Mathematics has established itself as a foundation for all quantifiable values. A lesser known concept behind this science is that it, essentially, revolves around all things infinite. The idea of infinity appears to be an inconceivable notion that many have a hard time wrapping their heads around. What many fail to understand is that we are surrounded by infinity in the form of everyday mathematics.
An apt example that we could take into consideration would be the Fibonacci sequence. This sequence was made during the 12th century by a man named Leonardo Fibonacci. The Fibonacci numerical sequence built the foundation for the mathematical relationship between Phi (Golden Ratio).
This recursive formula of begins with a 0 and 1. Every new number in the sequence is being the addition of the two numbers before it. Hence, creating a formula:
Fn+1= Fn + Fn-1
which leads to the formation of the series:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 …
As per observation, we can determine that this series is classified as an infinite series. In other words, it is a series that continues a similar pattern until it reaches infinity.
Alongside its tendency to continue up until infinity, we also observe other infinite limits to this sequence. Its ratio is known as ‘Phi’ or the Golden Ratio. Basically, Phi is a number that is given the value of an approximate 1.1618 (1+5√2). By bringing in limits, we obtain evidence for how the ratio of the two successive pairs of numbers tends to converge to this number.
The limit carries on until an infinite number of ratios obtained. Hence, this further proves how, in this case, the convergence, as well as the divergence of the series, both revolve around the concept of infinity. This can be proven further through its vast prevalence throughout nature in the form of flower petals, or even our DNA!
Hilbert’s Hotel Paradox
When we are talking about the countable infinity in Mathematics, the Fibonacci sequence is not the only series that converges to infinity.
The Hilbert’s Hotel Paradox provides an interesting insight into layers of countable infinity. The paradox discusses the manner of accommodating an infinite number of guests into a hotel with an infinite number of rooms.
The Three Situations
The first situation is that of a finite number of guests arriving, e.g. 3. To ensure they get a room in the infinite hotel, each person is asked to move from room n to room n+3.
This is followed by a situation wherein a bus filled with an infinite number of guests arrives at the hotel. To accommodate everyone, each existing guest is asked to move from room n to room 2n, thereby clearing all odd numbered rooms. Since there is an infinite number of odd and even numbers, every guest is accommodated.
The third situation presents an infinite number of infinitely long buses arriving at the hotel. Each bus consists of an infinite number of guests.
To accommodate all of them, each existing guest is asked to move from room n to room 2n, which, as previously mentioned, results in the vacancy of all odd-numbered rooms. Subsequently, a prime number is assigned to each bus, e.g. number 2 for the first bus, 3 for the second bus and so on. Despite there being an infinite number of buses, it can be assured that there shall never be a lack of prime number as, in 300 B.C., Euclid proceed the existence of an infinite number of prime numbers.
Based on these numerical designations, each passenger shall be allotted a room based on the formula:
where B = number assigned to passenger’s bus and n = passenger’s seat number.
With this formula and allotment, each of the infinite numbers of passengers will be able to get a room in the Hilbert Hotel.
It is in this manner that mathematical concepts and ideas have explored the idea of infinity. Generalizations and formulas indirectly serve the purpose of giving an infinite set of values. Every general formula is applicable until infinity. It is in this precise manner that the predominance of infinity in Mathematics can be postulated.