finish md book

docs/book/src/SUMMARY.md

@@ -1,9 +1,22 @@
 # Summary
 
-- [Install](./install.md)
-- [Usage](./usage.md)
-- [Functions](./function.md)
-- [Configuration](./configuration.md)
-- [Logic](./logic.md)
-- [Plot](./plot.md)
-- [Exact math](./exact.md)
+# Basic Usage
+- [Introduction](./install.md)
+- [Chapter I. Usage](./usage.md)
+
+# Advanced Usage
+- [Chapter II. Functions](./function.md)
+- [Chapter III. Configuration](./configuration.md)
+- [Chapter IV. Logic](./logic.md)
+- [Chapter V. Plot](./plot.md)
+  - [Section V.1. GUI](plot/gui.md)
+  - [Section V.2. Terminal](plot/term.md)
+
+# Algebra
+- [Chapter VI. Algebra](algebra/index.md)
+  - [Section VI.1. Vectors](algebra/vectors.md)
+  - [Section VI.2. Matrices](algebra/matrices.md)
+- [Chapter VII. Exact math](exact/index.md)
+  - [Section VII.1. Rational Reduction](exact/rational.md)
+  - [Section VII.2. Symbolic Reduction](exact/symbolic.md)
+  - [Section VII.3. Function differentiation](exact/diff.md)

docs/book/src/algebra/index.md

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+# Chapter VI. Algebra
+
+The calculator supports basic algebraic calculations, including
+
+- Vectors
+  - Dot product (operator `*` - with another vector.)
+  - Scalar product (operator `*` - with a scalar.)
+  - Norm (function `norm`)
+  - Addition (operator `+`)
+- Matrices
+  - Addition (operator `+`)
+  - Multiplication (operator `*` - with another matrix.)
+  - Scalar multiplication (operator `*` - with a scalar.)
+  - Transposition (function `transpose`)
+  - Determinant (via _LUP-decomposition_ function `det`)
+  - Inversion (via _LUP-decomposition_ function `invert`)
+
+The matrices (and vectors) are pretty printed (and aligned) (since `v2.11.5`)
+
+![matrix_pretty_printed](/assets/aligned_matrices.png)

docs/book/src/algebra/matrices.md

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+# Section VI.2. Matrices
+As of `v2.7.0` matrix algebra supports
+
+- Addition (operator `+`)
+- Multiplication (operator `*` - with another matrix.)
+- Scalar multiplication (operator `*` - with a scalar.)
+- Transposition (function `transpose`)
+- Determinant (via _LUP-decomposition_ function `det`)
+- Inversion (via _LUP-decomposition_ function `invert`)
+
+![matrix](/assets/matrix.png)

docs/book/src/algebra/vectors.md

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+# Section VI.1. Vectors
+Basic vector calculation supports
+
+- Addition (operator `+`)
+- Dot product (operator `*` - with another vector.)
+- Scalar product (operator `*` - with a scalar.)
+- Norm (function `norm`)
+
+![vector_calc](/assets/vector.png)

docs/book/src/configuration.md

@@ -1,4 +1,4 @@
-# Configuration
+# Chapter III. Configuration
 The calculator is completely configurable, you can change the general color,
 the greeting message, the greeting color, the prompt and prompt color
 in a toml file found in your config folder.

docs/book/src/exact/diff.md

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+# Section VII.3. Function differentiation
+As of `v3.2.0`, the calculator can differentiate both known functions (built-in)
+and user-defined functions constructed using known functions and operations.
+
+<h2 style="color:red;">Warning</h2> 
+
+Beware, as of `v3.3.3` there is some bugs to iron out. It doesn't 
+differentiate variables (for example `x*x` works but not `x²`). 
+And differentiation of function referencing each other 
+(for example `v(x) = f(x) + x*x`) doesn't work either.
+
+It's currently being fixed, and will be fixed before stable release `v3.4.0` 
+which is currently in `alpha` development-testing (it will implements basic languages
+features 
+such as flow control and condition.)
+
+The release `v3.3.4`, which correct those bugs is currently in `beta` dev-testing 
+and will be released before `v3.4.0`.
+
+## Examples
+### Built-in 
+![diff_bi](/assets/diff_builtin.png)
+
+### User defined
+![ud](/assets/diff_ud.png)

