Skip to main content

Math fundamentals and Katex

It was really tough for me to understand many articles about data science due to the requirements of understanding mathematics (especially linear algebra). I’ve started to gain some basic knowledges about Math by reading a book first.

The great tool Typora and stackedit with supporting Katex syntax simply helps me to display Math-related symbols.

Let’s start!

The fundamental ideas of mathematics: “doing math” with numbers and functions. Linear algebra: “doing math” with vectors and linear transformations.

1. Solving equations

Solving equations means finding the value of the unknown in the equation. To find the solution, we must break the problem down into simpler steps. E.g:

x24=45x24+4=45+4x2=49x=49x=7x=7 or x=7\begin{aligned} x^2 - 4 &= 45\\ x^2 - 4 + 4 &= 45 + 4\\ x^2 &= 49\\ \sqrt{x}&=\sqrt{49}\\ |x| &= 7\\ x=7 &\text{ or } x=-7 \end{aligned}

2. Numbers

Mathematicians like to classify the different kinds of number-like objects into sets:

  • The natural numbers: NN = {0,1,2,3,4,5,6,70, 1, 2, 3, 4, 5, 6, 7, … }
  • The integer: ZZ = { … , 3,2,1,0,1,2,3-3, -2, -1, 0, 1, 2, 3, … }
  • The rational numbers: QQ = {53{5}\over{3}, 227{22}\over{7}, 1.5,0.125,71.5, 0.125, -7, … }
  • The real numbers: RR = {1,0,1,2,e,π,4.94...-1, 0, 1, \sqrt{2}, e, \pi, 4.94..., … }
  • The complex numbers: CC = {1,0,1,i,1+i,2+3i-1, 0, 1, i, 1 + i, 2 + 3i, … }

Operations on numbers

  • Addition is commutative and associative. That means:
    a+b=b+aa + b = b+ a
    a+b+c=(a+b)+c=a+(b+c)a + b + c = (a + b) + c = a + (b + c)
  • Subtraction is the inverse operation of addition.
  • Multiplication is also commutative and associative.
    ab=a+a+a+...+ab times=b+b+b+...+ba timesab = \underbrace{a + a + a + ... + a}_{\text{b times}} = \underbrace{b + b + b + ... + b}_{\text{a times}}
    ab=baab = ba
    abc=(ab)c=a(bc)abc = (ab)c = a(bc)
  • Division is the inverse operation of multiplication. You cannot divide by 0.
  • Exponentiation is multiplying a number by itself many times.
    ab=aaa...ab timesa^b = \underbrace{aaa...a}_{\text{b times}}
    ab=1aba^{-b} = {{1}\over{a^b}}
    ana1n\sqrt[n]{a} \equiv a^{{1}\over{n}}

The symbol “\equiv” stands for “is equivalent to” and is used when two mathematical object are identical.

3. Variables

Variables are placeholder names for any number or unknown. Variable substitution: we can often change variables and replace one unknown variable with another to simplify an equation. For example:

65x=xu=x65u=u \begin{aligned} {6 \over{5 - \sqrt{x}}} = \sqrt{x}\\ u = \sqrt{x}\\ {6 \over{5 - u}} = u \end{aligned}

4. Functions and their inverses

The inverse function f1f^{-1} performs the opposite action of the function ff so together the two functions cancel each other out. For example:

  1. f(x)=cf(x) = c
  2. f1(f(x))=x=f1(c)f^{-1}(f(x)) = x = f^{-1}(c)
  3. x=f1(c)x=f^{-1}(c)

