the sweat. The result is that sweating, instead of giving any relief, makes us feel hot and sticky. High humidity makes us feel hotter, more uncomfortable and unable to lose heat our core temperature actually begins to rise.  The  body compensates by working harder to cool us down. The loss of water, salt and chemicals can lead to dehydration and chemical imbalances within the body leading to heat exhaustion.  
The heat index chart tells us quantitatively that in high humidity conditions the body feels hotter than the actual ambient temperature. For example, for an ambient temperature of 104°F (40°C) and a relative humidity of 40%, the water content in air is about 20 g/kg and it will feel like 119°F (48°C). But if the relative humidity increases to 55% (water content = 27 g/kg) the temperature will feel like 137°F (58°C)! 

At high water content (greater than 7 g/kg) fungus, molds and dust mites also survive and become a problem.  As I had mentioned earlier, for high ambient temperatures, it is not possible to reduce water in the air to less than 7 g/kg and one might need to use dehumidifiers. Maintaining a dehumidifier in a clean condition is another issue ...  

Final Word:    Mean global temperatures have risen over the past 100 years and are expected to rise in coming decades - mainly due to human activities.  The combination of higher temperatures (> 38C) and higher humidity can be dangerous and will result in increased number of fatalities.  Detailed sturdies have quantified such risks and identified regions of the Earth where high humidity and temeperatures will make life intolerable and survival difficult. 

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.