Understanding Why 0.999... Equals 1 and the Basics of Calculus
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Chapter 1: The Nature of Numbers
Let's delve into the elegant reasoning behind why mathematicians assert that 0.999... is indeed equal to 1.
(Note: The image above serves as an infographic on transfinite induction. A detailed explanation of transfinite induction will follow in another discussion!)
We often refer to the real number line, but how is it perceived by mathematicians? They typically start by defining basic elements before building upon them. Assuming you are familiar with fractions, let's examine sequences of rational numbers. Consider the sequence (1/2, 1/3, 1/4, 1/5, 1/6, …, 1/n, …).
If you had to assign a value to this sequence, what would it be? Take a moment to think about it. What does a sequence converge to?
If you pondered this, you might conclude that, if you had to give this sequence a value, it would be zero, as it approaches 0 infinitely.
Now, suppose you chose a different number, like 0.001. That might seem reasonable for a while, but as you progress to the term 1/100000000000 = 0.00000000001 in the sequence, it begins to stray significantly from that estimate!
Next, let’s compare two sequences of rational numbers. It makes logical sense to say that two sequences are ‘equal’ if the gap between them approaches zero.
For example, let's compare the constant sequence of 1s—(1, 1, 1, 1, 1, 1, 1, 1, 1, …)—with the sequence (0.9, 0.99, 0.999, 0.9999, 0.99999…).
The difference between these two sequences approaches zero quite rapidly! The gap between their first elements is 1 minus 0.9 = 0.1, and for their second elements, it's 1 minus 0.99 = 0.01. The difference between their Nth elements is: 0.00…01 = 1/10^n, which is N-1 zeros followed by a 1.
The Foundations of Mathematics
What we are discussing here is formally known as the ‘completion of a metric space’. We take mathematical objects like rational numbers, which possess desirable properties:
- Rationals are straightforward to construct.
- The distance between rationals can be easily defined.
- We can create sequences of rational numbers.
However, the challenge arises because a rational number does not necessarily exist that corresponds to a sequence of rationals.
This is evident with numbers like the square root of 2; assuming such a number exists, we can demonstrate that it is not a rational number! Therefore, if we have a sequence of rationals converging to this ‘root 2’, the sequence would not equate to a rational number.
Completing a ‘metric space’ introduces new elements defined by sequences within our original metric space. This is precisely what we have accomplished! We define a real number via a sequence of rational numbers and assert that two sequences are equal if their difference approaches 0.
This allows us to perform calculus in this new space, enabling discussions about limits while ensuring that the limit exists. Engaging in limit-based mathematics solely within the realm of rationals is problematic, as the limit of a sequence of rational numbers may not always yield a rational result.
Isn't it astonishing? We initiate with something as basic as integers and build a complex realm of mathematical objects, from which we can derive information about the integers we began with.
Chapter 2: Exploring the Equality of 0.999... and 1
This video titled "Why 0.999... is Equal to 1" delves into the fundamental concepts behind this intriguing mathematical assertion.
In this video, "I'm Settling This Math Debate Forever (.99 repeating = 1)," the presenter addresses and resolves the ongoing debate regarding the equality of 0.999... and 1.