Infinity – the final frontier of mathematics

In mathematics, we often come across statements such as “x tends to infinity” or “x -> ∞”. Most of the students read it and proceed to solve questions as per their understanding and defined process, tips, and tricks. Have you ever thought about infinity as concept? A great mathematician Rene Descartes (of Descartes rule of roots of polynomials) said that infinity is beyond the understanding of humans because we are finite. Whatever it meant the idea was to show that the very concept of infinity touches upon the existence of God.

Infinity – venture into unknown

Mathematically, infinity refers to a very large number, larger than any number you can think of. As soon as you think of any number, it is always possible to add a number and make it larger. There is no end to it. Hence statements such as n = ∞, or x takes the value of infinity really have no meaning. The correct statement is “n tends to infinity” which means n is a large number larger than any other number.

Essentially infinity refers to idea of unlimited such as space, time etc. Remember the quote of Einstein, “Only two things are infinity; the universe and the human stupidity. And I am not sure of the former”. At least, we humans have something in infinite supply.

Infinity as a concept creates some interesting situation in mathematics.

2 concentric circles with different sizes have the same number of points on the circumference. How? Draw a line from the centre intersecting the two circles. For every point common to the line and the bigger circle, you have a corresponding point on the smaller circle. Hence shouldn’t the number of points be same. There is no way to refute this.

2 sets of unlimited objects have the same size. Let’s say a set A is set of natural numbers while set B is set of real numbers between 1 and 2. Since both the sets have infinite number of elements, they have the same size. Who can refute this too?

We know that 2 parallel lines never meet. Well, here comes infinity… the place where they meet, far away from the madding limits of mathematics. Similarly asymptotes meet the hyperbola at infinity. Looks like infinity must be a nice place for meeting people torn apart by the limits of human mind. Our Bollywood too has adopted the concept of humanity where the young couple suffocated by traditions and limiting values swear that they will meet in next birth. This is the translation of mathematical infinity into something simpler to understand.

There are many more baffling examples.

You can create infinity too at some situation

We have infinity in our grip is when we draw the slope of tangent to a curve. Just draw a vertical line touching a curve and you have created a tangent with slope of infinity.

In a quadratic equation, put the coefficient of x2 equal to 0 and you have just created an equation with one root as ∞. Off course the second root is –C/A. If you put both the coefficient of x2 and x equal to 0, you have created a quadratic equation whose both roots are ∞.

Infinity, by its very nature, is unknown. The only person who ever tried to unravel it was none other than the extraordinary mathematician Ramanujan on whose life Hollywood made Oscar-winning movie called “The man who knew infinity” based on a book by the same name by Robert Kanigel. We can only try to understand a miniscule part of the infinite.

 

~ Pankaj Priyadarshi

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