Irrational Numbers - COSMOS

Bernabé 2In which both a and b are integers. Suppose that the rational fraction, a/b, is in its lowest terms,for this reason, we know that neither a nor b are even. Then square the equation above andsimplify to get:2 = 2 2 , 2 2 2 .The term 2b 2 denotes an even integer; therefore a 2 is an even integer, and hence a is an eveninteger. Say a = 2c, in which c is also an integer and replace a by 2c in the equation a 2 = 2b 2 toget2 2 , 4 2 , 2 .Now, the term 2c 2 denotes an even integer; therefore, b 2 is also an even integer, and hence b is aneven integer. The conclusion now states that both a and b are even integers, whereas a/b waspresumed to be simplified in lowest terms. This contradiction indicates that it is not possible toexpress √2 in the rational form a/b, and therefore helps us conclude that √2 is an irrationalnumber (Numbers: Rational and Irrational, 42).The number pi, also an irrational number, serves as the most important irrational numberin the world because it is utilized to find the circumference of a circle, and find the area andvolume of a sphere or square. Pi, along with the square root of two, are often used in theconstruction of buildings in our society. Pi was originally founded as an irrational number bymany mathematicians. One particular mathematician that provided a proof with the use ofcontinued fractions was Johann Heinrich Lambert. Lambert proved pi to be irrational by thefollowing continued fraction:

Bernabé 4Thus, the other terms are equal to the sum of two integers. Therefore they sum to an integer andin which we will call the remaining terms R.1! 1! 1 2! 1 3! ⋯ 11 1 1 2 1 1 2 3 ⋯ 11 1 1 1 1 … 11 1 1 1 1 1 ⋯1 1 11 1 11 1 1 1 Reflecting to the original definition of R, we can now see that since q is positive, so is R. Thus,stating R is an integer between 0 and 1 , contradicts our original assumption of e being a rationalnumber. Thus, e is an irrational number (Irrational Numbers, 57).Although irrational numbers are difficult to comprehend, they serve in a major portion ofour modern mathematical industry. Irrational numbers are often used in the daily lives of oursociety whether it is in the creation of buildings, chemical equations, or basic physics. Irrationalnumbers effectively help modern society to understand the workings of the natural and future ofour world.