Aashishium

Indian mathematician & computer programmer | Indian history geek | Photo-blogger | Foodie

[aashishium]

P&S 01: Experiments , sample space & axioms of probability

Date : 14th November 2018 (Children's day) So I started learning concepts of probability and statistics from Probabilistic system analysis & applied probability course from OCW Scholar series of Massachusetts Institute of Technology's Open Course Work...

Completed lecture 1 of unit 1 of “Probability system analysis & applied probability” from @mit OCW.

[aashishium]

Day 4 of trigonometry from Khan Academy

Such a great day !! Just completed the section of Trigonometry with general triangles and it’s unit test on KA (Here’s my profile:  https://www.khanacademy.org/profile/aashishium/ ). I learnt a lot many new things today such as -

1. Solving for an angle with the law of sines , accoring to which -

When there is a triangle with sides A , B & C and the angle opposite to the sides A , B & C are a , b & c. then -

sin(a)/A = sin(b)/B = sin(c)/C

2. Proof of law of sines (which was damn interesting).

3. Solving for a side and angle with the law of cosines , according to which -

When there is a triangle with sides A , B & C and the angle opposite to the sides A , B & C are a , b & c. then -

A^2=B^2 + C^2 - 2*B*C*cos(a)

4. Proof of the law of cosines. 

5. Review of the law of sines & cosines.

[aashishium]

Day 3 of trigonometry from Khan academy

:) Finally completed the section of Trigonometry with right triangles and started the section of Trigonometry with general triangles , which I hope to complete within 2-3 days . Things I learned today are -

1. Introduction to pythagorean trig identity through which we could prove sin(A)^2 + cos(A)^2 =1 .

2.Reciprocal trigonometric identities (cosecant , secant & cotangent) where cosec(A) = 1/sin(A) , sec(A) = 1/cos(A) & cot(A) = 1/tan(A).

3. Solving for a side with the law of sines. 

4. Reviewing Trig with right triangles

[aashishium]

Day 2 of trigonometry from Khan Academy

Today I almost completed the section of “Trigonometry of right triangles” from Khan Academy (except for two small sections which I am going to complete tommorow and start the new section “Trigonometry of general triangles”). Today I learnt about -

1. Trig to solve angles & sides of a right triangle.

2. Inverse trig ratios & solving for an angle in a right triangle using trig ratios.

3. Modelling with right triangle with angle of elevation & depression.

4. Review of right triangle trig -

sin(A)=opposite/hypotenuse

cos(A)=adjacent/hypotenuse

tan(A)=opposite/adjacent

where A represents an angle in a right triangle. The above rules can be remembered as soh-cah-toa.

5. Similarities between triangle & trig ratios

6. Sine & cosine of complementary angles : 

sin(A)=cos(90-A)

cos(A)=sin(90-A)

7. cofunctions : Two functions f() & g() are co-functions if -

f(A)=g(Ac-A)

g(A)=f(Ac-A)

where , A = value & Ac = Conjugate of A

8. Special triangles :

30-60-90 triangle : Ratio of sides of such a triangle would be 1:√3:2.

45-45-90 triangle : Ratio of sides of such a triangle would be 1:1:√2.

[aashishium]

Day 1 of trigonometry from Khan academy

These are the few things I have learned today in Trigonometry from Khan Academy -

1. Trigonometric ratios (meaning)

2. Hypotenuse , Opposite & Adjacent sides for a right triangle

3. Sine , Cosine & Tangential trigonometric ratio for a right triangle 

4. The famous soh-cah-toa rule where ,

soh represents that sine is the ratio of opposite side wrt hypotenuse 

cah represents that cosine is the ratio of adjacent side wrt hypotenuse

toa represents that tangent is the ratio of opposite side wrt adjacent 

[aashishium]

Block 1 : 23rd June 2018 : Group theory

Group theory is perhaps one of the most pleasurable mathematical subjects ever ...

[aashishium]

Building libraries for students and learners for STEM in C/C++ , Julia , Python and more ...

I wasn’t selected for Google Science Fair (Yup , that was a little bit sad for me because I had lot’s of hope for it , but it’s okay because there’s always a next time and I am gonna try for it next year) but the project I had applied for was “Creating Julia based libraries for STEM education“ and it wasn’t that bad.

Thinking a lot about it , I decided to start my own open-source project called as “numeg“ , through which I would build software libraries for educational purposes for software assisted learning for STEM .

