GUIDE.md

Usage

Types

Scalar

Scalars are 2x64-bit rational numbers - let's call this field $\hat{Q}$. They can be signed. Examples

10
-45
2/13
-75/3

Rational numbers are always normalized e.g. if number is in form $\frac{p}{q}$ it will be displayed as $\frac{p/g}{q/g}$, where $g=\text{gcd}(p, q)$. Also if $q/g = 1$ the number is displayed as an integer.

Scalars can be both added in shell or via GUI editor.

Matrix

Matrices are 2D arrays of Scalars. More precisely matrix $A \in \hat{Q}^{N\times M}$ is a matrix over field $\hat{Q}$ with width $N$ and height $M$.

Currently, the only way to create a matrix is to use GUI editor.

Warning

A matrix $M\in \hat{Q}^{1,1}$ is not a scalar. It is a matrix with one element.

Variables

Variables are supported, and they are calculated during their initialization. Let's say there is a variable called x and it stores the value $\frac{1}{3}$. Creating variable y = x will result in copying the value of x into y, so later changes to x will not apply to y.

Operations

• Addition - both Scalars and Matrices support addition operation.
  • For Scalars it is defined as $\hat{Q} \times \hat{Q} \to \hat{Q}$, and works as expected for rational numbers.
  • For Matrices it is defined as $\hat{Q}^{N\times M} \times \hat{Q}^{N\times M} \to \hat{Q}^{N\times M}$ and works element-wise.
• Subtraction - both Scalars and Matrices support subtraction operation.
  • For both Scalars and Matrices it is defined as $\hat{Q} \times \hat{Q} \to \hat{Q}$ and works like addition.
• Multiplication - both Scalars and Matrices support multiplication operation.
  • For Scalars it is defined as $\hat{Q} \times \hat{Q} \to \hat{Q}$, and works as expected for rational numbers.
  • For Matrices it is defined as $\hat{Q}^{N\times M} \times \hat{Q}^{M\times K} \to \hat{Q}^{N\times K}$ and works as expected for matrices.
  • For Matrices and Scalars it is defined as $\hat{Q}^{N\times M} \times \hat{Q} \to \hat{Q}^{N\times M}$ and works as expected for matrices and scalars.
• Division - only Scalars support division operation.
  • For Scalars it is defined as $\hat{Q} \times \hat{Q} \to \hat{Q}$, and works as expected for rational numbers.
• Inverse - only Matrices support inverse operation.
  • For Matrices it is defined as $\hat{Q}^{N\times N} \to \hat{Q}^{N\times N}$. Inverse $A^{-1}$ of matrix $A$ is defined as $A^{-1}A = AA^{-1} = I$, where $I$ is identity matrix.
• Echelon - only Matrices support echelon operation.
  • For Matrices it is defined as $\hat{Q}^{N\times M} \to \hat{Q}^{N\times M}$. Echelon form is defined here.
• Power - both Scalars and Matrices support power operation.
  • For Scalars it is defined as $\hat{Q} \times \mathbb{N} \to \hat{Q}$, and works as expected for rational numbers.
  • For Matrices it is defined as $\hat{Q}^{N\times N} \times \mathbb{N} \to \hat{Q}^{N\times N}$. Power $A^k$ of matrix $A$ is defined as $A^k = A \cdot A \cdot \dots \cdot A$ where $k$ is a positive integer.

Examples

x = 1/3
y = 2/3
M = [1 2; 5 3]  % This syntax is not supported yet.
N = [6 7; 3 1]  % Use GUI editor instead.
% Addition
z = x + y   % z = 1
P = M + N   % P = [7 9; 8 4]
% Subtraction
z = x - y   % z = -1/3
P = M - N   % P = [-5 -5; 2 2]
% Multiplication
z = x * y   % z = 2/9
P = M * N   % P = [18 10; 36 22]
% Division
z = x / y   % z = 1/2
% Inverse (only in GUI)
P = inv(M)    % P = 1/7 * [-3 2; 5 -1]
% Echelon (only in GUI)
P = echelon(M)    % P = [1 0; 0 1]
% Power
z = x^2   % z = 1/9
P = M^2   % P = [11 8; 20 19]

Shell

Shell is a command line interface for the calculator. It is used to input commands. Supported commands are:

• x = <expression> - creates a variable x and assigns it the value of <expression>.
• <expression> - evaluates <expression> and stores it in special variable $. Error messages are displayed as a popup notification toast.