Common functions and their inverses:
functionf(x)inversef1(x)x+2x22x12xx2±x2xlog2(x)3x+513(x5)axloga(x)exp(x)exln(x)loge(x)sin(x)sin1(x)arcsin(x)cos(x)cos1(x)arccos(x) \begin{aligned} function f(x) &\Leftrightarrow inverse f^{-1}(x)\\ x+2 &\Leftrightarrow x-2\\ 2x &\Leftrightarrow {1\over2}x\\ x^2 &\Leftrightarrow \pm{\sqrt{x}}\\ 2^x &\Leftrightarrow log{_2}(x)\\ 3x + 5 &\Leftrightarrow {1\over 3}(x-5)\\ a^x &\Leftrightarrow log{_a}(x)\\ exp(x) \equiv e^x &\Leftrightarrow ln(x) \equiv log{_e}(x)\\ sin(x) &\Leftrightarrow sin^{-1}(x) \equiv arcsin(x)\\ cos(x) &\Leftrightarrow cos^{-1}(x) \equiv arccos(x) \end{aligned}
The principle of “digging” (Bruce Lee-style) toward the unknown by applying inverse functions is the key for solving all these types of equations, so be sure to practice using it.

5. Basic rules of algebra

Given any three numbers a, b, and c we can apply the following algebraic properties:

  • Associative property: a+b+c=(a+b)+c=a+(b+c)a + b + c = (a + b) + c = a + (b+ c) and abc=(ab)c=a(bc)abc = (ab)c = a(bc)
  • Commutative property: a+b=b+aa + b = b + a and ab=baab = ba
  • Distributive property: a(b+c)a(b + c) = ab+acab + ac

Some algebraic tricks are useful when solving equations

  • Expanding brackets: (x+3)(x+2)=x2+5x+6(x + 3)(x +2) = x^2 + 5x + 6
  • Factoring: 2x2y+2x+4x=2x(xy+1+2)=2x(xy+3)2x^2y + 2x + 4x = 2x(xy + 1 + 2) = 2x(xy + 3)
  • Quadratic factoring: x25x+6=(x2)(x3)x^2-5x+6=(x-2)(x-3)
  • Completing the square: Ax2+Bx+C=A(xh)2+kAx^2 + Bx + C = A(x- h)^2 + k e.g: x2+4x+1=(x+2)23x^2 + 4x + 1 = (x + 2)^2-3

6. Solving quadratic equations

The solutions to the equation ax2+bx+c=0ax^2 + bx + c =0 are
x1=b+b24ac2aandx2=bb24ac2a x_1 = {{-b + \sqrt{b^2 - 4ac}}\over{2a}} \quad and \quad x_2 = {{-b -\sqrt{b^2-4ac}}\over{2a}}
Actually, we can use the technique completing the square to explain this formula.

7. The Cartesian plane

Vectors and points

  • Point: P=(Px,Py)P = (P_x, P_y). To find this point, start from the origin and move a distance PxP_x on the x-axis, then move a distance PyP_y on the y-axis.
  • Vector: v=(vx,vy)\overrightarrow{v} = (v_x, v_y). Unlike points, we don’t necessarily start from the plane’s origin when mapping vectors.

Graphs of functions

The Cartesian plane is great for visualizing functions, f:RRf: {R} \rightarrow {R}

A function as a set of input-output pairs (x,y)=(x,f(x))(x, y) = (x, f(x))

8. Functions

We use functions to describe the relationship between variables.

To “know” a function, you must be able to understand and connect several of its aspects including definition, graph, values and relations.

Definition: f:ABf: A \rightarrow B. Function is a mapping from numbers to numbers.

  • Function composition: fog(x)f(g(x))=zfog(x)\equiv f(g(x)) = z
  • Inverse function: f1(f(x))f1of(x)=xf^{-1}(f(x)) \equiv f^{-1}o f(x) = x
  • Table of values: {(x1,f(x1)),(x2,f(x2)),...}\{(x1, f(x1)), (x2, f(x2)), ...\}
  • Function graph: using the Cartesian plane
  • Relations: e.g: sin2x+cos2x=1sin^2x + cos^2x = 1

9. Function references

- Line

The equation of a line: f(x)=mx+bf(x) = mx + b and f1(x)=1m(xb)f^{-1} (x) = {{1\over m} (x-b)}
The general equation: Ax+By=CAx + By = C

- Square/Quadratic: f(x)=x2f(x) = x^2

- Square root: f(x)=xx12f(x) = \sqrt x \equiv x ^{1\over2}

- Absolute value: f(x)=x={xif x0,cif x<0.f(x) = |x| = \begin{cases} x &\text{if } x \ge 0, \\ c &\text{if } x \lt 0.\end{cases}

