### What is DigiMat?

DigiMat unifies math and programming in a unique way. Creativity, motivation and industrial impact are key elements. The program spans all levels, from pre-school to top academic and professional.

The method is based on world-leading mathematics research at KTH and Chalmers, together with didactic research at Stockholm University. Computation is the leading principle and music and visual art is part of the pedagogical concept.

By learning a few basic algorithms anyone can understand and carry out advances programming and physics simulations. No prior knowledge is needed. See the learning goals for details, and get started with ”Ada's World”!

### Try our DigiMat learning activities!

#### Pedagogical Editable App [Basic]

Enter Ada's World! Our pedagogical app is inspired by Ada Lovelace (1815-1852), mathematician and one of the first programmers in history. Here you learn the binary addition algorithm and time-stepping for simulation, by organizing a party.

"Time-stepping" activity in DigiMat Ada's World, allowing you to solve all mathematical models!

### The DigiMat Education Program

DigiMat is mathematics education for the digital world from basic school level through top academic and professional, and teachers:

1. for preschool, school, teacher's education and early university/professional. In the current version Basic consists of the app Ada's World together with a comprehensive program with rich text and interactive learning sessions.
2. advanced application - university and professional.

DigiMat unifies math and programming in a unique way. Creativity, motivation and industrial impact are key elements. The program spans all levels, from pre-school to top academic and professional.

The method is based on world-leading mathematics research at KTH and Chalmers, together with didactic research at Stockholm University. Computation is the leading principle and music and visual art is part of the pedagogical concept. By learning a few basic algorithms anyone can understand and carry out advances programming and physics simulations. No prior knowledge is needed. See the learning goals for details, and get started with ”Ada's World”!

Contact us (jjan@kth.se) and we will help teachers and students to get started! Accounts are free during the Corona crisis.

### Demo Learning Activity: Time stepping - compute any mathematical model

Press the </> button to edit, modify and run again!

Watch and play with the adjacent Digital Math game containing the essence of Calculus in few lines.

We can express the Fundamental Theorem of Calculus as:

1. Time stepping of $\frac{dx}{dt} = f(t)$ constructs the integral $x=\int f(t)\, dt$ of the integrand $f(t)$.
2. Differentiation of the integral $x=\int f(t)\, dt$ produces the integrand $f(t)$.
• Study experimentally by computing the dependence of $\int_0^T f(t)\, dt$ on the time step $dt$.
• ### Demo Learning Activity: Time stepping - trigonometric functions - oscillation

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Watch and play with the adjacent Digital Math game containing the essence of trigonometry and oscillation in just a few lines!

The trigonometric functions $x=cos(t)$ and $y=sin(t)$ appear as solutions to the initial value problem for $t>0$:

• $\frac{dx}{dt} = -y$ (1)
• $\frac{dy}{dt} = x$ (2)

or in time stepping form

• $x=x-y*dt$
• $y=y+x*dt$

with the following initial values for $t=0$:

• $x=\cos(0)=1$ and $y=\sin(0)=0$.

Watch and play with the code.

Understand that by construction with $D=\frac{d}{dt}$

• $D\cos(t) = -\sin(t)$
• $D\sin(t) = \cos(t)$

Observe that the differential equations express that the velocity vector $(vx,vy)=(\frac{dx}{dt},\frac{dy}{dt})$ and the position vector $(x,y)$ satisfies

• $vx* x + vy*y = -y*x + x*y =0$, (velocity is orthogonal to position)

which means that the point $(x,y)$ moves around a circle with radius 1, which give $\cos(t)$ and $\sin(t)$ a geometric meaning with t appearing as an angle. Find out the details of the geometry.

### Learning goals

DigiMat contains the following fundamental learning goals, which are the basis for all of Digital Math:

1. Number representation [Basic] in first binary form making representation and arithmetic algorithms easy to understand.
2. Arithmetic algorithms [Basic] constructed by repetition of the basic operation of +1 according the basic prototype of all computer programs of DigiMat in the form n = n + 1
3. Time-stepping [Basic-Pro] automatically solving all (ordinary) mathematical models in the form x = x + v*dt
4. Text programing [Basic-Pro] enabling the students to understand, modify and extend the algorithms and computer realizatons themselves.
5. Finite Element Method (FEM) [Pro] automatically solving all continuum mathematical models in the form $r(u, v) = 0, \forall v \quad in V_h$
6. Real Simulation [Pro] directly predicting First Principles models $R(u)$ such as aerodynamics with FEM with adaptive error control, stability, and in a Best Possible sense in the form $(R(U), v) + h(R(U), R(v)) = 0, \forall v \quad \in V_h$