These are the rules expressed in BNF:

<digit>      ::= "0" | "1" | ... | "9"
<integer>    ::= <digit>+
<letter>     ::= "a" | "ą" | "b" | ... | "ż"
<identifier> ::= (<letter> | "_") (<letter> | <digit> | "_")* | "$"
<unary_op>   ::= "+" | "-"
<binary_op>  ::= "+" | "-" | "*" | "/"
<expr>       ::= <integer> | <identifier> | <expr> <binary_op> <expr> | "(" <expr> ")" | <unary_op> <expr>

Examples

v = 1/3 + 4/15 - 4/19 * 2/3 - 4^5 * (3/4 - 2/3)
w = ((((4/3 + 5/2) * 14) - 44) / 2) ^ 2
N = M^14 - Z * 4 * (M - Z)    % where M, Z are square matrices
a = -v
very_simple_NAME_123 = 1/3

GUI

GUI is a graphical user interface for the calculator. All objects created in current environment are displayed on Objects list. Clicking on an object will open a new window with object's properties. In such window, the value can be edited. If the value is edited, the object will be updated. There are certain operations that can be performed on objects:

• Scalar
  • Inverse - calculates inverse of the scalar, copies its LaTeX representation to clipboard and stores the numerical value in $.
  • LaTeX - copies the scalar's LaTeX representation to clipboard.
• Matrix
  • Inverse - calculates inverse of the matrix, copies its LaTeX representation to clipboard and stores the numerical value in $.
  • LaTeX - copies the matrix's LaTeX representation to clipboard. If an error occurs during the operation, the error message will be displayed as a popup toast.
  • Echelon - calculates echelon form of the matrix, stores the numerical value in $ and copies all transitions in LaTeX to clipboard.

Echelon LaTeX example

Let's say we have a matrix

$$A = \begin{bmatrix} 1 & 2 & 3 \\ 11 & 67 & 2 \\ 8 & 1 & 34 \end{bmatrix}.$$

Echelon operation will result in LaTeX code representing this:

$$\left[\begin{array}{ccc} 1 & 2 & 3\\11 & 67 & 2\\8 & 1 & 34 \end{array}\right] \xrightarrow{\substack{w_{2} - 11w_{1}\\w_{3} - 8w_{1}}} \left[\begin{array}{ccc} 1 & 2 & 3\\0 & 45 & -31\\0 & -15 & 10 \end{array}\right] \xrightarrow{w_{2} : 45} \left[\begin{array}{ccc} 1 & 2 & 3\\0 & 1 & -\frac{31}{45}\\0 & -15 & 10 \end{array}\right] \xrightarrow{\substack{w_{1} - 2w_{2}\\w_{3} + 15w_{2}}} \left[\begin{array}{ccc} 1 & 0 & \frac{197}{45}\\0 & 1 & -\frac{31}{45}\\0 & 0 & -\frac{1}{3} \end{array}\right] \xrightarrow{w_{3} : \left(-\frac{1}{3}\right)} \left[\begin{array}{ccc} 1 & 0 & \frac{197}{45}\\0 & 1 & -\frac{31}{45}\\0 & 0 & 1 \end{array}\right] \xrightarrow{\substack{w_{1} - \frac{197}{45}w_{3}\\w_{2} + \frac{31}{45}w_{3}}} \left[\begin{array}{ccc} 1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1 \end{array}\right]$$

