## onsdag 2 december 2020

### Symbolic vs Digital Mathematics

Traditional mathematics eduction formed before the computer is symbolic mathematics based on manipulation of mathematical symbols in forms of symbolic computation. A typical example is to express the integral of an elementary function f(x) with a symbol (e g cos (x)) in terms of another elementary function F(x) with a symbol (e g sin(x)) by using the fact the DF= f ( Dsin(x)= cos(x)) where D = d/dx denotes derivation. F(x) is referred to as a primitive function of f(x) if DF=f.

A traditional Calculus book contains a list of primitive functions of elementary functions expressed in symbolic form, together with techniques of symbolic computation (e g integration by parts) allowing symbolic computation of not listed primitive functions. The symbolic computation can be very cumbersome and lengthy and is automated in software such as Mathematica. This is Symbolic Calculus.

In digital mathematics primitive functions/integrals are computed digitally by simply performing the summation building the integral digitally as a Riemann sum, with a certain precision. The computation is steered by a one-line code performing the summation term by term. By digital computation Calculus is thus reduced to a one-line code which the student writes and thus can understand and be allowed to use not simply as a black box. This is Digital Calculus. From experience with Digital Calculus the true meaning of symbols such as cos(x) can be learned and understood.

Another example is the following symbolic computation with fractions:
• 1/3 + 1/4 = 7/12
which takes the digital form
• 0.3333333…+ 0.250000… = 0.5833333…
Symbolic computation with fractions fills a large portion of school mathematics and is generally perceived as troubling, non-illuminating and destructive to enthusiasm for mathematics.

Summary:
• Symbolic Calculus is limited, tricky to use and hard to learn.
• Digital Calculus is unlimited, easy to learn and use.
Digital Calculus of DigiMat School can be taught in basic school and give many students a tool for life. Symbolic Calculus is a traditional core topic of university level, but only few can profit.

Symbolic Calculus in symbiosis with Digital Calculus gives a new tool ready to be used by many. Digital Calculus does not replace Symbolic Calculus (thought processes have symbolic not digital form), but gives Symbolic Calculus a richer meaning by supplying substance to symbols.