Arithmetics forms the core of basic school mathematics education. Arithmetics occurs in symbolic form as fractions p/q with p and q non-zero integers, and in digital form as decimal (or binary) expansions.
Symbolic computation with fractions fills a large portion of school mathematics and is generally perceived as troubling, non-illuminating and destructive to interest. See also Rational Numbers.
As a most basic example, the symbolic computation with fractions:
- 1/3 + 1/4 = 7/12 (1)
takes the digital form
- 0.3333333…+ 0.250000… = 0.5833333… (2)
The numerical values of the fractions are hidden behind the symbols in (1) and the manipulation of symbols to arrive at the result 7/12 is not directly obvious, while (2) is fully transparent as addition digit by digit.
Calculus forms the core of mathematics education beyond basic school. The Calculus you meet in standard text books (there are very many) is Symbolic Calculus which builds on a list/table of so-called elementary (or special) functions F(t) depending on a (time) variable t denoted by specific symbols, such as F(t) = sin(t), together with derivates DF(t) with D = d/dt, also appearing as elementary functions (see table page below vs Symbolic Differentiation) with C an arbitrary constant:
- D(t*t) = 2*t (same as ∫2*t dt = t*t +C)
- Dsin(t) = cos(t) (same as ∫ cos(t) dt = sin(t) +C)
- Dcos(t) = -sin(t) (same as -∫ sin(t) dt = cos(t) +C)
- Dexp(t) = exp(t) (same as ∫ exp(t) dt = exp(t) +C)
- Dlog(t) = 1/t (same as ∫ dt/t = log(t) +C)
- Dtan(t) = 1/cos(t)*cos(t) (same as ∫ 1/cos(t)*cos(t) dt = tan(t) + C
We understand that a symbol like sin(t) does not reveal the value of sin(t) for a specific value of t. For these values we need another table containing values of elementary functions, a table which is much longer since each elementary functions takes on many values as t varies. Symbolic Calculus is thus supported by long tables, of elementary functions and their derivatives (and their values).
The prime use of the table of derivatives of elementary functions is to allow solution in symbolic form of the basic initial value problem of finding, for a given function f(t) and given initial value x0, a function x(t) named integral of f(t) or primitive function of f(t) and denoted by ∫ f(t)dt, such that
- Dx(t) = f (t) for t > 0, x(0) = x0, (1)
simply by searching in the table for a function F(t) such that DF(t) = f(t), assuming f(t) occurs in the list, and declaring that the solution x(t) = F(t) + C with the constant C chosen such that x(t) = x0. For example, if f(t) = cos(t) and x0=0, we find from the table that x(t) = sin(t) knowing that sin(0)=0. Given f(t) we thus seek a function F(t) with derivative DF = f.
Integrating f (t) we get F(t) and differentiating (computing the derivative of) F(t) we get f(t). Differentiation is the inverse operation of integration and vice versa:
- Integration of f(t) into F(t) followed by differentiation of F(t) gives back f(t).
- Differentiation of F(t) into f(t) followed by integration gives back F(t).
The table allows us to solve (1) for a given (elementary) function f(t) simply by searching through the table with the hope of function a function F(t), a primitive function of f(t) or integral of f(t), which satisfies DF = f. If the table does not contain such a function F(t), then we are stuck. Examples include f(t) = exp(-t*t), sin(t*t), sin(t)/t and 1/log(t).
A traditional Calculus book starts with a definition of derivate followed by analytical computation of derivatives of some elementary functions such as polynomials and trigonometric functions sin(t) and cos(t) defined geometrically. Integrals are then introduced as primitive functions of functions for which derivatives have been computed, as shown above. No digital computation is involved, it is all symbolic with Dsin(t) = cos(t) and sin(t) = ∫cos(t)dt +C.
In Digital Calculus the initial value problem (1) is instead computed digitally by time stepping
- x(t+dt) = x(t) + f(t)*dt (2) (or dx = f(t)*dt or dx/dt = f(t))
with x(0) = x0. In this case we do not need the table of derivatives and we can solve for any f(t), not just f(t) with primitive function F(t) as an elementary function.
We understand that Digital Calculus is fundamentally different from Symbolic Calculus: A very long table of functions values for a very limited set of functions is replaced by a single line of code executed by digital computation which works for any function. The essence of Digital Calculus is captured in the proof of The Fundamental Theorem of Calculus in a form which is understandable and illuminating. Try it!
This is an enormous rationalisation as an expression of the power of DigiMat, which you can cash in. It is a revolution of Calculus, from Symbolic Calculus to Digital Calculus, from special functions to functions of any form:
- From a local library with limited selection of books to a Global Digital Library containing all books.
- From an alphabet of two letters cut in stone to a full alphabet with language capable of expressing anything.
- Invent your own metaphor for the difference between Digital and Symbolic Calculus.
In Digital Calculus we start with computing integrals ∫ f(t)*dt for any f(t) by time stepping as solutions x(t) to dx = f(t)*dt and obtain by construction that dx/dt = f(t). There is then no need to compute the derivate of x(t) symbolically: By construction it equals f(t). The essence is to integrate digitally. Compare with Symbolic Calculus starting with a list of derivatives in symbolic form.
Note that computing an integral by time stepping is directly feasible as it only involves successive addition and multiplication of f(t) with dt, while direct computation of a derivate as a limit
- (x(t+dt) – x(t))/dt = dx/dt with dt tending to zero,
is cumbersome because it involves division by dt which is small and inflates any little noise in the value of x.
- Symbolic Calculus: Symbolic computation of derivatives = special + tricky.
- Digital Calculus: Digital computation of integrals = general + easy.
Traditional mathematics eduction formed before the computer is symbolic mathematics based on manipulation of mathematical symbols in forms of symbolic computation.
- Symbolic Calculus is limited, hard to learn and tricky to use.
- Digital Calculus is unlimited, easy to learn and easy to 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.
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The Fundamental Theorem of Calculus has a clear meaning in Digital Calculus as expressed here (Basics):
- The initial value problem (1) can be solved by time stepping under a mild condition on f(t) (Lipschitz continuity). By construction Dx(t) = f(t).
In Symbolic Calculus the Fundamental Theorem of Calculus appears as the following seemingly trivial/self evident statement :
- The initial value problem (1) in u(t) can be solved by finding a function F(t) such that DF(t) = f(t) and setting u(t) = F(t) + C with the constant C so defined that u(0) = u0.
In other words, the equation Dx = f is solved by finding F such that DF = f and setting x(t) = F(t) + C with a proper constant C. This makes The Fundamental Theorem of Calculus into a triviality, which is very confusing to students expecting something substantial under such a grand a name, which is what Digital Calculus delivers.
The Fundamental Theorem of Calculus thus states that the differential equation Dx(t) = f(t) can be solved by time stepping which constructs the solution u(t) as an integral. The key ingredient is thus a constructive process to compute integrals, which is represented by a one-line code. Understandable. Works always.
In Symbolic Calculus, this turned around into instead symbolic computation of derivatives DF with the hope of finding F such that DF = f from a vast precomputed table. Tricky. Special. But when it works it (magically) constructs the solution seemingly without any computational work.
- Digital Calculus: Integrals are constructed by time stepping.
- Symbolic Calculus: Derivatives of elementary functions determined by symbolic computation.
Recall that Digital Math/Calculus can be viewed to encompass Symbolic Math/Calculus as a synthesis of both digital/constructive and symbolic math.
Recall that Mathematica was started out as software (computer program/code) for automatic symbolic computation of derivatives with the purpose of replacing the table of derivatives found in traditional Calculus books: