In Science and Mathematics, Infinity that denotes a quantity which is ever increasing, enlarging, getting bigger and bigger to the extent it can not be measured or mentioned by numerical figure, finds repeated use in Science and Mathematics. This quantity which is endless, numberless and not finite, was assigned the symbol ∞ by Great mathematician, John Wallis* to denote it for its use in problems of mathematics. How large infinity can continue to grow, is the extent of ones imagination of the the largest quantity and it surpasses that as it is bigger than what one imagined. It is larger than the largest.
Defining Infinity In Simple Terms
Consider an apple which is cut in pieces of one fourth size, each piece then can be distributed to four persons. If the apple is sliced to pieces of the size 1/100000, it can be distributed to 100000 persons. Likewise, if each piece is cut to very very small size, it can be given to very very large number. As the size of the piece reduces, number of persons served increases. If each piece is cut to very very small or to extent of zero size, it can be distributed to persons as large as ones imagination and beyond and that number would be infinite. Infinity is thus defined as a quantity divided by zero. Mathematically, ∞ = Q/0 where Q is any quantity other than 0.
Twin Jet Nebula Resembling Infinity
Multiplying Or Dividing infinity With Any Quantity.
Since infinity is beyond imagination and endless, nothing can be bigger than it, two times ∞ will again be endless or ∞.Three times ∞ will again be as endless and large as ∞. Or any quantity multiplied with infinity results in ∞. Conversely, infinity divided by any quantity is infinity.
Mathematically, n.x ∞ = ∞. And ∞/N = ∞ where N is any quantity other 0.
If anything is added to endlessly growing quantity, it will remain endlessly growing. Similarly, anything is subtracted from endlessly growing quantity, it is not affected and remains same. Mathematically, n + ∞ = ∞ and ∞ – n = ∞.
If endless quantity is multiplied with another endless quantity, once or twice or any number of times, it remains endless or infinity.
Mathematically, ∞ x ∞ = ∞, ∞ x ∞ x ∞ = ∞,∞ x ∞ x ∞ x ∞……….any number of times = ∞.
If infinity is divided by infinity, resultant quantity is in determinant, If infinity is subtracted from infinity, it does not result in zero, since infinity is endlessly growing and never attains a finite value and two endlessly growing quantities if subtracted from one another, results in any quantity to ones imagination. Therefore, subtraction of infinity from infinity is in determinate.
Mathematically, ∞/∞ = In determinate, ∞ – ∞ = In determinate.
However, negative infinity exists and will be explained in appropriate paragraph in this article.
Time Arrow Paradox
Zeno of Elea, a pre-Socratic Greek philosopher of southern Italy, considered when an arrow is shot from a point A1 to hit target A∞, at any instant, during its flight, arrow will be found stationary because the instant does not have any magnitude of time variation. No time variation means no displacement. Accordingly, if the arrow is observed at any other instant, it would again be found stationary or at rest. There will be such infinite instants during flight of arrow and at each instant, it would be observed as if it were at rest. But in fact, the arrow during its flight, always remains in motion and also has also specific time of flight. But there would be infinite instants during flight of arrow as each instant has zero time duration. If that be so, its time during fight should be zero as instants of zero time duration added infinite times would always be zero. But there is finite time of flight. This self contradicting situation is called Time Arrow Paradox.
In the example of slicing of apple in to infinite parts, each part had zero size. In that case also, if one adds all parts of zero size, these must restore to the original size of apple from which these were cut but zero added infinitely remains zero. Infinity defined in foregoing paragraph was any quantity divided by zero. Also infinite parts of size zero, do not add up to result in original quantity. There is then something amiss and it needs further explanation to define infinity and solve Time Arrow Paradox.
