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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