Comment on I just cited myself.
sp3tr4l@lemmy.zip 5 days agoThe explanation I’ve seen is that … is notation for something that can be otherwise represented as sums of infinite series.
In the case of 0.999…, it can be shown to converge toward 1 with the convergence rule for geometric series.
If |r| < 1, then:
ar + ar² + ar³ + … = ar / (1 - r)
Thus:
0.999… = 9(1/10) + 9(1/10)² + 9(1/10)³ + …
= 9(1/10) / (1 - 1/10)
= (9/10) / (9/10)
= 1
Just for fun, let’s try 0.424242…
0.424242… = 42(1/100) + 42(1/100)² + 42(1/100)³
= 42(1/100) / (1 - 1/100)
= (42/100) / (99/100)
= 42/99
= 0.424242…
So there you go, nothing gained from that other than seeing that 0.999… is distinct from other known patterns of repeating numbers after the decimal point.
Tomorrow_Farewell@hexbear.net 4 days ago
The ellipsis notation generally refers to repetition of a pattern. Either as infinitum, or up to some terminus. In this case we have a non-terminating decimal.
0.999… is a real number, and not any object that can be said to converge. It is exactly 1.
In what way is it distinct?
And what is a ‘repeating number’? Did you mean ‘repeating decimal’?
sp3tr4l@lemmy.zip 3 days ago
Ok. In mathematical notation/context, it is more specific, as I outlined.
Ok. Never said 0.999… is not a real number. Yep, it is exactly 1 because solving the equation it truly represents, a geometric series, results in 1. This solution is obtained using what is called the convergence theorem or rule, as I outlined.
0.424242… solved via the convergence theorem simply results in itself, as represented in mathematical nomenclature.
0.999… does not again result in 0.999…, but results to 1, a notably different representation that causes the entire discussion in this thread.
I meant what I said: “know patterns of repeating numbers after the decimal point.”
Perhaps I should have also clarified known finite patterns to further emphasize the difference between rational and irrational numbers.
Tomorrow_Farewell@hexbear.net 3 days ago
It is not. You will routinely find it used in cases where your explanation does not apply, such as to denote the contents of a matrix.
Furthermore, we can define real numbers without defining series. In such contexts, your explanation also doesn’t work until we do defines series of rational numbers.
In which case it cannot converge to anything on account of it not being a function or any other things that can be said to converge.
A series is not an equation.
What theorem? I have never heard of ‘the convergence theorem’.
What do you mean by ‘solving’ a real number?
In what way does it not ‘result in 0.999…’ when 0.999… = 1?
You seem to not understand what decimals are, because while decimals (which are representations of real numbers) ‘0.999…’ and ‘1’ are different, they both refer to the same real number. We can use expressions ‘0.999…’ and ‘1’ interchangeably in the context of base 10. In other bases, we can easily also find similar pairs of digital representations that refer to the same numbers.
What we have after the decimal point are digits. OTOH, sure, we can treat them as numbers, but still, this is not a common terminology. Furthermore, ‘repeating number’ is not a term in any sort of commonly-used terminology in this context.
The actual term that you were looking for is ‘repeating decimal’.
No irrational number can be represented by a repeating decimal.
sp3tr4l@lemmy.zip 3 days ago
www2.kenyon.edu/Depts/Math/…/GeomSeriesCalcB.pdf
Here’s a standard introduction to the concept of the Convergence/Divergence Theorem of Geometric Series, starts on page 2.
Its quite common for this to be referred to as the convergence test or rule or theorem by teachers and TA’s.