Basic infinitesmal doubt: Can there be a negative infinitesmal?

[tellmesomething]

Homework Statement

Can there be a negative infinitesmal?

Relevant Equations

None

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:

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?:

 

[tellmesomething]

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?

 

[FactChecker]

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.

 

[tellmesomething]

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?

 

[Erland]

Of course, if ##\varepsilon## is a positive infinitesimal, then ##-\varepsilon## is a negative infinitesimal.

 

[Mark44]

Sorry whats epsilon representing here?

A positive real number, usually one that is close to zero.

 

[tellmesomething]

A positive real number, usually one that is close to zero.

I see makes sense. Thankyou

 

[WWGD]

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.

 

[FactChecker]

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.