Hyperreal numbers are an alternate way of conceiving of infinite quantities. Infinite numbers, in this system, behave exactly like very large numbers. So large that any finite number becomes insignificant in comparison.
Likewise infinitesimal numbers behave exactly like very small numbers. So small that they are insignificant.
What makes hyperreals so appealing is that they match how calculus is used in the real world. To a mathematician $ dq $ might mean zero charge but in the real world there are no charges that small. In the real world $ dq $ would mean a single electron. The charge of a single electron is definitely not zero but by itself it is insignificant to most calculations.
Moreover, many calculations depend on $ dq $ being either positive or negative which is impossible if it is made so small that it is zero.
Increment theorem
In non-standard analysis, a field of mathematics, the increment theorem states the following: Suppose a function y = f(x) is differentiable at x and that Δx is infinitesimal. Then
- $ \Delta y = f'(x)\,\Delta x + \varepsilon\, \Delta x $
for some infinitesimal ε, where
- $ \Delta y=f(x+\Delta x)-f(x). $
If $ \scriptstyle\Delta x\not=0 $ then we may write
- $ \frac{\Delta y}{\Delta x} = f'(x)+\varepsilon, $
which implies that $ \scriptstyle\frac{\Delta y}{\Delta x}\approx f'(x) $, or in other words that $ \scriptstyle \frac{\Delta y}{\Delta x} $ is infinitely close to $ \scriptstyle f'(x) $, or $ \scriptstyle f'(x) $ is the standard part of $ \scriptstyle \frac{\Delta y}{\Delta x} $.
References
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