Basic infinitesmal doubt: Can there be a negative infinitesmal?

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Homework Statement
Can there be a negative infinitesmal?
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What I mean is on a coordinate plane like below we take the positive x axis measure a certain distance x on it and take the infinitesmally small quantity dx next to it in the positive direction:

IMG_20240502_022456.jpg



Now can I do the opposite as in can I measure a distance negative x on the negative x axis and take an infinitesmal quantity?:
IMG_20240502_022507.jpg
 
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I ask so because in things like mechanics to find out the equilibrium condition of a particle
If :
Df/dx=infinitesmally small change in Force < 0, the particle is in stable equilibrium
Ive gotten an expression of force which is a function of x. But I have ofcourse used some sign conventions for the force too since there were multiple therefore if my dx is negative I think ill get a different answer compared to if it was positive . Mainly I do not know how to incorporate a negative dx in calculus do I just use normally And take it to the numerator?
 
Yes, there are often negative infinitesimals in mathematics. Don't be fooled by the common usage. The most common use of infinitesimals is as a positive ##\epsilon \gt 0## that is compared to an absolute value like ##|x-x_0| \lt \epsilon##. In that case, ##\epsilon## is positive because it is being compared to an absolute value. But the ##x-x_0## inside the absolute value could be positive or negative. Although ##\epsilon## is usually stated to be positive, there are other uses of ##-\epsilon## or ##\pm \epsilon##. Those are also infinitesimals.
 
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FactChecker said:
Yes, there are often negative infinitesimals in mathematics. Don't be fooled by the common usage. The most common use of infinitesimals is as a positive ##\epsilon \gt 0## that is compared to an absolute value like ##|x-x_0| \lt \epsilon##. In that case, ##\epsilon## is positive because it is being compared to an absolute value. But the ##x-x_0## inside the absolute value could be positive of negative. Although ##\epsilon## is usually stated to be positive, there are other uses of ##-\epsilon## or ##\pm \epsilon##. Those are also infinitesimals.
Sorry whats epsilon representing here?
 
Of course, if ##\varepsilon## is a positive infinitesimal, then ##-\varepsilon## is a negative infinitesimal.
 
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tellmesomething said:
Sorry whats epsilon representing here?
A positive real number, usually one that is close to zero.
 
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Mark44 said:
A positive real number, usually one that is close to zero.
I see makes sense. Thankyou
 
Well, the Hyperreals are a field. Then every element, including infinitesimals, must have an additive inverse. EDIT: That means pure Infinitesimals, i.e., those with Real part =0, must have an additive inverse.
 
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If you have one equation with ##dx## and ##y=-x##, then it is often stated that ##\frac {dy}{dx} = -1## and the substitution of ##-dy## for ##dx## is made. If ##dx## is positive, then ##dy## is negative.
 
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