DigiMat and the Multiplication Table

DigiMat is new reformed mathematics education. DigiMat starts by constructing the natural numbers (1, 2, 3, 4,.., in decimal notation) by repetition of the basic operation +1, starting from 0:

• 1 = 0 + 1
• 1+1 (= 2 )
• 1+1+1 (= 3 = 2+1)
• 1+1+1+ 1 (= 4 = 3+1)
• 1+1+1+1+1 (= 5 = 4+1)
• and so on

DigiMat starts with a binary representation instead of the usual decimal representation, because it is more basic and simpler (and therefore preferred by the computer): Instead of the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 of the decimal system, the binary system has only two digits 0 and 1, which when used in a position system gives the following representation 

• 1 = 1 (decimal)
• 10 = 2 (decimal)
• 11 = 3
• 100 = 4 
• 101 = 5
• 110 = 6
• 111 = 7
• 1000 = 8
• 1001 = 9
• ... 

In DigiMat for preschool different binary systems can quickly be mastered such as for example 

• one red (apple) is 1
• two red (apples) is one yellow (apple)
• two yellow is one green
• two green is one blue
• two blue is one violet
• ... 

with the following representation 

• 1 =  red (apple)
• 2 = yellow (apple)
• 3 = yellow and red (apples)
• 4 = green (apple)
• 5 = green and red (apples)
• 6 = green + yellow (apples)
• 7 =  green + yellow + red (apples)
• 8 = blue (apple)
• 9 = blue and red (apples)
• ... 

Alternatively, a one-bead abacus (column of beads with bead left = 0 and bead right = 1) is quickly mastered.  See Basics on DigiMat School.

With such binary representations any preschool child quickly learns to (without effort) perform the basic operations of addition, multiplication (as repeated addition) and subtraction of natural numbers, followed by division.  This is carefully described on the DigiMat School site.

This is to be compared with standard school mathematics, where the decimal multiplication table is both corner stone/top catch and greatest hurdle. The discussion on the necessity of learning the multiplication table by heart is filling endless texts on teaching school mathematics, with no consensus after 100s of years. Some experts claim that knowing by heart that 7 times 8 is 56, is immensely useful, while others claim this blocks true understanding, which is better expressed in a computation of the form 

• 7 x 8 = (10 - 3) x (10 -2) = 10 x 10 - 3 x 10 - 10 x 2 + 2 x 3 = 100 - 30 - 20 + 2 x 3 =  50 + 2 x (2 + 1) = 50 + 2 x 2 + 2 x 1 = 50 + 2 x (1+1) + 1 + 1 = 50 + 2 x 1 + 2 x 1 + 1 +1 = 50 + 1 + 1 + 1 + 1 + 1 +1 = 56  

requiring familiarity and understanding of in particular the distributive law. Here is a recent article presenting this conundrum with the shocking title:

• The Danger of Memorising the Multiplication Table (in Swedish) 

with references to authorities such as Arthur Baroody, Jo Boadler and Gina Kling.  

DigiMat lifts school mathematics out of this paralysis. Recall that the binary multiplication table is so simple, that it is not necessary to memorise: 0 x 0 = 0, 0 x 1 = 0, 1 x 0 = 0, 1 x 1 = 1.

Nor the (slightly more complex) addition table: 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 10.

DigiMat meets the World

DigiMat is new mathematics education for the digital society consisting of  

• Ada's World (app for pree-school introducing basic concepts for all ages)
• DigiMat School (from elementary school to university)
• DigiMat Pro (top academic and professional level)

DigiMat is now being launched on a broad international scale for school, university and continued education. Take a look and check out what DigiMat can offer to you, as student, teacher or professional. 

 

The Pressure Equation of Incompressible Flow

The standard mathematical model for incompressible flow of unit density with vanishing viscosity takes the form of the incompressible Euler equations:

• $\frac{\partial u}{\partial t}+u\cdot\nabla u + \nabla p = 0$,
• $\nabla\cdot u=0$,

where $u(x,t)$ is flow velocity, $p(x,t)$ is pressure and $(x,t)$ are space-time coordinates. 

The first equation expresses Newton's 2nd Law (conservation of momentum), while the second equation is not a physical law but a side condition as a stipulation that the velocity is to be divergence free. 

When solving the incompressible Euler equations computationally the second equations is typically replaced by an elliptic equation for the pressure of the form 

• $-\delta*\Delta p = - \nabla\cdot u$,      (1)
• where $\delta \approx h$ with $h$ mesh size.

In a variational setting this equation results from regularisation of the pressure as Lagrange multipler for the side condition $\nabla\cdot u=0$, which can be seen as a form of relaxation. The choice of $\delta =h$ brings $\delta\Delta$ in parity with $\nabla\cdot$. The equation (1) can be seen as a physical law connecting pressure to the divergence of velocity. 

Let us now give more perspective on the elliptic pressure equation by extending to compressible ideal flow where the equation for the internal energy $e$

• $\frac{\partial e}{\partial t}+\nabla\cdot (eu)+p\nabla\cdot u = 0$,

can be seen as an equation (expressing a law of physics) for the pressure $p=\gamma e$ with $\gamma >0$ a gas constant:

• $\frac{Dp}{Dt}=-\gamma p\nabla\cdot u$    (2)
• with $\frac{Dp}{Dt}=\frac{\partial p}{\partial t}+\nabla\cdot (pu)$,

With suitable normalisation the Mach number is $\sqrt{\gamma}$ with a switch from supersonic over transonic to subsonic flow as $\gamma$ increases beyond 1. 

This equation suggests a parabolic variant of the incompressible pressure equation of the form

• $\frac{\partial p}{\partial t}-\delta*\Delta p = - \nabla\cdot u$  (3)

where the right hand side of the compressible pressure equation $-\gamma p\nabla\cdot u$ is replaced by $-\nabla\cdot u$ for the incompressible case (and (2) is correspondingly regularised). Again, (3) can be seen as a law of physics connecting pressure to the divergence of velocity. It allows direct time stepping of the flow equations as an expression of physicality with finite speed of propagation/sound , compared with the elliptic equation (1) with infinite speed of pressure variation/sound.  

Choosing $\gamma$ small gives compressible flow with large variation of $\nabla\cdot u$ corresponding to moderate variation of density, while choosing $\gamma$ large will keep the variation in density small. 

We can thus view the compressible solver as a unified flow solver for both compressible and incompressible flow depending on the choice of the single parameter $\gamma$, with a natural connection to standard incompressible solvers. 

The unified flow solver can be explored as item 98 in the Model Workshop of DigiMat BodySoul including comparison a with standard incompressible solver.

Remark. Integrating (2) along streamlines gives

•  $p \sim \exp(-\gamma\int \nabla\cdot u\,dt)$ 

which expresses an exponential connection between the pressure $p$ and $-\gamma\nabla\cdot u$ compatible with $\nabla\cdot u$ being small for $\gamma >>1$.