What does it mean to assign a value to a number? Isn't the value of a number the number itself? I cannot understand this remark at all. Thus the need for these infinitely small. the area under any continuous curve can be obtained between any two limits. Ultimately, Cauchy, Weierstrass, and Riemann reformulated Calculus in terms of limits rather than infinitesimals. Limits and infinitesimals help us create models that are simple to use, yet share the same properties as the original item (length, area, etc.). > it means smaller than any positive number to which you can assign a valueĬan you give an example or an explanation of a positive number to which you can't assign a value? I can't imagine what that could possibly mean. a revolutionary new approach to mathematics: infinitesimal calculus. Note that the term 'indefinitely small' is meaningless for numerical analysis for obvious reasons. I don't use the word 'arbitrary' because in numerical analysis (unlike in regular calculus) it means something different to 'indefinite' - namely that the minimum value of the increment may be arbitrary (alternatively, depending on the functions concerned, it may have to meet certain criteria). Wtf: I did think carefully about word choice for the thesis, in particular I would never use the term 'infinitely small' to describe an infinitesimal - 'indefinitely small' does not mean smaller than 'every' positive number, it means smaller than any positive number to which you can assign a value (this is similar to Kant's ideas about the infinite as discussed by Bell). Another important question to ask when looking at functions is: What happens when the independent variable becomes very large For example, the function f(t)e.
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