Achilles And Tortoise Paradox
Zeno considered another case of a footrace between Achilles** and tortoise. Achilles challenged tortoise of a footrace and gave him a head start of half distance and both started at the same instant Achilles from point A1 and tortoise from mid point A2. When Achilles reached point A2, the tortoise, in the mean time, had gone ahead to point A3. When Achilles reached point A3, the tortoise had gone ahead to point A4. In this way, as Achilles reached the point left by the tortoise, the tortoise had gone ahead in the intervening period. Thus, there were infinite such points left by the tortoise which had to be covered by Achilles to overtake the tortoise. Crossing infinite points left behind by the tortoise was not physically possible as infinite points being larger than the largest number, can never be covered by Achille . But Achilles practically overtook the tortoise. There was thus contradiction between theory and practice. This contradicting situation is called Zeno paradox.
The situation that Achilles would overtake the tortoise as correct, means Achilles had crossed infinite points left by the tortoise. Is crossing infinite points by Achilles correct when it is known that infinite means larger than the largest which is beyond our imagination?
Mathematical Examination Of Achilles Tortoise Paradox
This paradox is examined mathematically by considering Achilles runs the distance 1/2, in the mean time, the tortoise runs 1/4 ahead, Achilles crosses that ¼ left by the tortoise, the tortoise in the meantime runs 1/8 more so on and it continues up to infinity.
A1——————->A2———–>A3——–A5—->A6– -A ∞
The distance covered by Achilles can be written mathematically as
1/2 + 1/4 + 1/8 + 1/16 + ——————- so on up to infinity.
Inspection of these terms reveals that it is a Geometrical Progression whose terms are decreasing in magnitude endlessly. As second term 1/4 is less than first term 1/2, third term 1/8 is less than fourth 1/16 so on and has a common ratio r, ½ which is less than 1, such an infinite terms series is called an infinite convergent series and has a finite sum.
If a computer programme for adding these terms ab initio without using mathematical formula for summation of such series, is fed, the computer will take infinite time to solve it because it will calculate the magnitude of each terms which extend to infinity and then will add up. Calculation of magnitude of endless terms will take endless time and endless time means an unaccomplishing job. The computer will continue calculating endlessly. If the computer is commanded to state the intermittent status, it will display, “calculating.” That means, summation of infinite terms abinitio without use of mathematical formula, is impossibility for computer and is in conformity with prediction of Zeno.
Summation Of Convergent Infinite Series Using Mathematical Formula
However, such an infinite convergent series can be summed up by simple formula a/(1 -r) where a is the first term and r is mean ratio. In the above series, a = 1/2 and r = 1/2. Therefore its sum or distance covered is 1/2. [1/(1-1/2)] which equals1. In other words, impossibility of crossing infinite points by Achilles is solvable by a simple mathematical formula. Zeno paradox if examined from the point of view of performing infinite acts, is a physical impossibility but it is not so, from the point of view of mathematics. That leads to the concept, though infinity can not be determined yet under certain situations infinite acts can be performed resulting in finite output.
There are number of infinite convergent series which sums up to finite value. One example is given below.
Natural number, e = 1+1/2!+1/3!+1/4!+……………..up to infinity.
Here also infinite terms are summed up to give value e where ! is sign of factorial. All decreasing infinite series not necessarily adds up to finite sum. A series given below does not have a finite sum although magnitude of terms is decreasing.
1+1/2+1/3+1/4+1/5………………….so on up to infinity. It does not have finite sum as sum of its terms is infinity.
But series 1-1/2+1/3-1/4+1/5………so on up to infinity, is a convergent series and sums up to log 2. Here also infinite terms have been summed up to give finite result. Thus endless repetitive acts or infinite acts are possible under certain circumstances.
Infinitely Repetitive Number And Infinity.
Coming to the case of Achilles and the tortoise with regard to summation of the terms of the series by a computer, if the computer is commanded to give result at the stage where it was, the result displayed would be 0.999999999999999999999999—— so on and it will never be 1. But it is seen above that summation of this series results in one. Sum as 1 will only be achieved if the computer is let continue summation till infinite time. Achieving infinite time is impossible. In ideal case, sum or distance covered as displayed by computer would be
0.9999999 repeating 9 up to ∞.
D = 0.99999—————– up to ∞, where D is the total distance.
Or D = 0.9 + 0.0999999999999 —– up to ∞
Or D = 0.9 + D/10
Or 0.9.D = 0.9
Or Distance D = 1.
