The abstract idea of infinity is used to define something as limitless or without limits. Relevant applications can be found in the fields of mathematics, cosmology, physics, computers, and the arts.
1. The Infinite Symbol
Infinity has its own special symbol: ∞. The symbol, sometimes called the lemniscate, was introduced by clergyman and mathematician John Wallis in 1655. The word "lemniscate" comes from the Latin word lemniscus, which means "ribbon," while the word "infinity" comes from the Latin word infinitas, which means "boundless."
Wallis may have based the symbol on the Roman numeral for 1000, which the Romans used to indicate "countless" in addition to the number. It's also possible the symbol is based on omega (Ω or ω), the last letter in the Greek alphabet.
The concept of infinity was understood long before Wallis gave it the symbol we use today.
Around the 4th or 3rd century B.C.E., the Jain mathematical text Surya Prajnapti assigned numbers as either enumerable, innumerable, or infinite. The Greek philosopher Anaximander used the work apeiron to refer to the infinite. Zeno of Elea (born circa 490 B.C.E.) was known for paradoxes involving infinity.
2. Zeno's Paradox
The paradox of the Tortoise and Achilles is the most well-known example of Zeno's paradoxes. An ancient Greek paradox is a tortoise who dares the Greek hero Achilles to a race, on the condition that the tortoise is given a little advantage. The tortoise reasons that he will come out on top because by the time Achilles catches up to him, he will have already gone a little further.
Picture yourself pacing across a room by taking two steps instead of one. Now that you've travelled halfway there, you still have half to go. A quarter is the next unit of measurement after the half. We've come three-quarters of the way but still, have a ways to go. Followed by 1/16th, then 1/8th, and so on. You can approach the far side of the room indefinitely, but you'll never make it there. Or at least you would if you walked forever.
3. Pi as an example of Infinity
The number pi, or, is another well-known illustration of infinity. Because it would take too long to write down the actual value of pi, mathematicians instead employ a symbol to represent it. There is no limit to the number of digits in Pi. It's hard to put down the whole number, thus it's usually rounded to 3.14 or 3.14159.
4. The Monkey Theorem
The monkey theorem may be used as a conceptual framework for thinking about infinity. If you give a monkey a typewriter and an unending length of time, the theory predicts that the animal would, at some point, produce a version of Hamlet written by Shakespeare.
Mathematicians consider the theorem as proof of how uncommon some events are, contrary to the interpretation that some people have of it, which is that it suggests everything is conceivable.
5. Fractals and Infinity
An abstract mathematical entity known as a fractal, fractals are often employed in art and to replicate natural events. The vast majority of fractals are not capable of being differentiated when written as mathematical equations.
This indicates that a picture of a fractal may be zoomed in on, allowing for the discovery of previously unseen details. In other words, the size of a fractal may be multiplied indefinitely.
One fascinating example of a fractal is the snowflake known as the Koch snowflake. A triangle with equilateral sides serves as the basis for the snowflake. In each iteration of the fractal, the following are true:
1. Each section of the line is cut into thirds such that they are all equal.
2. The base of an equilateral triangle is the part at the centre of the triangle, which points away from the triangle.
3. A cut is made through the portion of the line that previously served as the base of the triangle.
It is possible to carry out this procedure an endless number of times. The snowflake that was produced as a result has a definite perimeter, despite the fact that its area is limited.
6. Cosmology and Infinity
The study of the cosmos as well as theorising about infinity is what cosmologists do. Do the reaches of space never come to a close? This remains an unresolved question.
Even if the physical universe as we know it is contained by some kind of barrier, the multiverse hypothesis should still be taken into consideration. That is, it's possible that our world is only one among an endless number of others like it.
7. Different Sizes Of Infinity
Although it has no limits, infinity may be measured in a variety of ways. Both the positive numbers, which are those that are bigger than 0, and the negative numbers, which are those that are smaller than 0, may be thought of as infinite sets of the same size.
Nevertheless, what results do you get if you merge the two groups? You will receive a set that is twice as huge. Another illustration would be to think about all of the even numbers (an infinite set). This is an illustration of an infinity that is one-half as large as all of the whole numbers.
One more illustration of this would be just adding one to infinity.
The number ∞ + 1 > ∞.
Dividing By Zero
In standard mathematics, you should never divide a number by zero. In the grand scheme of things, the division of 1 by 0 does not make sense and cannot be defined. It goes on forever. It's a code for an error. Having said that, there are times when this is not the case.
In extended complex number theory, the not-automatically-collapsing form of infinity known as 1/0 has been given a definition as a form of infinity. To put it another way, there are several approaches to solving mathematical problems.
Gowers, Timothy; Barrow-Green, June; Leader, Imre (2008). The Princeton Companion to Mathematics. Princeton University Press. p. 616.
Scott, Joseph Frederick (1981), The mathematical work of John Wallis, D.D., F.R.S., (1616–1703) (2 ed.), American Mathematical Society, p. 24.
Helmenstine, A. M. (2017, November 13). 8 Facts About Infinity That Will Blow Your Mind. ThoughtCo. https://www.thoughtco.com/infinity-facts-that-will-blow-your-mind-4154547