Will keep an update .. Bye bye.

To join me , for this noble effort , you can contact me at - [email protected]

[aashishium]

Solving biquadratic equations

Well we already know about quadratic equations which are equations of the form -

ax^2+ bx+c=0

But biquadratic equations are of the form -

ax^4+ bx^2+c=0

So there are two ways to solve / find solution of a bi-quadratic equation -

1. By standard substitution - 

Where x^2=y

And so the equation ax^4+ bx^2+c=0 becomes ay^2+ by+c=0.

2. By special transformation method -

Here the bi-quadratic equation is transformed into two individual quadratic equations.

[aashishium]

Some basic rules of numerical integration (Trapezoidal , Simpson's 1/3rd rule , Simpson's 3/8th rule & Weddle's rule)

[aashishium]

Number raised to an irrational power

Well , we are well versed with the idea of a number raised to a rational power . For example when we come across 5^0.6666... , we can clearly analyse that 0.6666.... is 1/3 and so 5^0.66666.... is equivalent to 5^(2/3) and we can explain that 5^2/3 can be expressed as cube root of 5^2=25 or square of cube root of 5 .

But what about 5 raised to square root of 2 or 5^(sqrt(2)) ?

"Well it's quite difficult to contemplate the value of a number raised to a irrational power because it is not intuitive ."

So , how are we going to find value of a number raised to an irrational power and answer comes from a concept called as sequence of numbers.

So let's take example of 5^(sqrt(2)) (5 raised to square root of 2) . For Solving it we can consider a sequence X={1.4, 1.41 ,1.414 ,1.4142 , ..... } such that x belonging to X tends to sqrt(2).

So x belonging to X tends to sqrt(2)

Hence , 5^x where x belongs to X has a limit and it tends to 5^(sqrt(2)) and by using limit of the sequence we can find the value of a number raised to an irrational power.

😊😊😊😊😊😊

[aashishium]

Beauty of commutativity

While growing up we have always been taught about algebraic systems over which commutativity always holds , one such example would be natural numbers or real numbers over addition where -

x1 + x2 = x2 + x1

But as you grow up you will realize that the algebraic systems we have been taught about almost our entire life just covers a small fraction amongst vast number of algebraic systems like Groups where commutativity doesn't comes by default and in fact commutative groups fall under a special category called as "Abelian groups".

For example , lets consider G to be a group under matrix multiplication such that -

G={Group of all 3*3 matrices where all elements belong to real numbers}.

So if we consider two matrices A and B belonging to G then A×B may not be equal to B×A in all conditions.

"We are so intricately linked with idea of commutativity that it's inexistent sometimes makes it feel like a paradox , but that's not the case."

But commutativity has it's own mathematical beauty and often leads to beautiful properties.

[aashishium]

A different way to look at divisibility theorem

Well we know about the divisibility theorem, which can be described as -

If a, b are Integers such that b>0 them there exists two unique integers q and r such that -

a=bq+r | 0<=r<b

Where , q is called as quotient

& r is called as remainder

So, if a>0 then remainder is the smallest non-negative integer obtained when a is subtracted from b multiple number of times and quotient is the number of times b is subtracted from a to obtain remainder and if a<0 then b is added to a.

Example 1 >

a=27 , b=5

1> a-b= 27-5=22 , q=+1

2> a-b-b=a-2b=27-10=17, q=+2

3> a-2b-b=a-3b=27-15=12 , q=+3

4> a-3b-b=a-4b=27-20=7 , q=+4

5> a-4b-b=a-5b=27-25=2 , q=+5

6> a-5b-b=a-6b=27-30=-3

So, here the smallest non-negative number is 2 and the number of times b has been subtracted from a to obtain 2 is 5 and so ,

Remainder= +2

Quotient= +5

And now let's check whether our view about divisibility theorem matches the actual theorem or not -

a=27,b=5

q=5,r=2

So , bq+r=5*5+2=25+2=27=a

Yeah , it works !!!

I hope looking in this way , the concept became too easy to understand !!

[aashishium]

Some basic terminologies & concepts in programming language -

Well most of the beginners when learning programming languages would basically come across two kinds of errors -

1. Compile-time error / Syntax error

Syntax error or compile-time error occurs when the user has used a wrong syntax or a syntax which the compiler cannot understand. Syntax basically refers to a combination of symbols which represents correctly structured document format.

