Is ‘0.99999…’ Really Equal To ‘1’?

If you are required to assign a precise value to the infinite decimal ‘0.99999…’, what would you do? Mathematical intuition says that it could be approximately equal to ‘1’. But if you are the curious type like me, the following series of questions would arise:

1. Is ‘0.99999…’ really just approximately equal to ‘1’?

2. If so, why?

3. If not, why?

4. Besides, what is the precise value of ‘0.99999…’?

In this essay, I try to investigate answers to these questions. To begin, I approach the problem mathematically to figure out what is going on. Then, I dive deeper into the philosophical challenges hiding behind the mathematical constructs.

At the end of this article, you will gain a broader perspective of how this problem showcases the lively, evolutionary nature of mathematics.

This essay is supported by Generatebg

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Simple Algebraic Proofs

The first thing to note about ‘0.99999…’ is that it is an infinite decimal. And it is definitely not the only infinite decimal in the world of numbers. Consider the fraction 1/3:

This is a fairly undisputed standard result. Sure, if one needs to approximate to the n-th decimal place, a ‘4’ shows up at the corresponding point. But if one is not interested in stopping at the n-th decimal place, then we get recurring 3’s infinitely. Let us now multiply the above expression by 3 on both sides.

That is an interesting result. It appears that ‘1’ is precisely equal to the infinite decimal ‘0.99999…’. Could there be something fishy going on with our approach?

Let us change the starting point to something more relatable to our problem. We shall assume that the value of ‘0.99999…’ is unknown. Let this unknown value be x. Then, we get the following expression:

After multiplying both sides by 100, we get the following result:

This expression can further be simplified as follows:

Again, in this case, we arrive at the same conclusion as before.

Summing up, we just proved algebraically that ‘1 = 0.99999…’. But hang on! There is at least one more factor we need to consider before we can celebrate.

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A Potential Paradox

Let us say that our algebraic proof is undisputed, and we all agree that ‘1 = 0.99999…’. Then, the following expression would also hold:

This, in turn, would lead to the following question:

For our algebraic proof to hold, we require the value of this expression to be zero. In other words, ‘1’, ‘0.99999…’, and ‘1.00000…’ are (by requirement) three different ways of representing the same value. They are all one and the same.

However convenient this definition might be, if you are the curiously rigorous type, you would have a lingering feeling that there would definitely be a residual remaining after an infinite number of zeroes on the right-hand side of the following expression (resolved from the previous one):

So, how do we handle this?

Infinitesimals to the Rescue

It turns out that mathematicians who were faced with profound problems such as calculus used such expressions to define infinitely small numbers called infinitesimals.

An infinitesimal is a quantity that is closer to zero than any standard real number, but that is not zero. If this sounds vague and non-mathematical, rest assured that the history of infinitesimals has been one of the most controversial in all of mathematics.

For a long time, infinitesimals were not accepted as valid mathematical entities. But later on, mathematicians invented Nonstandard analysis, where infinitesimals are treated rigorously with the help of systems such as the hyperreal number system. If you are interested in the full story of calculus and/or infinitesimals, check out the links at the end of this article.

But for now, even though infinitesimals give us an answer of sorts to the question ‘(1.00000…) — (0.99999…) = ?’, we still don’t appear to be much closer to calculating the precise value of ‘0.99999…’. 

Surely, mathematics must have figured out a way to handle this, right?

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Is ‘0.99999…’ Really Equal to ‘1’?

The answer to this question is:

It depends…

To anyone operating outside the world of mathematics, this might come across as a shock. Mathematics is supposed to be the world of ultimate precision and clear-cut definitions. Yet, how can there be uncertainty as to how the precise value of ‘0.99999…’ is calculated?

Well, the question we need to be asking is not how to calculate the precise value of ‘0.99999…’. We should instead focus on what we define ‘0.99999…’ as!

If we define the definite whole number value of ‘0.99999…’ as something other than ‘1’, then our entire algebraic system starts to break down. On the other hand, if we define ‘0.99999…’ as ‘1’, there are no drawbacks in the realm of real numbers.

When we require ‘infinitesimal’ precision, then we turn to the hyperreal number system (for example), and the definition of ‘0.99999…’ is no longer necessarily ‘1’.

The Grey Areas of Mathematics

While concepts like algebra are perfectly and indisputably defined, there exist quite a few grey areas until this day in mathematics. This very challenge of ‘0.99999…’ just showcases one of those grey areas where everything is not clear-cut.

It is important to realise that mathematics is not a complete science that has been fully solved by former mathematicians. It is a living and growing field where boundaries are pushed every single day. Many of these boundaries are deeply related to the limits of human logic.

At the limits of human logic, things tend to get fuzzy. Mathematics tries to embrace the fuzziness whilst trying to achieve clarity at the same time. In this regard, some of the most fundamental building blocks of mathematics are fuzzy blocks.

The fact that human beings have built such powerful mathematical systems out of fuzzy fundamental blocks is a testament to what we are capable of. With evolution in play (both technical as well as biological), who knows what the limits are?

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Credit: The proof in this essay was inspired from the work done by Mr. Jordan Ellenberg.

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Further reading that might interest you: The Thrilling Story Of Calculus and What Really Happens When You Invent Infinite Infinities?