docs/book/src/exact/index.md

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+# Chapter VII. Exact math
+
+Exact math has been added between version `v2.11.0` to version `v3.3.3`

docs/book/src/exact/rational.md

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+# Section VII.1. Rational Reduction
+
+As of version `v2.11.0` rational exact math reduction was added, it supports.
+
+- Rational operations
+  - `+`
+  - `-`
+  - `*`
+  - `/`
+- Rational reductions
+- Float rationalization (with 10 digits of precision)
+
+## Operations
+![operations](/assets/exact_rationals.png)
+
+## In matrices
+![matrix](/assets/exact_inverse.png)

docs/book/src/exact/symbolic.md

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+# Section VII.2. Symbolic Reduction
+
+As of `v3.0.0` the support for symbolic reduction has been added to calc. 
+It supports multi-variables multi-operations reductions, you can reduce 
+expressions using it, fair warning, it uses _a lot_ of parentheses
+
+![symbolic reduction](/assets/multi_variable.png)

docs/book/src/function.md

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-# Functions
+# Chapter II. Functions
 
 ## Built-in functions
 The following functions are currently (as of `3.3.0`) built-in

docs/book/src/logic.md

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-# Logic
+# Chapter IV. Logic
+
+## Infix Operators
+
+- `or` (alias `||`)
+- `and` (alias `&&`)
+- `geq` (alias `>=`)
+- `leq` (alias `<=`)
+- `gt` (alias `>`)
+- `lt` (alias `<`)
+
+![example](/assets/logic.png)
+

docs/book/src/plot.md

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-# Plot
+# Chapter V. Plot
+
+You can plot, provided you have `gnuplot` installed, 
+which the backend relies on.
+
+It works great on Linux and macOS, but it's not tested on Windows.
+
+## Help
+As the `plot` function is a little difficult you can call it without 
+parameters to show the help `plot()`
+
+![plot_help](/assets/plot_help.png)

docs/book/src/plot/gui.md

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+# Section V.1. GUI
+
+## Default 
+
+To plot a function `f` you just need to do `plot(f)`
+
+![plot_default](/assets/plot_cos_default.png)
+
+## Options
+You can pass parameters to the function.
+
+```
+> plot(sin,-pi,pi,0.01,"sin","x(rad)","y","line")
+```
+
+Let's unpack, we have ploting the `sin` function, from `x=-pi` to `x=pi`
+with a step of `0.01 rad`, the title of the graph is `sin`, the x_axis 
+label is `x(rad)`, the y_axis label is `y` and we want a line.
+
+![plot_custom](/assets/plot_sin_custom.png)
+
+## User-defined functions
+You can plot your own functions with the same call. 
+Here is the example for `f(x) = x*x`
+
+![plot_f](/assets/plot_f.png)
+

docs/book/src/plot/term.md

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+# Section V.2. Terminal
+You can also plot right into the terminal, without need for the `gnuplot`
+backend.
+
+## Default
+The best example to show it to you (before the auto-scaling) is the square
+function, from `x=-5` to `x=5` with a step of `0.1`.
+
+![plot_term](/assets/plot_term_x_squared.png)
+
+## Options
+Like the GUI plotting, the terminal supports options
+
+![plot_term_option](/assets/plot_term_x_squared_labels.png)
+
+## Auto scaling
+Both the `x` and `y` axis supports _auto-scaling_ 
+
+![plot_term_scaling](/assets/termplot_cos.png)

docs/book/src/usage.md

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-# Usage 
+# Chapter I. Usage 
 
 ## Basic Operators