Prologue

I place one tweet from my page @lal_shiban to set the tone and tenor of the discussion that follows.

1.What is junk math?

Telling you that I have decoded the secret of
Napier’s log/anti-log tables. So, I know a little
bit of Mathematics.

Now, Sin x is related to a Rt. Angled Triangle
Log sin x to none.
Sin x/log sin x are numbers between 0 & 1
Differentiate     y = log sin x is an example of junk math.

  1. What is misleading math? [Feedback address: shibhanlalpandita@gmail.com]

d/dx sin x = cos x  &  Sin (90 – x) = cos x
So, d/dx sin x = Sin (90 – x) = Cos x
This is an example of misleading math
Here static & dynamic expressions are in a romantic hug!

  1. What is corrupt math

p/q is supposed to be a number. In fact, it can
be anything but a number. It is a field comparison
expression. Forget about it. When our kids are told
that p: q can be expressed as p/q it turns math on its head
p/q is the correct math but not a number.

  1. What is fictitious math?

When a sine curve is made a progressive motion
It is an example of fictitious math.
While field above the line is +ive & below -ive
the direction is linear. Negative, kids are taught is
when direction is reversed. Here direction doesn’t change.
The AC current is also a sine wave. Our domestic supply
has a frequency of 50 cycles. It means in one second the
direction is reversed 50 times.
And when area under the curve is calculated, all
records of cons are surpassed. I see no meters involved.
And when curvature is worked out as 1/r, I simply get exhausted
because I can’t measure meter inverse!

_____________________________________________________________________
Thus, spake Nature’s Geometrician:

I shall now settle the parameters/values of plane geometry. Observations by humans is limited to the inclined plane as such three dimensional geometry is only a figment of imagination. The curvature of the earth limits the extent of vision of the humans.

  1. Take any 3 points P1, P2, P3 in space. The curvature of the space is so designed that there is a 4th point O, which is equidistant from these 3 points. It means that the three points lie on a curve whose centre is the 4th point.

As a corollary, these three points independently serve as centres of three independent curves. Such is the complexity yet permissivity of the Space. There is nothing straight in space. In the future when humans will come up with aeroplanes, they too have to follow a curved path or paths depending upon the flight exigencies while flying from point A to point B.

  1. If a straight line is drawn through the centre, then the curve and the line shall have two common points CP1 & CP2 as shown. This line divides the curve/circle in two halves.
  2. If these two points are joined by any random point on the locus of the points, then we shall have a triangle and the following values will hold:
  3. a) The value of the angle at periphery is equal to the sum of the two angles linked to the diameter.

Or the two angles courtesy the diameter, x & y sum up the value of the third angle opposite to the diameter which is one side of the triangle.

  • b) The line O-RP is the radius as well as the median of the triangle. On this line, 1/3rd of the length of the radius measured from centre O is the point A through which the other two medians shall pass.
  • c) The diagonals of the three angles shall pass through one common point which shall be the centroid of another circle. To find the radius of this circle, we shall draw a perpendicular to any side of the triangle. In this way the three sides of the triangle shall serve as tangents of the in-circle. So the field of tangents lies within the field of two eccentric circles. The diameter of the original circle shall necessarily serve as on tangent to the in circle.
  • d) The random point RP on the periphery which we shall call the circumference, shall serve as the crank point of a system of vector transfer. It means if we have to convert linear motion into circular motion, an arrangement aligned with the diameter and connected to the crank point through another arrangement shall accomplish the desired result.
  • e) The area of the triangle shall be worked out simply by multiplying the two sides CP1- RP & CP2- RP and dividing the value by 2. The maximum area of the triangle is fixed as square of the radius.
  • f) The circumference of the semi-circle was supposed to be 3 times the radius but here the space geometry did not oblige me. It stretched the circumference approximately by 1/7th of the value of the radius. So the altered value is (3r +1/7r).
  • g) Likewise the area of the circle was supposed to be 3r2 but the space fixed it as (3 + 1/7) r2 . Now this magnanimity shown by the space did in the long run tempt the botanical life to settle for tubular/circular cross section when the geometry of the botanical life was fine-tuning structural details. (Pl see the teaser 8 Part 3, Ch 1).
  • h) Last but not the least, the most important equation accompanying the triangle in the semi-circle shall always hold true.

(CP1  – CP2)2 =  (CP1-RP)2 + (CP2-RP)2

Or Diameter square = The sum of squares of the other two sides of the triangle.

From the above it is clear that a triangle should always be treated while keeping in view the circle in which the triangle is trapped!

