Whether 0.999 equals 1 is the subject of bitter dispute in countless online forums
Countless debates in classrooms, lecture halls and online forums have swirled around the question of whether 0.999. equals 1. Teachers, professors and math-savvy Internet users repeatedly affirm that it does.
Countless debates in classrooms, lecture halls and online forums have swirled around the question of whether 0.999... equals 1. Teachers, professors and math-savvy Internet users repeatedly affirm that it does.. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today. They come up with all kinds of explanations and proofs, some of which are plausible. But as polls and field reports have shown, many others still refuse to believe them. So let’s dig into it. First, we should think about how we present numbers. In school, we learn to represent numbers in several ways. We start with counting our fingers and later learn formal notation. We learn to express rational numbers as fractions or decimals. And we discover that the decimal representations of some fractions are infinite, such as 1/3. But the digits after the decimal point in these cases are not totally pattern-less—instead they start repeating after a certain point: for example, 1/7=0.142857142857.... Meanwhile irrational numbers, such as pi or √2, have an infinite number of decimal places without a periodic pattern, and they cannot be expressed as fractions. To represent them exactly, one therefore chooses a symbol because a decimal notation would only approximate the actual value.So how should we think about 0.999...? Some experts argue that we can start with the fact that the rational number 1/3 corresponds to the decimal number 0.333.... You can multiply it by 3 to get 0.999.... They reason that because 1/3 × 3=1, then 1 and 0.999... must be the same. And there are a few other proofs that prove that 0.999... is equal to 1. As one example, start by writing out the periodic number in decimal notation to the. Now you can factor out 0.9 because it appears before each summand.). You can rewrite the 0.9 as 1 – 1/10 to get an even nicer formula: × is equal to 1. So there seems to be a clear winner in the discussion: the camp defending 0.999...=1. But not so fast. Even though mathematics is a subject in which you can derive correlations exactly, with minimal room for interpretation, it’s still possible to argue about fundamentals.For example, one could simply specify that by definition, 0.999... is smaller than 1. Mathematically speaking, this kind of proposal is allowed—but when you examine it, you will discover some unusual consequences. For instance, typically, if you look at the number line and pick any two numbers, there are always infinitely many more between them. You can calculate the mean value from both, then the mean value from this mean and one of the two numbers, and so on. But if you assume that 0.999... is smaller than 1, then there is no further number that lies between the two values. You have found a break in the number line. And that gap means calculations can get weird. Because 1/3 + 2/3=1 also holds in this system, correspondingly, 0.333... + 0.666...=1. As soon as you calculate a sum, you have to round up if you end up with a result in the strange space between 0.999… and 1. This rounding up also applies to multiplication, such that 0.999... × 1=1, which means a basic rule of mathematics, that anything multiplied by 1 is itself, no longer applies. And there are other approaches to getting rid of the ambiguity of 0.999... For instance, you can dabble in the realms of nonstandard analysis, which allows for so-called infinitesimals, or values closer to zero than any real number. This shift in framework makes it possible to distinguish between 1 and 0.999... if they differ by one infinitesimal. And it does not lead to any contradictions . But it’s complicated in ways that mean most mathematicians don’t consider it a true alternative. So yes, there is still a debate whether 0.999...=1. On the one hand, working with the numbers and calculation familiar to most of us, the equation is undoubtedly true. But you can explore other versions of mathematics to get a different answer—provided you can also consider the curious consequences.Scientific American has served as an advocate for science and industry for 180 years, and right now may be the most critical moment in that two-century history.subscriber since I was 12 years old, and it helped shape the way I look at the world.always educates and delights me, and inspires a sense of awe for our vast, beautiful universe. I hope it does that for you, too., you help ensure that our coverage is centered on meaningful research and discovery; that we have the resources to report on the decisions that threaten labs across the U.S.; and that we support both budding and working scientists at a time when the value of science itself too often goes unrecognized.
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