**BASIC CONCEPT BEHIND NUMBERS, ZERO AND INFINITY**

Most scientists and mathematicians have failed to understand
the implication of zero and infinity, and invariably make mistakes. Hence, I
will explain the basic concept behind numbers, zero and infinity.

Number is a property of everything in the universe by which
we differentiate between similars, just like we differentiate between events through
time and placement of objects through space. If there are no similars, it is
called one. If there are similars, it is called many, which can be 2, 3, 4, ….
N, depending upon the sequential perception of “one’s”. There is a science
behind the designation of numbers 2, 3, 4, …. N, depending upon the nature of
perception. However, I am not going into that now. Thus, numbers have no
physical existence like the objects represented by them, but are mental
constructs that are assigned to the objects based on our perception of “one’s”.

Fractions (also decimals, where the denominator varies at a
uniform rate) are division of one into many new “one’s” indicated by the
denominator and taking out some “one’s” indicated by the numerator. Linear
accumulation of “one’s” is addition. Linear reduction of “one’s” is subtraction.
Non-Linear accumulation of “one’s” is multiplication. Non-Linear reduction of
“one’s” is division. Here, non-linear means related - partly similar and partly
different. Consider five apples each in six rows. If you count all of them as “one’s”,
it is addition. If you count number of apples in a row and number of rows
(where row is common to both, but number of apples in a row and number of
apples in a row are related, but different), that operation is multiplication.

All our actions are done at here-now. Thus, perception of
numbers are also possible at here-now. All actions not at here-now can only be
mental constructs – imagination based on memory. Zero is the non-perception
(absence) of something known in memory to have existed at some time, but not
present at here-now. Unless something exists in your memory, you cannot feel
its absence – hence call it zero. Thus, zero is not a very small number, but
removal of something either at once or gradually (reducing numbers till
everything vanishes) from here-now. Since all operations are done at here-now,
linear accumulation or reduction (addition or subtraction) using zero leaves
the number unchanged. For the same reason, non-linear reduction (division) of a
number by zero also leaves the number unchanged (not infinity, as will be
discussed below).

But non-linear accumulation (multiplication) by zero is
different. Here, zero becomes the non-linear (related, but different) part.
Since zero is not present at here-now, we cannot do any operation involving it.
But since it is partly related to the number, the whole number becomes zero,
because we cannot relate something at here-now with something that is not at
here-now. Thus, any number multiplied by zero becomes zero.

While number is a universal property of all substances,
there is a difference between its application to objects and quantities. Number
is related to the object proper that exist as a class or an element of a set in
a permanent manner, i.e., at not only “here-now”, but also at other times.
Quantity is related to the objects only during measurement at “here-now” and is
liable to change from time to time. For example, protons and electrons as
separate classes can be assigned class numbers 1 and 2 or any other permanent
class number. But their quantity, i.e., the number of protons or electrons as
seen during measurement of a sample, can change. The difference between these
two categories is a temporal one. While the description “class” is time
invariant, the description quantity is time variant, because it can only be
measured at “here-now” and may subsequently change. The class does not change.
This is important for defining zero, as zero is related to quantity, i.e., the
absence of a class of substances that was perceived by us earlier (otherwise we
would not perceive its absence), but does not exist at “here-now”. It is not a
very small quantity, because even then the infinitely small quantity is present
at here-now. Thus, the expression: lim n → ∞, 1/n = 0 does not mean that 1/n
will ever be equal to zero.

Infinity is not a very big number – n. It is like one –
without similars, but with a difference. To perceive as different from others,
we must have a total picture of the object – we can perceive only limited
things. In other words, we can perceive something as one, only if we can
perceive its full and finite dimensions. But where the total dimensions cannot
be perceived, we cannot determine or assign a number to it. In such a case, it
is called infinite (infinity). There are only four infinities in the universe:
space, time, coordinates and consciousness.