This may not look useful, as it produced an identity matrix, but when we take a different matrix

$$B = \left[\begin{array}{cccc}1 & 4 & 0 & 15\\6 & 11 & 8 & 4\\-1 & 3 & 6 & -6\end{array}\right]$$

which represents a system of linear equations, we get a much more useful result:

$$\left[\begin{array}{cccc}1 & 4 & 0 & 15\\6 & 11 & 8 & 4\\-1 & 3 & 6 & -6\end{array}\right] \xrightarrow{\substack{w_{2} - 6w_{1}\\w_{3} + w_{1}}} \left[\begin{array}{cccc}1 & 4 & 0 & 15\\0 & -13 & 8 & -86\\0 & 7 & 6 & 9\end{array}\right] \xrightarrow{w_{2} : \left(-13\right)} \left[\begin{array}{cccc}1 & 4 & 0 & 15\\0 & 1 & -\frac{8}{13} & \frac{86}{13}\\0 & 7 & 6 & 9\end{array}\right] \xrightarrow{\substack{w_{1} - 4w_{2}\\w_{3} - 7w_{2}}} \left[\begin{array}{cccc}1 & 0 & \frac{32}{13} & -\frac{149}{13}\\0 & 1 & -\frac{8}{13} & \frac{86}{13}\\0 & 0 & \frac{134}{13} & -\frac{485}{13}\end{array}\right] \xrightarrow{w_{3} : \frac{134}{13}} \left[\begin{array}{cccc}1 & 0 & \frac{32}{13} & -\frac{149}{13}\\0 & 1 & -\frac{8}{13} & \frac{86}{13}\\0 & 0 & 1 & -\frac{485}{134}\end{array}\right] \xrightarrow{\substack{w_{1} - \frac{32}{13}w_{3}\\w_{2} + \frac{8}{13}w_{3}}} \left[\begin{array}{cccc}1 & 0 & 0 & -\frac{171}{67}\\0 & 1 & 0 & \frac{294}{67}\\0 & 0 & 1 & -\frac{485}{134}\end{array}\right]$$

Producing a row echelon form is a very tedious task, but with this calculator, it is as easy as clicking a button. It may be very useful for students, as it can be used to create LaTeX for their homework.

Another useful application is to calculate the inverse of a matrix. Let's say we have a matrix

$$Z = \left[\begin{array}{cc}1 & 2\\6 & 1\end{array}\right]$$

If we produce a matrix

$$Z' = \left[\begin{array}{cc|cc}1 & 2 & 1 & 0\\6 & 1 & 0 & 1\end{array}\right]$$

and perform Echelon operation on it, we get

$$\left[\begin{array}{cccc}1 & 2 & 1 & 0\\6 & 1 & 0 & 1\end{array}\right] \xrightarrow{\substack{w_{2} - 6w_{1}}} \left[\begin{array}{cccc}1 & 2 & 1 & 0\\0 & -11 & -6 & 1\end{array}\right] \xrightarrow{w_{2} : \left(-11\right)} \left[\begin{array}{cccc}1 & 2 & 1 & 0\\0 & 1 & \frac{6}{11} & -\frac{1}{11}\end{array}\right] \xrightarrow{\substack{w_{1} - 2w_{2}}} \left[\begin{array}{cccc}1 & 0 & -\frac{1}{11} & \frac{2}{11}\\0 & 1 & \frac{6}{11} & -\frac{1}{11}\end{array}\right]$$

and as surprising as it may seem, the inverse of Z is

$$\left[\begin{array}{cc}-\frac{1}{11} & \frac{2}{11}\\\frac{6}{11} & -\frac{1}{11}\end{array}\right]$$

We just inverted the matrix Z by performing elementary row operations on the augmented matrix and got all transformations in LaTeX. Finally, the Inverse operation will also produce LaTeX code representing all the transformations.

GUI editor

GUI editor is a graphical interface for creating matrices and scalars. To open it click on Add matrix or Add scalar button. A new variable has to have a name and a value, that can be evaluated using existing environment variables. If provided value is invalid, an error message will be displayed and new variable will not be created.

Features

If you get bored with plain background and want to spice things up, you can turn fft feature on. It will draw an image of a Fourier transformed image provided in assets/. The other way to change the background is to turn clock feature on. It will draw a fractal clock in the background. If both fft and clock are turned on, fft will prioritize clock - only if image file is missing the clock will be drawn.