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Opposite Of
Infinity
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Infinite triangles right angle triangle and calculation
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What Is The Opposite Of
Infinity?
Akash Peshin
No, the answer isn’t zero. Infinity
is the largest number there is, so the opposite of infinity would be the
smallest number there is.
Zero would mean nothing, so what
we’re looking for is a number just greater
than zero.
However, as we’ll find out,
determining this number isn’t as simple as pointing to the number 1.
Infinities are
weird
Infinity has baffled humanity since
antiquity. One must realize that infinity is not a concrete number, but rather
an idea; it exists only in abstraction.
Infinity cannot be a concrete
number, say, x, because we can, by the logic of addition, add 1 to x and create
a new infinity. We can then add another 1 to create a larger infinity.
We can, in fact, add infinity to
infinity to create perhaps the infinity of all infinities, but then we can add
to this infinity another 1 and … you know the drill.
The microscopic
realm isn’t any different. The opposite of infinity is called infinitesimal,
and its nature is equally bizarre.
Unlike whole
numbers, real numbers aren’t rigid. Their splintered nature allows us to find
and create infinite numbers between any two numbers.
A number can be
combined as many times as it can be divided. There could be a hundred numbers
between 0 and 1, from 0.01-0.99, or even millions, one just has to add zeroes
after the decimal point — divide it increasingly to create new numbers. So,
while 0.00000000000000001 seems infinitesimal, one can just divide it by 10 to
create a new infinitesimal — 0.000000000000000001.
So, infinitesimal, like infinity,
exists only in abstraction, yet its uncertain nature is very disconcerting not
just for mathematicians, but also physicists.
Infinitesimal
Errors
Mathematics is the language we use
to express our ideas in physics, so an inconsistency in math translates to an
inconsistency in physics, in our knowledge of nature – of reality.
The inconsistency arises from our
uncertainty of the value of infinitesimal, which has been used to derive many
crucial formulae.
In fact, an entire branch of
mathematics is based on infinitesimal, without which, progress in physics would
have been sluggish.
One formula I can think of is the
area of a circle. Kepler calculated the area of a circle by dividing it into
triangles.
The area of the circle, therefore,
would be the sum of areas of each triangle.
A circle can be divided into four
triangles with two diameters, however, the sides of these triangles do not
approximate the curves properly, excluding some space, so the area calculated
is erroneous.
To reduce this error, we can draw more diameters to
create more triangles of shorter sides. However, the error is reduced in
this way, but it is still finite.
So, we further divide the circle into more and more
triangles until there is no space left excluded. However, to eliminate this
error entirely, we must divide it into an infinite number of triangles.
Now, because a line can be interpreted as part of a huge
circle, we can say that our circle is composed of infinite lines, which
are approximated by the infinitesimal bases of our infinite triangles.
One may notice that the sequence of triangles is
reminiscent of a Chinese fan. All the triangles occupy an equal area, but we
can convert the fan into a large right-angled triangle by distributing or
stretching this area.
Their perimeters have changed, but the total area
remains the same. The height of this triangle, with the summit being the
circle’s center, is the length of the fan — the radius of our circle, and the
base, the circle’s circumference.
The area is ½ times base times height, which is ½ times
r times 2πr, or πr².
This is, of course, the correct answer, but the
outcome is still erroneous.
The bases must be truly infinitesimal,
so even though Kepler draws really, really, really thin triangles, we know that
he could have
drawn more.
The moment he stops drawing triangles, he leaves behind
spaces, albeit really, really, really small ones, but still, finite.
The curves are then inadequately approximated and
the calculation of the area of the circle is slightly erroneous.
While this might make a mathematician
uncomfortable, the majority “ignore” such differences for, as we have seen,
the results procured aren’t incorrect.
Calculus, either invented or
discovered by Leibniz and Newton independently, was also founded on
infinitesimals.
That branch of mathematics is
concerned with change, with curves.
For instance, when we integrate
a function, we essentially calculate the area under the curve it draws.
However, like calculating the
area of a circle, we calculate it by approximating the curve with
infinitesimally thin rectangles. The thinner the rectangles, the smaller the
error.
The area of one rectangle is the
product of its length — the value on the Y-axis at that point on the curve and
its breadth – the infinitesimal unit we call ‘dx’.
We calculate the area of each
rectangle and sum them to determine the area under the curve.
This is very useful in physics; for
instance, the area under the velocity curve of a body gives the value of
its displacement, but then shouldn’t the outcome be erroneous, just like the
area of the circle was?
This ineradicable, insoluble
problem bothered mathematicians for two centuries following the advent of
calculus until the concept of limits was refined.
Limits were implicit in Newton
and Leibniz’s work, but they were modified and redefined later in the early
1800s.
The new ideas were
mathematically rigorous and consistent. The details are beyond the scope of
this article, but limits allowed mathematicians to finally get rid of
infinitesimals for good.
What we still haven’t gotten
rid of is the absurdity that is infinity.
Akash Peshin is an Electronic Engineer from the University of
Mumbai, India and a science writer at ScienceABC. Enamored with science ever
since discovering a picture book about Saturn at the age of 7, he believes that
what fundamentally fuels this passion is his curiosity and appetite for wonder.
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Infinity
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