- Polynomial functions: f(x)=a0+a1x+a2x2+a3x3+...+anxnf(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + ... + a_nx^n

- Sine: f(x)=sin(x)f(x) = sin(x)

- Consine: f(x)=cos(x)=sin(x+π2)f(x) = cos(x) = sin(x + {\pi\over2})

- Tangent: f(x)=tan(x)sin(x)cos(x)f(x) = tan(x) \equiv {sin(x)\over{cos(x)}}

- Exponential: f(x)=exexp(x)f(x) = e^x \equiv exp(x)

- Natural logarithm: f(x)=ln(x)=loge(x)f(x) = ln(x) = log_e(x)

- Function transformation

Vertical translation: g(x)=f(x)+kg(x) = f(x) + k

Horizontal translation: g(x)=f(xh)g(x) = f(x-h)

Veritcal scaling: g(x)=Af(x)g(x) = Af(x)

Horizontal scaling: g(x)=f(ax)g(x) = f(ax)

- General quadratic function: f(x)=A(xh)2+kf(x) = A(x-h)^2 + k

- General sine function: f(x)=Asin(2πλxϕ)f(x) = Asin({2\pi\over\lambda}x - \phi)

10. Polynomials

In general, a polynomial of degree nn has the equation

f(x)=anxn+an1xn1+...+a2x2+a1x+a0k=0nakxkf(x) = a_nx^n + a_{n-1}x^{n-1} + ...+ a_2x^2 + a_1x + a_0 \equiv \displaystyle\sum_{k=0}^na_kx^k

11. Trigonometry

Pythagoras’ theorem
adj2+opp2=hyp2adj2hyp2+opp2hyp2=1sin2(θ)+cos2(θ)=1 \begin{aligned} |adj|^2 + |opp|^2 &= |hyp|^2\\ {{|adj|^2}\over{|hyp|^2}} + {{|opp|^2}\over{|hyp|^2}} &= 1\\ sin^2(\theta) + cos^2(\theta) &= 1 \end{aligned}

12. Trigonometric identities

sin2(θ)+cos2(θ)=1sin^2(\theta) + cos^2(\theta) = 1

sin(a+b)=sin(a)cos(b)+sin(b)cos(a)sin(a+ b) = sin(a)cos(b) + sin(b)cos(a)

cos(a+b)=cos(a)cos(b)sin(a)sin(b)cos(a+b) = cos(a)cos(b) - sin(a)sin(b)

And, more …

13. Geometry

A: area, P: perimeter, V: volume

Triangles: A=12ahaA = {1\over2}ah_a, P=a+b+cP = a + b + c

Sphere: A=4πr2A=4\pi r^2, V=43πr3V={{4\over3}\pi r^3}

Cylinder: A=2(πr2)+(2πr)hA = 2(\pi r^2) + (2\pi r)h, V=(πr2)hV = (\pi r^2)h

14. Circle

Radians: 2π[rad]=36002\pi [rad] = 360^0

15. Sovling systems of linear equations

a1x+b1y=c1a2x+b2y=c2 \begin{aligned} a_1x + b_1y = c1\\ a_2x + b_2y=c_2 \end{aligned}
There are some approaches to sovling it:

  • Solving by substitution
  • Solving by substraction
  • Solving by equating

[1]. Ivan Savov, “No bullshit guide to linear algebra”.


  1. I like your blog, I read this blog please update more content on python, further check it once at python online training


Post a Comment

Popular posts from this blog

Applying pipeline “tensorflow_embedding” of Rasa NLU

According to this nice article, there was a new pipeline released using a different approach from the standard one (spacy_sklearn). I wanted to give it a try to see whether it can help with improving bot’s accuracy.

