Uses of Infinity in Mathematics

In your college days you must have heard about infinity in mathematics. So what is it exactly, A number, a concept or something else. For someone who is not familiar with infinity it looks like a sideways eight (Its symbols resemble of sidewise 8).

According to Wikipedia it is defined as: Infinity is an abstract concept describing something without any limit and is relevant in a number of fields, predominantly mathematics and physics. The English word infinity derives from Latin infinitas, meaning "being without finish", and which can be translated as "unboundedness", itself calqued from the Greek word apeiros, meaning "endless"

Practically you can think that the earth have finite number of grains of sands, but its impossible to count by humans. Although they are countable but human cant do that. So this is the infinity. It’s not too big, it’s just uncountable.

So you might wonder that what is infinity used for in maths?
Infinity is not fictitious, it exists. Infinity is very useful concept in math and physics. The conception is beneficial in arithmetic, calculus and almost every field of physics.

For example in calculus if you like to look at the slope of a point on a curve, you have to take two points that are infinitely near to each other (1/∞ apart). Another example is in range theory if you add the numbers 1+1/2+1/4+1/8..... It gets nearer and nearer to the number 2, however never reaches it. But the concept is if you go to the infinity of calculation, the sum will be 2 exactly. In mathematics you investigate how a function behaves when it goes ∞ and -∞ because it past all of the ups and downs.

Limits at Infinity

Vocabulary

limit at infinity, infinite limit, horizontal asymptote, indeterminate expression, infinite limit at infinity

Objectives

                1. Explain what is meant by equations such as  or  .

                2. Recognize indeterminate expressions.

                3. Sketch the graph of a function with specified limit conditions.

                4. Find the value of certain limits at infinity and infinite limits.

                5. Determine the horizontal and vertical asymptotes of a given function.

Lecture Outline

In this section we consider the end behavior of functions. That is, we examine the behavior of  as  increases without bound or decreases without bound.

Definition of Limit at Infinity

Let  be a function which is defined on some interval (  ). Then  means that the values of  can be made arbitrarily close to  by taking  sufficiently large.

Often when  is a rational function, the graph of  will behave like a line as  increases or decreases without bound. When this happens, the line which the graph of  approaches is called a horizontal asymptote .

Infinity

Infinity (symbol: ∞) refers to something without any limit, and is a concept relevant in a number of fields, predominantly mathematics and physics. Having a recognizable history in these disciplines reaching back into the time of ancient Greek civilization, the term in the English language derives from Latin infinitas, which is translated as "unboundedness".

In mathematics, "infinity" is often treated as if it were a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as the real numbers. In number systems incorporating infinitesimals, the reciprocal of an infinitesimal is an infinite number, i.e., a number greater than any real number. Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called cardinalities). For example, the set of integers is countably infinite, while the set of real numbers is uncountably infinite.

Long-tailed Duck

Adults have white underparts, though the rest of the plumage goes through a complex moulting process. The male has a long pointed tail (10 to 15 cm) and a dark grey bill crossed by a pink band. In winter, the male has a dark cheek patch on a mainly white head and neck, a dark breast and mostly white body. In summer, the male is dark on the head, neck and back with a white cheek patch. The female has a brown back and a relatively short pointed tail. In winter, the female's head and neck are white with a dark crown. In summer, the head is dark. Juveniles resemble adult females in autumn plumage, though with a lighter, less distinct cheek patch.

Their breeding habitat is in tundra pools and marshes, but also along sea coasts and in large mountain lakes in the North Atlantic region, Alaska, northern Canada, northern Europe and Russia. The nest is located on the ground near water; it is built using vegetation and lined with down. They are migratory and winter along the eastern and western coasts of North America, on the Great Lakes, coastal northern Europe and Asia, with stragglers to the Black Sea. The most important wintering area is the Baltic Sea, where a total of about 4.5 million gather.

The Long-tailed Duck is gregarious, forming large flocks in winter and during migration. They feed by diving for mollusks, crustaceans and some small fish. Although they usually feed close to the surface, they are capable of diving to depths of 60m (200 feet).

In North American English it is sometimes called Oldsquaw, though this name has fallen out of favour under influence of negative connotations of the word squaw in English usage. Some biologists have also feared that this name would be offensive to some Native American tribes involved in the conservation effort. The American Ornithologists' Union (2000) stated that "political correctness" was not sufficient to change the name, but "to conform with English usage in other parts of the world", it officially adopted the name Long-tailed Duck.

The males are vocal and have a musical yodelling call Ow, ow, owal-ow.

Long-Tailed Duck

Long Island, NY, March 2007. By Tony Phillips.

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The Long-tailed Duck is one of the species to which the Agreement on the Conservation of African-Eurasian Migratory Waterbirds (AEWA) applies.