○ A war of words occured in Europe in the seventeenth century, when the Catholic priests refused to teach their students the concept of infinitesimals, claiming that God will never create anything that cannot be conceived and defined.

○ Years have passed since then, and today we have all the tools needed to give a strictly mathematical definition of the infintesimal. I use a Dirac-knife to make Dedekind-cuts on the real line and use a Hyper-clock to count the cuts.

○ The definition shows that studying infinesimals and infinities amounts to the same thing. Like infinities, there are an infinite number of infinitesimals.

I have often maintained that the best way to learn

number theory, complex analysis, and transform

theory is through Riemann zeta function. This

article adds matrix theory to the list.

Frequency and time-domain analyses are dual concepts

used by electrical engineers extensively, and we take this

as the inspiration for defining a complex variable

tau = t + is.

The value of Dirac delta function on the real line is zero except

at the origin, while the value of complex delta function on the

entire complex plane is zero except at the origin.

Two concepts, numerals and numerosities, are

introduced to facilitate the analysis of infinite

subsets of natural numbers.

Riemann Hypothesis is viewed as a statement

about the charge accumulated on a capacitor

in an electrical network.

Euler product formula is represented by a

Riemann zeta function, written as an exponential

with several zeta functions as exponents.

Studying infinitesimal and infinity amounts to the

same thing. They can be considered as dual of

each other. Infinitesimal has the advantage that

you can keep it in front of you.