It is deduced from above that computer would display the distance as 0.999999999999 —– repeating 9 very large times. If infinity were achievable, the display would have been 1. Infinity thus corresponds to the number of times 9 repeats after decimal point to make the figure1.
Irrational Number And Infinity
It is not true with the figure 0.9999 —— up to infinity only to define infinity but it can also be extended to any irrational number. An example of irrational number 1/3 is given here. 1/3 is generally written as 0.33 or 0.33333 but its actual value is different. Strictly speaking, 1/3 is 0.33333——-with 3 repeating up to infinity. Again infinity can be defined here as the number of times 3 must repeat after decimal point to make the figure equal to 1/3.
Consider another case where say y = 3.535353535 ———up to infinity. On examination, it is clear that 35 is repeating up to infinity. If y is divided by 100, then y/100 = .0353535——–35 repeating up to infinity
Or y – y/100 = 3.5 or y = 350/99. Though 35 was appearing in 3.5353535 infinitely but it has a finite value equal to 35/99. Infinity can therefore be defined as number of times 35 should repeat after 3.5 to make the results ant figure equal to 35/99. In this way, infinity can be defined by each irrational number.
Defining Infinity From Mathematical Terms Extending Infinitely
Consider a fraction 2 + 1/(2 +1/(2+ 1 which continues infinitely. Here also, infinite operations are required to solve it and it is known performance of infinite operations is impossibility.Therefore, this fraction involving infinite terms can not be solved by the method of elementary additions.
However this can be written as y = 2 + 1/(2 +1/(2+ 1 continuing infinitely
Or y = 2 + 1/y since the terms of y are repeating infinitely and are equal to y itself.
Or y^2 – 2.y – 1= 0 where sign ^ indicates raised to power. This quadratic equation is solvable for y and gives values as 1+ square root of 2 or 1 – square root of 2. Value 1 – square root of 2 being negative, will have to be ignored as y can not have negative value. That leaves us with solution as y equal to positive value ie 1+ square root of 2. Therefore fractions involving infinite numbers are solvable.
Consider another case where x is less than 1 and is raised to power ∞.
Let y = x^∞ = x.x.x.x.x.—————up to infinity where x is positive and less than 1. As x is multiplied by x, it becomes x^2 which has value further less than 1. Again if x^2 is multiplied by x, resultant x^3 has value further less than 1. If it x is continuously multiplied by x up to infinity, value of product x^∞ will diminish to zero. Infinity can thus be defined as that power of a positive quantity less than 1 which reduces it to zero.
In mathematics (algebra, calculus, integration etc) operations of ∞ in certain cases are allowed and are possible and give finite result also. But operations of ∞ are not permissible as rule in all cases.
Negative And Positive Infinity
Negative and positive infinity is used in solving mathematical and physics problems. Negative infinity is different from positive infinity in the sense that it is assumed to extend in the direction opposite to the direction of positive infinity. Physically, positive infinity with regard to length of a line can be visualised as a line going vertically up endlessly and negative infinity as a line going downward endlessly. If magnetic field due to a current carrying line conductor of length l is to be calculated at a point situated at a perpendicular distance d from the centre of conductor, an infinite small length delta x is considered along the conductor to calculate its magnetic field, then the magnetic field due to length l is calculated by integrating the field with respect to distance varying from -1/2.l to +1/2.l. But if the length is infinitely long, the limits will change to from – ∞ to + ∞.