Division of two numbers ‘a’ and ‘b’ is the reduction of
dividend ‘a’ by the divisor ‘b’ or taking the ratio a/b to get the result
(quotient). Cutting or separating an object into two or more parts is also
called division. It is the inverse operation of multiplication. If: a x b = c,
then ‘a’ can be recovered as a = c/b as long as b ≠ 0. Division by zero is the
operation of taking the quotient of any number ‘c’ and 0, i.e., c/0. The
uniqueness of division breaks down when dividing by b = 0, since the product a
x 0 = 0 is the same for any value of ‘a’. Hence ‘a’ cannot be recovered by
inverting the process of multiplication (a = c/b). Zero is the only number with
this property and, as a result, division by zero is undefined for real numbers
and can produce a fatal condition called a “division by zero error” in computer
programs. Even in fields other than the real numbers, division by zero is never
allowed.

Now let us evaluate (1+1/n)^n for any number n. As n
increases, 1/n reduces. For very large values of n, 1/n becomes almost
negligible (but still a small number). Thus, for all practical purposes,
(1+1/n) = 1. Since any power of 1 is also 1, the result is almost unchanged for
any value of n. This position holds when n is very small and is negligible.
Because in that case we can treat it as near zero and any number raised to the
power of zero is unity. There is a fatal flaw in this argument, because ‘n’ may
approach ∞ or 0, but it never “becomes” ∞ or 0.

On the other hand, whatever be the value of 1/n, it will
always be more than zero, even for large values of n. Hence, (1+1/n) will
always be greater than 1. When a number greater than zero is raised to
increasing powers, the result becomes larger and larger. Since (1+1/n) will
always be greater than 1, for very large values of n, the result of (1+1/n)n
will also be ever bigger. But what happens when n is very small and comparable
to zero? This leads to the problem of “division by zero”. The contradicting result
shown above was sought to be resolved by the concept of limit, which is at the
heart of calculus. The generally accepted concept of limit led to the result:
as n approaches 0, 1/n approaches ∞. Since that created all problems, let us
examine this aspect closely.

Bhaaskaraachaarya – II (1114 AD), in his algebraic treatise
“Veeja Ganitam”, had used the “chakravaala” (cyclic) method for solving the
indeterminate equations of the second order, which has been hailed by the
German mathematician Mr. Henkel as “the finest thing achieved in the theory of
numbers before Lagrange”. He used basic calculus based on “Aasannamoola”
(limit), “chityuttara” (matrix) and “circling the square” methods several
hundreds of years before Newton and Leibniz. “Aasannamoola” literally means
“approaching a limit” and has been used in India since antiquity. Surya
Siddhanta, Mahaa Siddhanta and other ancient treatises on astronomy used this
principle. The later work, as appears from internal evidence, was written
around 3100 BC. However, there is a fundamental difference between these
methods and the method later adopted in Europe. The concepts of limit and
calculus have been tested for their accuracy and must be valid. But while the
Indian mathematicians held that they have limited application in physics, the
Europeans held that they are universally applicable.

Both Newton and Leibniz evolved calculus from charts
prepared from the power series, based on the binomial expansion. The binomial
expansion is supposed to be an infinite series expansion of a complex
differential that approached zero. But this involved the problems of the
tangent to the curve and the area of the quadrature. In Lemma VII in Principia,
Newton states that at the limit (when the interval between two points goes to zero),
the arc, the chord and the tangent are all equal. But if this is true, then
both his diagonal and the versine must be zero. In that case, he is talking
about a point with no spatial dimensions. In case it is a line, then they are
all equal. In that case, neither the versine equation nor the Pythagorean
Theorem applies. Hence it cannot be used in calculus for summing up an area
with spatial dimensions.

Newton and Leibniz found the solution to the calculus while
studying the “chityuttara” principle or the so-called Pascal’s differential
triangle. To solve the problem of the tangent, this triangle must be made
smaller and smaller. We must move from x to Δx. But can it be mathematically
represented? No point on any possible graph can stand for a point in space or
an instant in time. A point on a graph stands for two distances from the origin
on the two axes. To graph a straight line in space, only one axis is needed.
For a point in space, zero axes are needed. Either you perceive it directly
without reference to any origin or it is non-existent. Only during measurement,
some reference is needed.

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