The Crux of the Confusion When People Think 0.999 is Less Than 1
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The whole “how is 0.999999… equal to 1, it’s obviously less” thing comes up from time to time on here.
The crux of the confusion is usually that they’re thinking “every item of the sequence {0.9, 0.99, 0.999, …} is less than 1, so the limit of the sequence must also be less than 1.”
Which happens to be false – just because every term in a sequence has property X, does NOT imply the limit has property X. Here, the limit is not a term of the sequence, so it does not automatically inherit properties of the terms of the sequence.
This may feel a bit counterintuitive but it’s true and it’s an idea that comes up over and over in higher math.
If you go far enough down the rabbit hole on this you get to some really weird results, e.g., if you add together a series of continuous functions (nothing crazy, just sines & cosines), and take the limit, the limit can end up being a DIScontinuous function.
Specifically, every function in the series sin(x) + sin(3x)/3 + sin(5x)/5 + … is continuous. But the limit is DIScontinuous. It’s a square wave.
That came as a shock to many mathematicians about 200 years ago, but it’s true, and it helped spur a movement in the 19th-century mathematical community to reformulate previous findings in more rigorous fashion, which is taught today as Real Analysis.
To quote the book “How We Got From There to Here: A Story of Real Analysis” by Eugene Boman and Robert Rogers (pp.78):
“In some sense, the nineteenth century was the ‘morning after’ the mathematical party that went on throughout the eighteenth century.”
Follow-Up Questions
Q: So if 0.99… is taken to the limit, it’s =1? We no longer have to assume it’s equal to 1, it’s actually 1?!
A: Yes, the limit of the sequence 0.9, 0.99, 0.999, …, is actually 1.
The limit is the number that you can always get as close as you want to by looking far enough in the sequence.
Want to get within 0.01 of 1? Look at the 2nd term (0.99).
Want to get within 0.00001 of 1? Look at the 5th term (0.99999).
If you tell me how close you want to get I can always tell you how far you have to look in the sequence to get at least that close.
For the sequence 0.9, 0.99, 0.999, …, the number 1 is the only number with the property described above.
Does the sequence ever actually reach the limit? No. But that doesn’t matter if we’re talking about limits in the mathematical sense. Limits aren’t about what number you can attain. They’re about what number you get arbitrarily close to.
You can say “but I can keep writing 9’s forever and by the end of my life it will still be less than 1,” and that’s totally true, but that’s not really what we’re talking about when we discuss limits and equality in the mathematical sense.
Q: 0.999… is less than 1. THE LIMIT of 0.999… is 1. Those are not the same statements.
A: The quantity 0.999… represents a limit. By definition.
The notation $a_1 + a_2 + …$ is shorthand for the limit of $a_1 + a_2 + … + a_N$ as $N$ approaches infinity.
E.g., $0.999…$ is shorthand for the limit of $0.9 + 0.09 + … + 9(0.1)^N$ as $N$ approaches infinity.
Q: Problem is with () and []. (0,1) Max is 0.999… and [0,1] Max is 1. I’m sure if (), and [] are used in real math, that concept is definitely taught in schools.
A: The set (0,1) contains no max. A max is a number in the set, such that no other number in the set is bigger. You can pick any number in (0,1) and I’ll make a bigger number by tacking on another 9 to the end of the decimal. So the set has no max.
What you’re thinking of is a called a supremum, the smallest number (not necessarily in the set) such that no other number in the set is bigger. The supremum of (0,1) is 1.
So the set (0,1) has a supremum of 1 but does not have a max.
While the set [0,1] has a supremum of 1 and a max of 1.
Q: Actual progress in mathematics won’t come from teaching the student to discard his intuition. Actual progress will come when the student thinks through what can be built on that student’s intuition.
Actual progress in mathematics will not happen without actually learning the fundamentals.
Many people are prone to jump straight into philosophical debate without even having spun up on the fundamentals (in this case, the basics of real analysis).
They think they’re onto something interesting/novel when it’s actually made trivial by foundational knowledge.
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