After applying done, I gave an evaluation of “tensorflow_embedding”. It seemed to work better a bit. For example, I defined intents “greet” and “goodbye” with some following messages in my training data.
## intent:greet- Hey! How are you? - Hi! How can I help you? - Good to see you! - Nice to see you! - Hi - Hello - Hi there ## intent:goodbye- Bye - Bye Bye - See you later - Take care - Peace In order to play around with Rasa NLU, I created a project here. You can have a look at this change from this pull request. Yay!

When I entered message “hi bot”, then bot with “tensorflow_embedding” could detect intent “greet” with better confidence scores rather than bot with “spacy_sklearn”. The following are responses after executing curl -X POST localhost:5000/parse -d '{&qu…

Creating a Chatbot with RiveScript in Java

Motivation"Artificial Intelligence (AI) is considered a major innovation that could disrupt many things. Some people even compare it to the Internet. A large investor firm predicted that some AI startups could become the next Apple, Google or Amazon within five years"
- Prof. John Vu, Carnegie Mellon University.

Using chatbots to support our daily tasks is super useful and interesting. In fact, "Jenkins CI, Jira Cloud, and Bitbucket" have been becoming must-have apps in Slack of my team these days.

There are some existing approaches for chatbots including pattern matching, algorithms, and neutral networks. RiveScript is a scripting language using "pattern matching" as a simple and powerful approach for building up a Chabot.
Architecture Actually, it was flexible to choose a programming language for the used Rivescript interpreter like Java, Go, Javascript, Python, and Perl. I went with Java.

Used Technologies and ToolsOracle JDK 1.8.0_151Apache Maven 3.5…

[Snippet] CSS - Child element overlap parent

I searched from somewhere and found that a lot of people says a basic concept for implementing this feature looks like below:
HTML code:
<div id="parent">  <div id="child">  </div> </div> And, CSS:
#parent{   position: relative;   overflow:hidden; } #child{   position: absolute;   top: -1;   right: -1px; }
However, I had a lot of grand-parents in my case and the above code didn't work. Therefore, I needed an alternative. I presumed that my app uses Boostrap and AngularJs, maybe some CSS from them affects mine. I didn't know exactly the problem, but I believed when all CSS is loaded into my browser, I could completely handle it.

I tried to create an example to investigated this problem by Fiddle. Accidentally, I just changed: position: parent; to position: static; for one of parents -> the problem is solved.
Look at my code:

<div class="modal-body dn-placeholder-parent-position"> <div clas…

How I did customize "rasa-nlu-trainer" as my own tool

Check out my implementation here Background I wanted to have a tool for human beings to classify intents and extract entities of texts which were obtained from a raw dataset such as's conversation, Maluuba Frames or here. Then, the output (labeled texts) could be consumed by an NLU tool such as Rasa NLU.

rasa-nlu-trainer was a potential one which I didn't need to build an app from scratch. However, I needed to add more of my own features to fulfill my needs. They were:

1. Loading/displaying raw texts stored by a database such as MongoDB
2. Manually labeling intents and entities for the loaded texts
3. Persisting labeled texts into the database

I firstly did look up what rasa-nlu-trainer's technologies were used in order to see how to implement my mentioned features.
At first glancerasa-nlu-trainer was bootstrapped with Create React App. Create React App is a tool to create a React app with no build configuration, as it said. This tool is also recommended by the …

The HelloWorld example of JSF 2.2 with Myfaces

I just did by myself create a very simple app "HelloWorld" of JSF 2.2 with a concrete implementation Myfaces that we can use it later on for our further JSF trying out. I attached the source code link at the end part. Just follow these steps below:

1. Create a Maven project in Eclipse (Kepler) with a simple Java web application archetype "maven-archetype-webapp". Maven should be the best choice for managing the dependencies, so far. JSF is a web framework that is the reason why I chose the mentioned archetype for my example.

2. Import dependencies for JSF implementation - Myfaces (v2.2.10) into file pom.xml. The following code that is easy to find from with key words "myfaces".

<dependency> <groupId>org.apache.myfaces.core</groupId> <artifactId>myfaces-api</artifactId> <version>2.2.10</version> </dependency> <dependency> <groupId>org.apache.myfaces.core</groupId&g…