Students remain mind-boggled and confused with the mathematical fact that 0.9999 = 1. Many pre-calculus arguments strengthen the truth behind this heavily debated mathematical problem. With the seemingly infinite distance between 0.999 and 1, one can understand the difficulty with its comprehension.
The repetition of 9 proves to close the distance between the two figures. As every 9 is added at the end of 0.999…, the number gets nearer and nearer to 1, eventually leading to its convergence.
The Definite Convergence of 0.999 to 1
Mathematics is a generally confusing topic for most high school students, especially the truthful 0.9999= 1 equation. Most people assume that there would be a minuscule number that would hinder the two from being equals. If one wants to understand why the seemingly infinite numbers of .9 equate to 1, one needs an understanding of long decimal expansions.
Terminating and finitely-long decimals are very direct in manner. They are fractions that have a power of ten as their denominator. 0.25 or the fraction 25/100 equates to 1/4, 0.9 equates to 9/10, these terminating and finitely-long decimals can serve the purpose of extending non-terminating decimals.
Meanwhile, infinitely long decimals denote a compressed finitely-long decimal sequence, meaning that 0.9999 and so on represents a succession of 9s after the decimal point. The first number is 0.9, the following is 0.99, and the next is 0.999, and it continues. For every following number, one adds a number 9 at the end of the preceding figure.
With that said, 0.99999 and so forth is not an overwhelming and never-ending sequence of numbers but a contribution of many terminating decimals. The addition of each 9 to the whole number closes its distance to 1 every time as the interval between two different numbers is just the difference between the bigger number and the smaller number.
As the overall figure gets nearer and nearer to 1, one can see that every integer closer to 1 denotes a finite amount of 9s after a decimal point that is nearer to closing its distance to 1. One can see that as the amount of 9s after the decimal point gets longer and longer, the sequence converges to 1 as the limit of its sequence. (Source: Business Insider)
The Skepticism Behind The Equation
The consistent doubtfulness of the students of the 0.999 = 1 truth despite the many arguments proving its correctness has a historical basis. Experts have studied this phenomenon and determined various underlying causes behind the skepticism of the students.
Some students tend to perceive 0.999 as a figure with constant change, as they see the number as getting nearer and nearer towards 1, with the space between the two figures never diminishing. This perception can be associated with Aristotle’s concept of potential infinity that states that 0.999 and so on is only a potentially infinite decimal expansion.
Researchers have identified two categories of the students’ reasoning of acceptance or rejection. The Sameness by Proximity and the Infinitesimal Difference have different conclusions even though they use the same line of reasoning. While Sameness by Proximity concludes that the values can be the same as the difference is infinitely tiny, Infinitesimal Difference states that the values will always be different because of the infinitely small distance between the values. (Source: The Mathematics Educator)