# When can I treat infinitesimals as numbers? [duplicate]

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Even if many mathematicians don't like the notation, I have found in many rigorous math books things like

$$\frac{dy}{dx}=Ay$$

so

$$\frac{dy}{y}=Adx$$

What I don't understand is this: under which conditions is ok to treat infinitesimal as numbers ? (multiplying for example both sides of an equation by dx).

Edit:

Not only, what allows me to integrate both sides of the last equation ???

## marked as duplicate by Mark S., Hans Lundmark, Kemono Chen, Gibbs, DylanFeb 11 at 14:34

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## 1 Answer

If you want to make this rigorous you can introduce a number system similar to that of dual numbers, viz. $$dx^2=0$$. (See also here.) Depending on the calculus task at hand in more advanced examples, you might change these axioms slightly. For example, a metric $$ds^2=g_{\mu\nu}dx^\mu dx^\nu$$ would use $$dx^\mu dx^\nu dx^\rho=0$$, while Brownian noise in stochastic calculus may be taken to satisfy $$dW_t^2=dt,\,dW_t^3=0$$.