What's greater than -1

Jun 2019
Or: 1 - 0.9recurring
Except this is 1 - 9/9 = 1 - 1 = 0 (exactly). It's the same as \(\displaystyle \lim_{n\to\infty} 10^{-n}\), which is the same as \(\displaystyle \lim_{n\to\infty} \frac{1}{n}\). They're all zero.

I'd like to throw \(\displaystyle \varepsilon\) and \(\displaystyle \hbar\) into the ring. :p
Aug 2012
I'd like to throw \(\displaystyle \varepsilon\) and \(\displaystyle \hbar\) into the ring. :p
There is no smallest positive real number. This fact remains true if you extend the reals to the hyperreals of nonstandard analysis. It's not commonly understood that even in the hyperreals, there is no smallest positive real and no smallest positive infinitesimal. This is easily proven by noting that if you claim $\varepsilon > 0$ and $\forall x \in \mathbb R$, we have $0 < \varepsilon < x$; then $\frac{\varepsilon}{2}$ is also positive and strictly smaller than $\varepsilon$. So $\varepsilon$ wasn't the smallest positive real after all; and since $\varepsilon$ was an arbitrary positive infinitesimal, there is no smallest positive infinitesimal in the hyperreals. I only mention this because I've seen the contrary asserted on too many message boards.

ps -- How do we know that $\frac{\varepsilon}{2}$ exists and is a positive hyperreal number? Because the reals, and the hyperreals, are a field. That's a number system in which you can always divide as long as the divisor isn't zero.

So "the smallest positive real" is simply not well defined, and you can't assign it a variable in a meaningful way. Any such assignment is meaningless since it does not refer to anything even in the abstract mathematical world. There's no mathematical context where the idea makes sense.

The symbol $\varepsilon$ is universally understood in math to be an arbitrary small positive real number; and not a specific one. Your proposed overloading would be incorrect in the context of established math.

As for $\hbar$, that is a well-known physical constant whose meaning is universally understood as such by pretty much everyone, mathematicians and physicists alike.

Moreover it is strictly larger than zero, and moreover there is no claim by physicists that it's the smallest unit of measure in the world; only that it's the scale below which our equations don't work. The Planck scales (time and space) are statements about the limitations of our theories; and not statements about the world. The Planck limits are epistemological and not ontological.

So it's bad notation to propose overloading a symbol that the physicists own by universal agreement and that doesn't mean what you'd like it to mean. Let alone that it wouldn't refer to anything.

Well I hope you don't mind my opinionated opinion but your post inspired me to let it all hang out. I noted your smiley so this is offered in a similar lighthearted vein.
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Dec 2015
\(\displaystyle \epsilon \rightarrow 0\) is an infinitesimal variable with properties of number 0 against a different variable and has properties of a number with itself.

Imagine the derivative of a function involving time which states dx decreasing over time and tending to 0 as time moves .In other words , dx is changing.
So it is not a mistake to say \(\displaystyle \frac{change}{change} =1.\)
A change that has properties of an elimination.(x•0=0)
A small change to any varying quantity.
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Jun 2019
Actually, I was thinking of computer programming. Some languages have an eps function or similar that returns the value of the LSB of a given floating point number.

And I am by no means an expert in quantum physics. I just know in thermodynamics we use Planck's constant to quantise the number of possible distinct microstates of a system -- which is how we assign a physical value to entropy.

My point was in the world of mathematics there is no smallest positive number, but in worlds ruled by the laws of computers or quantum physics the concept of smallest meaningful values does come into play.