You would have first come across infinity when you are thinking of the universe and existence, how everything is either finite of infinite within its existence. You next encounter with infinity would have been in your GCSE mathematics lessons, when an odd symbol represented a never-ending solution to a complex problem.

Infinity, is known to be a concept within mathematics and philosophy, referring to a never-ending quantity. The word is derived from Latin origin, infinitas, with the literal translation meaning “unboundedness”.

Infinity in Mathematics

In mathematic subjects such as maths A-levels the term infinity is treated as though it was a number, as in an infinite number of terms. Despite this belief, infinity is not the same type of number as real numbers (i.e. 1, 10, etc.) and has been described by mathematicians as different things.

During the late 19 th and early 20 th centuries Georg Cantor, a German mathematician, formalised a variety of ideas that were related to infinity and infinite sets. He discovered that there are different types of infinite sets, leading to the concept of cardinality. For example, the set of integers is known to be countably in infinity, whereas the set of real numbers is uncountably infinite.

Sound confusing? Well infinity develops to new heights of confusion. You can have sets of elements that are Dedekind-infinite, which is when the set has a seemingly paradoxical quality, or has a subset of its elements that are able to be matched up on a one-to=one basis to each individual element within the set.

This paradoxical nature of infinity can be illustrated with a hotel with infinitely many rooms, where each is occupied by a guest but can still manage to accommodate a new visitor by moving each guest over one by one to the next available room.

The concept and nature of infinity has existed throughout numerous ancient cultures, with various ideas concerning its nature. For example the Isha Upanishad of the Yajurveda (4 th -3 rd century BC) states “if you remove a part from infinity or add a part to infinity, still what remains is infinity”. Indeed the Indian mathematical text Surya Pranjnapti (400 BC) classified numbers into three possible sets: enumerable, innumerable and infinite respectively.

The ancient Indians and also the ancient Greeks were unable to codify infinity in formalised mathematical terms and instead approached the subject as a philosophical concept. To correspond with Aristotle’s traditional view, Hellenistic Greek civilisation distinguished the potential infinity from actual infinity, for example instead of claiming that there are an infinity of primes, Euclid states that there are more prime numbers than contained in a collection of prime numbers.

Even Buddhist imagery has depicted infinity, with the deity Chenrezig being pictured holding a mala that is twisted in the middle, forming the figure 8, representing the endless cycle of existence, or infinity.

The infinity symbol, ∞, is often called the lemniscate, a word derived from the Latin lemniscus or “ribbon”. This symbol was introduced in 1655 by John Wallis following his derivation from the Roman numeral1000, which was in turn derived from the Etruscan 1000 numeral, which meant many. Indeed infinity does not only define a limit but is also a value in the affinely extended real number system.

Infinity is used with mathematics, physics, geometry, computing, cosmology, philosophy and even the arts. The best thing to remember is that the lemniscate represent a boundless sum. So if you are faced with ∞ in a mathematical problem, you will understand.

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