Limitation Of Length
While defining infinity earlier, cutting apple endlessly till size of each piece reduces to zero thus making the number of pieces infinite was considered. Mathematically, it is possible to cut the piece to the size as small as zero but laws of physics do not allow to reduce the size of a piece further below a fixed length. Below that limit, length is indivisible. This limiting length is Planck length which is equal to 1.6 x 10^-35 m. Therefore, physics does not allow division of length infinitely but stops further subdivision at Planck length. However, mathematics allows subdivision of length below Planck length even up to zero. Also in case of Achilles, Tortoise race, when the tortoise ultimately runs ahead by Planck length and Achilles thereafter covers this length, the tortoise is stopped by laws of Physics to run further. That point is the finishing point and the act of running ahead by the tortoise is not infinite. But what Zeno said was mathematical, not precluded by Physics. According to mathematics, acts of reaching the point left by the tortoise were infinite as is evident from infinite terms of the series. But Physics limits it to that number where the final distance to be covered by the tortoise is reduced to Planck length. Therefore length consists of small units of indivisible Planck length. Limiting the minimum achievable length to that of Planck, makes the number of acts finite. But mathematics does not have any such limitations and the acts can be infinite but such infinite acts are possible as the resultant convergent infinite series was summable.
Limitation Of Time Duration
Again coming to time arrow paradox, it is submitted, Zeno considered an instant (of time) is of zero second and infinite instants constitute the time of Arrow flight as of zero time. As instant has zero time variation, the arrow could not be seen moving in zero time variation and would appear stationary. Therefore at all such instants of time during flight, the arrow should be stationary. Physics does not allow division of time below the limit of Planck time, 5.39 x 10^-44 second. Duration less than Planck time will tantamount to zero duration of time. Therefore, in this case, time duration of instant can be minimised to Planck time but can never be zero. Thus time of flight is quantised or consists of smallest duration of time which is Planck time that is indivisible. Number of instants would be finite equal to time of flight divided by Planck time and will not be infinite. However, mathematics allows infinite instants where each instant corresponds to zero time. But Physics is based on laws of nature and allows motion of Arrow in an instant of time which can nit be less than Planck time.
Infinity And Metaphysics
By now it is abundantly clear, mathematics allows endless quantity bigger than the biggest one can imagine of. No matter infinity is endless, it can be visualised even in small number because, it is perpetual endless continuation of an act. For example, even between 0 and .1, infinite fractions exist like .01, .001, .0001, .00001, .000001,………….so on endlessly. Infinity can be visualised as extending endlessly beyond observable universe of 13.8 billion light years. Infinity can also be considered as endless points of zero volume confined to the volume of photon as infinite points of zero volume do not constitute any volume. Nevertheless, infinity pertaining to infinite points which can be accommodated in a photon is no smaller than infinity pertaining to infinite length extending beyond the observable universe. Thus infinity from the perspective of Infinite points those can be accommodated in a minuscule particle is as good an infinity as that from the point of of view of endless length beyond observable universe. Infinity does not have beginning and also does not have end, it is endless. It can not be eliminated, it remains same if it is multiplied with, divided by, added to, or subtracted from any quantity, it is always unaffected. No physical parameter can change it.
These are all the qualities which one and only one possesses and He is Almighty who is omnipresent, indestructible, unborn, immortal, unaffected by space or time. Infinity is synonymous with God. Nature represents it in the form of infinity nebula. Infinity will continue endlessly unstoppably, unfathomably, unmeasurably even when the time stops.
1 * John Wallis was an English mathematician who gave symbol ∞ to denote infinity. He also defined 1/∞ as infinitesimal. He also gave formula to calculate the value of pi.
2** Achilles was a great Greek warrior and had his heel as vulnerable part of his body. His mother dipped him with feet in her hands in holy river Styx to immortalise him. The feet which remained out of holy water became his vulnerable body parts where he was hit by Alexander Paris. Archives died of wounds at his heels and that is why vulnerable part is called, “Achilles’ heel.”
3*** Tortoise is a personification of sluggishness and slowness.
4. Image Courtesy https://en.m.wikipedia.org/wiki/List_of_planetary_nebulae#/media/File%3APlanetary_Nebula_M2-9.jpg
5. I have not dealt bigger infinity and smaller infinity from the point of view of set theory as propounded by Cantor.
Writer is an Electronics and Electrical Communication Engineering graduate and was earlier Scientist, then Instrument Maintenance Engineer, then Civil Servant in Indian Administrative Service (IAS). After retirement, he writes short stories and also on subjects, Astronomy, Mathematics, Yoga, Humanity etc.