II

India exported zero as a helper to the world – gratis. And the world started counting sensibly. But Europe was not that intelligent. Dr C K Raju tells us that it took them five hundred years to learn the place value advantage of numbers. And in turn they created a new mathematics and a new misery.

Macaulay introduced us to European education and India imported Zero, evolved and appreciated, as a number! Simultaneously India imported misery for her own population.

I had pinned two equations on my Twitter Page and requested the mathematicians to verify the equations. I also kept prize money for the successful candidate. This tweet received about 12 thousand hits till date, but no one could dare to touch them.

I have scouted the math books published by the NCERT but all have avoided touching this derivation. It means I am not the first person who has discovered this major fault line in calculus. The first mathematician who derived didn’t have basic intelligence to note the two graphs were separated by a disconnect, measuring 900.

So NCERT abetted a wrong and continued with retailing wrong knowledge. The fear that a superstructure of the European math will collapse and squeeze the space of employability of mathematicians must have weighed on their minds. Tragedy!

Now I shall combine equations in the following manner.

d/dx sinx = sin(90 – x) = cosx

d/dx cosx = cos(90 -x) = sinx

What do the above equations tell us in reality? Can mathematicians tell us!

II

Ramanujan- Equation or Inequation!

This happened on 13th Dec 2016. I was scanning the home page of Twitter and one tweet from one Cliff Pickover read as follows:

Indian Mathematicians Ramanujan was fond of stating astounding formulas.

 

Pickover is a mathematician who has authored 50 books as per the profile punched on his twitter page. He seems to be a great wizard of making and breaking numbers. This equation what Pickover called awe and shiver really shook me up. I was mesmerised by the beauty of the expression. So I started flirting with this equation. I made some mental calculations, joined some dots and lo and behold, I became doubtful of the correctness of the equation! So I set about solving the LHS of the equation.

We can solve the LHS by three methods. We simply use log table and substitute the values as follows.

(cos 40)1/3  =  (0.7660)1/3  =  0.92

(cos 80)1/3  =  (0.1736)1/3  =  0.56

(cos 20)1/3  =   (0.939)1/3    = 0.975

The point to be noted here is that the cube root is greater than the field value. Absurdity!

LHS  =  0.92  +  0.56  –  0.975  =  0.505

RHS =  1.155   [ when solved]

So LHS ≠ RHS

We shall move into the geometric domain and check the validity or otherwise of this equation.

 

Not drawn to scale

 

The angles have been plotted in the unit circle as shown.

/_AOX= 800, /_BOX = 400  & /_COX = 200. The projections of points A, B, C meet the radius OX      at G, F, E.

Now cos 80 = OG/OA =OD/1 =OG   similarly cos 40 =OF & cos 20 = OE.

Let us examine cos 80 = OG which is a length segment. With this length we can make a cube. We know that a cube has 6 faces and 24 edges when we actually construct it. Therefore it follows that the cube root of cos 80 = 1/24 OG.

Therefore the LHS can be reproduced as:

1/24[ OG + OF – OE]

1/24[ cos 80 + cos 40 – cos 20]

1/24[ 2(cos )(cos ) – cos 20]

1/24[ 2. Cos 60.Cos 2o – cos 20]

1/24[  cos 20 –cos 20]  = 1/12[cos20-cos 20] = 0     [cos 60 =1/2]

So the equation doesn’t hold. We can now take another route to prove that the equation doesn’t hold.

We draw ZY perpendicular on CE.  Now the triangles OGD and YZC are similar. Why? Because angle c is common to OEC & ZYC.  Means angle DOG = angle CZY.  All the angles are equal. We have to prove that they are congruent as well. If we can prove that YZ = OG.

Now YZ = EF.  And EF = OE – OF

EF = cos 20 – cos 40 = – 2 sin 30. Sin(-10) = sin 10.

OG = cos 80 = cos(90-10) =sin 10

So the two triangles are congruent. Therefore EF= OG &ZY = FE

Cos 80 +cos 40- cos 20= OG +OF – OE = OG +OG+GF-OG-GF-FE = OG –FE = OG-OG =0

It requires mathematical eagle eye to prove that the RHS will never equal to zero!

The final confirmation will prove that the equation doesn’t hold. We have to draw the angles and take the measurements.

II

So we have proved that mathematicians can make mistakes. Even physicists! Trigonometry is based on the principal of comparison of similar right angled triangles. Nature’s geometrician uses the same unit circle to create a right angled triangle. In fact there is no difference between a semi-circle and a right angled triangle! Think! The diameter of a semi-circle serves as the hypotenuse of a right triangle. In fact nature has trapped the right triangle within the domain of the circle. The unit circle is not the domain of counting but the domain of geometry. It is in fact the exclusive domain of geometry. Counting domain lies outside the domain of unit circle.

