Mathematicians can’t decide whether 0.999… is equal to 1.

Some experts argue that the rational number 1⁄3 corresponds to the decimal number 0.333, and that it can be multiplied by 3 to obtain 0.999…. Since 1⁄3 × 3 = 1, then 1 and 0.999… must be the same. There appears to be a clear winner in the argument. But let’s be calm…

Strange as it may seem, countless debates in classrooms, lecture theaters, and online forums have revolved around the question of whether 0.999… is equal to 1. Teachers, professors and Internet users with solid knowledge of mathematics they repeatedly say yes.

They present all kinds of explanations and evidence, some of which are plausible. But as surveys and field reports have shown, many other scholars continue to refuse to believe them.

So let’s delve deeper into the issue, proposes . Firstly, we must think about how we represent the numbers.

At school, we learn to represent numbers in various ways. We started counting on our fingers and later learned formal notation. We learn to express rational numbers as fractions or decimals. And we discovered that the decimal representations of some fractions are infinite, as is the case of 1⁄3.

But the numbers after the comma in these cases are not completely devoid of pattern — on the contrary, begin to repeat after a certain point: for example, 1⁄7 = 0,142857142857….

Nonetheless, irrational numbers such as pi (π) or √2have an infinite number of decimal places without a periodic pattern and cannot be expressed as fractions. To represent them exactly, we therefore choose a symbolbecause a decimal notation would only approximate the real value.

Some explanations

So how should we think about 0.999…? Some experts argue that we can start with the fact that the rational number 1⁄3 match the decimal number 0.333…. You can multiply it by 3 to get 0.999…. They reason that, like 1⁄3 × 3 = 1, then 1 and 0.999… must be the same.

And there are some other proofs that demonstrate that 0.999… is equal to 1. As an example, let’s start by writing the periodic number in decimal notation up to the nth digit after the decimal point: 9 × 1⁄10 + 9 × 1⁄100 + 9 × 1⁄1000 + … + 9 × 1⁄10n + 1. Now 0.9 can be highlighted because appears before each installment.

This gives: 0,9 × (1 + 1⁄10 + 1⁄102 + … + 1⁄10n). You can rewrite 0.9 as 1 – 1⁄10 to get an even more elegant formula: (1 – 1⁄10) × (1⁄10 + 1⁄102 + … + 1⁄10n).

In other words, we have what is called a geometric seriessomething that mathematicians have known how to solve for several hundred years. In this case, we will then have: 1 – 1⁄10n + 1. Also 0,9999…9with 9 up to the nth position, corresponds to 1 – 0.00…01, with 1 in position (n + 1).

If we now consider the full number 0.999…whose nines are repeated infinitely, then n returns if infinite. In this case, the term 1⁄10n becomes zero. The range between 0.999…9 and 1 has been shifted to infinity. This example is just one of many proofs that demonstrate that 0.999… is equal to 1.

In fact, it can be shown in a similar way that 0,8999… = 0,9and that 0.7999… = 0.8, and so on. And even if we change our number system, these patterns remain.

For example, if we move to the binary notationwhich consists of only 0s and 1s, the same problem arises: 0.111… (which corresponds to 1 × 1⁄2 + 1 × 1⁄4 + 1 × 1⁄8 + …) is equal to 1. There therefore appears to be a clear winner in the debate: the field that defends 0.999… = 1.

But calm. Although mathematics is a discipline in which correlations can be derived exactly, with minimal room for interpretation, it is still possible discuss the fundamentals.

New rules for the game

For example, one could simply specify that, by definition, 0.999… is less than 1. Mathematically speaking, this kind of proposal is permitted—but when we examine it, we discover some unusual consequences.

For example, normally, if we look at the number line and we choose any two numbers, there are always infinitely more numbers between them. You can calculate the average value of both, then the average value of this average and one of the two numbers, and so on.

But if we assume that 0.999… is less than 1it also happens that there is no no other number that falls between the two values. We found a break on the number line. And that range means the calculations can get weird.

As 1⁄3 + 2⁄3 = 1 also holds in this system, correspondingly, 0.333… + 0.666… = 1. As soon as a sum is calculated, it must be rounded upwards if you end up with a result in the odd space between 0.999… and 1.

This rounding up also applies to multiplicationsuch that 0.999… × 1 = 1, which means that a basic rule of mathematics, according to which anything multiplied by 1 is itself, no longer applies.

And there is other approaches to disambiguating 0.999…. For example, you can venture into domains of non-standard analysiswhich allows the so-called infinitesimaisor values ​​closer to zero than any real number.

This change of framework makes it possible distinguish between 1 and 0.999…se differ by an infinitesimaland does not lead to any contradictions (or no more than conventional calculation). But it is complicated in ways that lead most mathematicians to not consider it a real alternative.

So yes, there continues to be a debate about whether 0.999… = 1. On the one hand, working with the numbers and calculations familiar to most of us, the equation is undoubtedly true.

But you can explore other versions of mathematics to get a different answer — as long as you can also consider the curious consequences.