We can understand this by a simple example -

Let's assume that a user wants to print a string Hello ,World! , so according to the syntax provided in C language , one could print the above string as -

printf("Hello ,World!");

So basically if you make any changes in -

printf(" ");

Then there will be an error (even the slightest change like changing printf to Printf) and this is just because you have not made program as per the syntax or rules provided by the C language. Think of syntax as laws provided by C language similar to the laws provided by government.

Even in other languages like python or ruby or other languages, they have a prescribed syntax and if you don't follow those then there will be error.

2. Segmentation fault / Access violation -

Well this a kind of fatal error which lead to abrupt closing of your program's executable. This occurs because of a program called as core and basically occurs when the program you have built is trying to read or write upon an illegal memory location where the program has no access previlages then there is occurence of segmentation fault or access violation.

Let's first understand what are access previlages ??

Have you ever came across a DVDs/CDs/Blu-ray disks , which basically says read-only when try to read or write anything on these disks ? Well even if you haven't , let me explain what happens in these disks, Earlier a lot of commercial game developers used to ship their games in DVDs or Blu-ray disks , but as they didn't wanted the content of the disks to be replaced , or removed - they inserted a constraint of "read-only" meaning that the user's program could only read contents of a disk but not modify it or right on it and so there are three kinds of access previlages which a program can have -

1. Read

2. Write

3. Modify

And their meaning could be understood from the words itself.

So , access violation occurs when the program is trying to access to a section of computer's memory which it doesn't has access to !

3. Prompt and Interactive computing -

Let's take an example from python language -

a=input("Enter value of a:");

This statement when executed produces an output like this -

Enter value of a:

After which you as user could enter an input and this is called as a prompt because this prompts the user to enter an input and such kind of computing , even though it's least Interactive is called as Interactive computing.

You can come across Interactive computing almost everywhere from your videos games to a Lot of other applications ...

[aashishium]

Playing with escape sequences in C language -

Well , escape sequences hold a really important part in C programming language and are often used in formatted control strings in library functions like printf() and scanf().

An escape sequence is a sequence of characters which is used inside formatted control strings and it produces an output which is difficult or almost impossible to produce directly. (for example system bell is almost impossible to produce directly and so we use an escape sequence for producing it.)

                                 Usage of escape sequences

Let’s try to understand some of the most used escape sequences one by one -

\a (alarm) - These are used for creating system bells or alarm and can be used to alarm user about occurrence of certain event. For example , in the following code when the variable will reach value of 10 in the for-loop then user’s  will be indicated by a system bell -

for(int i=0;i<=10;i++) {

if(i==10)

{ printf(”\a”); }

}

\n (newline) - These are used for moving the cursor to the next line. For example - For the following code the output will be produced as -

A

A

L

Code-

#include<stdio.h>

int main(void){

printf(”A\nA\nL”);}

\t (Horizontal tab) - Using \t will move the code to a distance horizontally which is equivalent to a tab space to the right. For example - For the following code the output will be produced as -

A          A          L

Code-

#include<stdio.h>

int main(void){

printf(”A\tA\tL”);}

\v(Vertical tab) - Using \v will move the position of the cursor in vertical tab space.For example - For the following code the output will be produced as -

Aashish

             Loknath

                           Panigrahi

Code-

#include<stdio.h>

int main(void){

printf(”Aaashish\vLoknath\vPanigrahi”);

}

\b(Backspace) - Using \b will exactly produce the same effect as what would be produces when you will enter the backsapce button in your keyboard and so it \b actually helps in removing previous character of a string.For example - For the following code the output will be produced as -

h

A

Code-

#include<stdio.h>

int main(void){

printf(”A\ba\bs\bh\bi\bs\bh\n”);

printf(”h\bs\bi\bh\bs\ba\bA”);

}

\? (Question mark) - Using \? will help in producing question mark in the formatted control string as shown below -

#include<stdio.h>

int main(void){

printf(”What is your name\?\t:”);

}

The above code will produce output as - What is your name?      :

\’ and \” (single and double quotation marks) - \’ when used in a formatted control string will produce a single quotation mark and \” will produce a double quotation mark. The reason why we need \” is that in argument we already have are using ”” and using another ” will produce an error and so we need this.