What is a circle? It is the story of two points, one static and the other dynamic. The distance between the two points remains constant in condition of dynamics. And every point on the path traced by the dynamic point, circumference, makes a right angle with the two end points of the diameter. Every point on the circumference represents 1. So if a circle is expanding, when the dynamic point is fixed, the three sides will stretch proportionally. The concept of proportionality has been corrupted by the mathematicians to such an extent that the concept has degenerated into an equation which it is not. It is this law of proportionality that gave birth to trigonometry.

The circumference of the unit circle is also the path of infinity! Imagine a metallic ring is suspended in space by some magnetic arrangement and an ant is placed on it. The ant will start moving around over and over again but reach nowhere. When the sense of direction is lost, the dynamic body is in an infinite state. All heavenly bodies suffer from this syndrome infinity! They move without reaching any destination. And when some lucky celestial body finds it, like the comet Shoemaker Levy-9, it reached Jupiter on 1992 and we were lucky to witness the fireworks which happened on that day.

So an infinite series is a number in circular motion. It acquires an incremental value at each complete revolution. The incremental value is fixed, just like the circumference of the unit circle. So the series which the chronometer attached with a robotic ant moving on the hypothetical ring suspended in space will record the series: πd,2πd,3πd,4πd,……. And so on. At each point of inflexion an increment will accrue to the original number which is fixed by the rigour of the unit circle.

We shall continue with understanding the domain within the unit circle. In the forgotten past, in India, the proof of what is popularly known as Pythagoras theorem was provided in the same unit circle, which doubled up as providing the space for writing the programme of a birth chart.

As shown in the figure above, AC & BD can be two right diagonals. The diagonals have not been drawn only the points have been retained to avoid unnecessary loading of the figure. We join the points to produce the square ABCD. As is evident the square is also made up of two right angled triangles on either side of a diameter. Or two semicircles, two rt triangles. We make another smaller square within the bigger square which can be drawn. Now EFGH is constructed by positioning the points E,F,G &H such that CG=DH=AE= BF = X and EB=CF=DG=AH= y. This leaves us with the square HEFG whose side is z. We can make the square by drawing a concentric circle with the radius OE. So we have two concentric circles and two squares whose diagonals pass through the centre.

It is quite evident that area square ACBD = area of square EFGH + area of 4 rt triangles.

Or ( x + y )2 = z2 + 4( ) x y

Or x2 + y2 + 2 x y = z2 + 2 x y    => x2 + y2 = z2

In this way the ancients in India proved that the square of the hypotenuse is equal to the sum of the squares of the base and altitude.

And when the diagonals of the two squares are drawn, we have 12 houses which serve the astrologers with the space for preparation of birth charts.

So when we talk about squares and cubes, we are in an ambiguous situation. The statements can pertain to geometrical shapes or pure numbers. Beyond the exponent 3, we are sure that we are exclusively counting numbers and nothing more.

If we look at the stem of a tree or the trunk of a tree, both have cylindrical x-section. It appears, especially in case of a tree the growth happens by creating a telescoping arrangement-cylinder within a cylinder. The x-section of a tree trunk reveals annular rings. By counting the rings we get the age of a tree. It happens by accumulating and arranging mass in concentric circles.

The counting field lies outside of the geometric field. We can count leaves, branches, flowers, fruits.

 And what is zero? It is the unit circle we have been talking about in the foregone paragraphs!

Zero is a circle of unit length. It is a two dimensional plane. 1 lies at its circumference, as if liberated from the trap by the nature’s geometrician. 1 is a one dimensional length segment. A symbol! The symbols from 2 to 9 are the repository values of number of one’s just like in paper currency. A ten rupee note contains ten one rupees. And number 6 contains six ones. All symbols have their face value.

And a sphere! It is the inflated version of a dot. It is unit sphere. Earth is unit sphere. Every point on its surface represents 1. We have one earth. The diameter of the earth circle is X meters. Zero lies at the centre of the earth.

I have no idea what Dr Raju means by the phrase zero limit calculus. But I think mathematicians can put his theories to test. Who knows Mathematics may glide on more assured wings than the present hypothetical gliders.

The ball is now in the court of the world of mathematics. When the subject was faked it was actually a desire of the mathematicians to keep pace with the expanding horizons of knowledge in other fields. The time has come when the entire math syllabi should be thoroughly subjected to critical analysis. The students at higher levels need better smarter pathways to grasp the spirit of mathematics.

Meanwhile, at primary and secondary levels I have the necessary road map to make this subject interesting, utilitarian and student friendly.

I want to see happiness and not fear on the faces of kids!

S L Pandita