Ivan Gil G. Oledan 202007944 Bachelor of Science in Accountancy Research Report 1: Understanding Infinity Infinity is described as a concept that pertains to something smaller or larger than the indicated number, it is usually referred as something that is boundless or without any limit as to how large or small the number system can be, but is it? Is it really bound to nothing as endless as it would seem to be? This concept is widely used in both fields of Physics and Mathematics. Should there be a one-to-one correspondence between a set and a subset of itself, the set of numbers listed can be identified as infinite. Taking as an example is x+1 = x, this will only happen if x is identified as an infinite number. Moreover, adding 1 to the equation won’t result in budging the original number. Another example in showing an infinite number is 1/x, when x→0. Well, to understand the concept of infinity better, let’s say x as a representation of a real number wherein -∞ < x < ∞, the negative infinity is less than the real number whereas the positive infinity is greater than the real number. Infinity focuses on two fields, the first one is the infinite divisibility or infinitesimally small numbers, wherein it talks about an object or a thing that can be divided into smaller or teensy bits of parts. Usually, it refers to mathematical analysis as to lines, surfaces, and other continuous objects. The other field of infinity pertains about infinitely large collections or infinitesimally large numbers, it just merely talks about the contrary on what was defined on infinitesimally small numbers. There is a need to understand the concept of infinity with regards to infinitesimally small or large numbers, for in that way we are able to identify and know such phenomenon, may it be in Mathematics or Science in general. It gives us an idea that something such as a sequence could be prolonged for anyone as he likes. It is infinite in the sense that it is bound to nothing. An example that best describes this idea is our number system. The order of our natural numbers goes as “1,2,3,4,5, and so on” this is considered as endless, and for that we can say it as an infinite sequence. We can go as large as we can, but this sequence will never reach to termination. Second is the concept of nanoseconds or the one billionth of a second. Small as it is, we can really think that seconds can go further small in partitioning it farther. This is usually done in the field of Science wherein nanosecond is divided into infinite number of infinitesimal time intervals. 1|Page Ivan Gil G. Oledan 202007944 Bachelor of Science in Accountancy Another example is the fraction 1/3. When the concept of division is performed, we arrive to the quotient of 0.3333…, where the number “3” is repeated indefinitely. By this way, it tells us another idea that shows how infinities could really work. Last example is the basic idea of lines. We are certainly aware that lines are made up of infinite number of points. It extends infinitely far in two different opposing directions. Having zero width, zero height, but an infinite length. The totality of the points that consists in the line is in any case infinite. The concept of infinity is definitely important because without it we won’t be able to identify and know certain phenomena, may it be in Mathematics or Science in general. In Mathematics to be specific, without infinities we won’t be able to understand Calculus and Set Theory better. Without infinities in Geometry, we will also not be able to analyze infinitely close points, and the same instance also with Inequalities, we will fail to analyze the effect of an infinitely small change as well as other tracks where the effects of an infinitely small change can be analyzed. With all those things being said, it only shows how the concept of infinities work, it touches a wide scope of everything especially in the field of Mathematics. However, should infinities really stand as it is, to go without further termination? As quoted by Aristotle with regards to this, “Nor does this account of infinity rob the mathematicians of their study; for all that it denies if the actual existence of anything so great that you can never get to the end of it.” In today’s new era of Mathematics, where knowledge can be acquired better, technologies and computer being more advanced, there is a good reason to believe, that our inclination to completed infinities may once overcome, to execute calculations of tremendous length will make inevitable the distinction between tremendously hefty calculations and such that are never intended to cease. It gives us the idea of an enriched knowledge towards studies that are way better understood that would create more help and make us believe on phenomena of certain works in field. 2|Page Ivan Gil G. Oledan 202007944 Bachelor of Science in Accountancy References: The Use of Infinity in Mathematics. St. John’s College, Annapolis, 12 November 2004. https://math.nyu.edu/faculty/edwardsd/stjohn.pdf. Accessed 12 May 2021. Infinity in Mathematics and Logic. Encyclopedia of Philosophy. Encyclopedia.com. 16 April 2021. https://www.encyclopedia.com/humanities/encyclopediasalmanacs-transcripts-and-maps/infinity-mathematics-and-logic. Accessed 12 May 2021. Infinity. BYJU’S. https://byjus.com/maths/infinity/. Accessed 12 May 2021. Infinity. Brilliant.org. https://brilliant.org/wiki/infinity/. Accessed 12 May 2021. 3|Page