What is infinitesimal times infinity?

What is infinitesimal times infinity?

In non-standard analysis, an infinitesimal times an infinite number can have various values, depending on their relative sizes. The product can be an ordinary real number. But it can also be infinitesimal, or infinite.

Is an infinitesimal equal to zero?

In mathematics, an infinitesimal or infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the “infinity-th” item in a sequence.

Did Leibniz use infinitesimals?

In calculus, Leibniz’s notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent infinitely small (or infinitesimal) increments of x and y, respectively, just as Δx and Δy represent finite increments of x and y, respectively.

Is infinitesimally small redundant?

‘Infinitesimally small’, to me, is an unequivocal redundancy, since ‘infinitesimal’ means ‘immeasurably or incalculably small’, as in ‘an infinitesimal difference’.

What is the symbol for infinitesimal Epsilon?

Even in analysis this varying variety of epsilon is generally interpreted as “as small as you can get it, but greater than 0.” There is no mathematical symbol that means “infinitesimal”. Variables that are thought of as infinitesimal are typically called ϵ or δ.

What is the product of an infinitesimal times an infinite number?

In non-standard analysis, an infinitesimal times an infinite number can have various values, depending on their relative sizes. The product can be an ordinary real number. But it can also be infinitesimal, or infinite. Similarly, the ratio of two “infinite” objects in a non-standard model of analysis can be an ordinary real number, but need not be.

Is infinity a really large positive number?

With infinity this is not true. With infinity you have the following. In other words, a really, really large positive number ( ∞ ∞ ) plus any positive number, regardless of the size, is still a really, really large positive number.

Is there such a thing as an infinite number?

In the ordinary calculus, there are noinfinitesimals. Abraham Robinson and others, from the $1950$’s on, developed non-standard analysis, which does have infinitesimals, and also “infinite” number-like objects, that one can work with in ways that are closely analogous to the way we deal with ordinary real numbers.