An example representing this has been shown below -

#include<stdio.h>

int main(void){

printf(”\’Hello world\’\n”);

printf(”\”Hello world\”\n”);

}

The above code will produce an output -

‘Hello world‘

“Hello world“

\r (carriage return) - Carriage return allows to cursor of the program back to the beginning of the line but it also removes the text before. You can understand this by the following example -

#include<stdio.h>

int main(void){

printf(”A\ra\rs\rh\ri\rs\rh\n”);

printf(” A\ra\rs\rh\ri\rs\rh”);

}

The above code will produce an output -

h

hA

Why in the 2nd line is the output hA whereas the output in the 1st line is h ?? It’s because in the first line every time when the cursor moves to front the previous character is deleted hence output is h where as in the second printf statement , the 1st character is not A but it is space hence while performing carriage return  space will be replace by the character ahead and at the end the output will be hA.

[aashishium]

A small experimentation with array initialization ....

Well, indeterminate values in C are really nasty and you would find one if you forgot to assign a certain value for an element for an array as you would find if you execute the following program -

Which you can also find at - https://github.com/aalpanigrahi/clang/blob/master/ARRAY/array7.c

Which has the output as shown below -

But if you do the initialization of an array as shown below , then you will find out that unlike indeterminate values of C which you will finding the output to be 0 -

Which you can find at - https://github.com/aalpanigrahi/clang/blob/master/ARRAY/array9.c

And apparently after asking this question on stackoverflow. I learnt that -

If an object that has static or thread storage duration is not initialized  explicitly, then:

— if it has pointer type, it is initialized to a null pointer;

— if it has arithmetic type, it is initialized to (positive or unsigned) zero;

— if it is an aggregate, every member is initialized (recursively) according to these rules,  and any padding is initialized to zero bits;

— if it is a union, the first named member is initialized (recursively) according to these  rules, and any padding is initialized to zero bits;

[aashishium]

The good example ...

Good examples are very much necessary for beefing up our understanding of the subject and while learning about open, closed, perfect and bounded sets - I just found a perfect example of a set which is closed set as well as open set as well perfect set but not bounded set.

This example helped me to understand that closed and bounded sets are necessarily not the same all the times.

And here it goes -

[aashishium]

What I learnt about LaTeX ?

LaTeX is a powerful document generation language which one could use for creating books , reports , articles , scientific literature , ... etc and on it’s own right , it’s a markup language like HTML, but with a strong language constraints ( silly errors don’t go well with LaTeX ). 

 So, I jumped on to learn LaTeX and it was pretty much fun and I created a document  with a title “Introduction to fractions using equivalence class” which is available on scribd for download :-   https://www.scribd.com/document/361099343/Understanding-fractions-in-the-perspective-of-equivalence-class . I have learnt a lot of concepts in it and I don’t think I will be able to squeeze all of it in just a single article. 

so , let’s understand it’s basics -

For beginning we have to mention what type of document you are trying to create and it can be an article , a report , a book , .. etc and you can do this by using -

\documentclass{article}        

[If you want to create a document for a different class you can do that as well].

Moving ahead , just like <HTML> & </HTML> , in LaTeX we have something similar to it -

\begin{document}

\end{document}

All the content of the document just stays inside it. 

Here are the different statements one could use for beautifying their text -

\textbf{} [For creating bold text]

\textit{} [For creating text in italics]

\underline{} [For underlining the text]

and you can also give different types of sizes for the text -

\small  [For creating text of smaller size]

\normalsize  [For creating text of normal size]

\footnotesize [For creating text of size of footnote]

\large  [For creating text of size larger than \normalsize ]

\Large [For creating text of size larger than \large ]

\huge [For creating text of size larger than \Large ]

\Huge [For creating text of size larger than \huge ]

For using mathematical equations we have to add a package called as “amsmath” which is basically the package for mathematical equations provided by American Mathematical Society. and you can add the package to the .tex file at it’s preamble which is basically in between the \documentclass{article} and \begin{document} by \usepackage{amsmath} and you can actually consider it to be similar to adding to preprocessors to C/C++ program.

And there are a lot of cool stuff you can do with this package like -

$a^b$ [Creating superscripts where it will represent a raised to b]

$a_b$ [Creating subscripts where it will represent a subscript b]

$frac{a}{b}$ [Creating fractions where it will represent a/b]

Well there are literally sooooooo many concepts that I have learnt over a past few days that I don’t think that I could explain all of it in one post like adding images using package graphicx or creating mathematical equations and symbols like \epsilon [which represents epsilon ] or \alpha [which represents alpha ] using package amssymb.