Tips for Improving Students' Math Skills | MATH TIPS

Many Students think that math is tough because somehow they are weak in math.

In this blog i post some tips and technique to improve your math skills.But before I am going to discuss about some problems in our study life: (Must read the below section)

➡️Our Study Life:

Most of the students study only for marks and nothing else.This is not the fault of the students, this is family pressure.

  I mostly says about India. Look at some example of this.

Example:

               Father(Family):- Son just secure 90% above marks in your 10th board and after that there will be no pressure for study.

      ( After Securing 85% in 10th board)

Father(Family):- Shame* on you. You can't secure just 90% in your exam.What will you do in your future . You will beg for money from house door to house door. Even if no one will give his daughter to you.

And  your friend Rahul(Son of Ramlal Sharma) got 93% , learn something from him.

          In our days we had to walk 10Km to go to school. If we got the facility that you have , then we must be an IAS or IPS Officer.

 And a interesting fact that is The same story will repeat after 12th board,Engineering/Medical entrances,job entrances and will go on.

After hearing this some students think that Marks is everything .Then they start study only for marks and  memories everything without understanding.

So Marks is not everything. We All know that mark is important but understanding is more important than marks.

If You really Want to Develop your math skills you should go on. Or Just Leave from here.

➡️1.Understanding the Concepts:

You can memorize formulae and rules to complete many math problems, but this doesn't mean that you understand the underlying concepts behind what you're doing. This makes it harder to successfully solve problems, as well as making it nearly impossible to easily absorb new information. Taking the time to make sure you understand why you're doing what you're doing can help your math skills immensely.

➡️2.Find New Concepts and Practice Problems:

Jumping directly into solving problems can lead to frustration and confusion. Try to study your textbooks and pay attention in class. You should also work on any practice problems your teachers assign before completing any assignments. This gives you a chance to absorb what you're learning.

➡️3.Work On Additional Exercises:

Practice makes perfect, even with math. If you are struggling with a particular kind of problem, you can improve by working on solving additional problems. You can start out with simplified problems of the same type, and move up in difficulty as you become more comfortable with finding the solutions.

➡️4.Make math part of your life:

Take the time to apply math in common situations. For example, if, say, a sweater that’s regularly $38 is on sale for 30% off, what is the sale price? Or if you need to double a recipe that calls for 3/4 cup of flour, how much flour will you need?

➡️5.Play math games.

Math games are good tools for honing your math skills and are designed to let you have fun while doing it.

There are some math Games for Android and PC

• 2048.(Android)
• Math Games.(Android)
• Math Land.(Android)
• Khan Academy Kids.(Android)
• Math Master.(Android)
• DragonBox5+(PC)
• Prodigy(PC)
• Minecraft(PC/Android)

➡️6.Study From Different Sources:

Friendship is the Great resource.You may hire one of your good friend as your study partner.

   Once Dr. APJ Abdul Kalam Sir Said “One best book is equal to hundred good friends but one good friend is equal to a library.”

Internet is now the most cool and intelligent source of anything.You may google something about your theory. 

My recommended  websites are

• MIT open courseware
• Khan Academy
• TedEd(Youtube Channel)
• wolframalpha
• Wikiversiry

CONVERGENCE OF SEQUENCE AND SERIES (Part -1)| REAL ANALYSIS

INTRODUCTION:

All of you must read the 1st part of real analysis.If you are not then just CLICK HERE TO READ.

Though I repeat some basic knowledge about real analysis.

Real Sequence:
A real sequence is a map ∱: N ➝ R
If we write ∱(n)= x_n , n ∈ N 
then it is customary to denote the sequence
∱ by (x_n)_n=1 or sometimes (x₁ ,x₂ ,x₃ ,...,x_n )

The set {x_n : n ∈ N } is the range of the sequence.

A sequence in R is a list or ordered set: (a₁ , a₂ , a₃ , ... ) of real numbers.

Bounded Sequence:

The sequence  (x_n) is said to be bounded above if there exist a real number 'M' such that x_n ≤ M  for all n ∈ N .

       It is said to be bounded below if there exist m∈ R such that x_n ≥ m  for all n ∈ N.

A sequence is said to be bounded if it is bounded above and below.
So, simply (x_n) is bounded if and only if there exist a constant K > 0 such that

      |x_n| < K for n ∈ N
i.e;   -K < x_n < K

Convergence:

A sequence x_n  is called convergent if there exist a real number ' a ' such that for every є > 0 ,there exist n₀ ∈ N  depending upon є such that for n > n₀

                                         |x_n - a| < є

that is,

                               a - є < x_n < a+ є ,

Simply,

$$

Definition: We say x_n→∞ as n→∞ if for every M in R there is a natural number N so that x_n≥ M for all n≥N. We say x_n→−∞ as n→∞ if for every M in R there is a natural number N so that x_n≤ M for all n≥N.

Example: 1, 1/2, 1/4, 1/8, 1/16 ,.........

This Sequence converge to a a finite point so it is Convergence Sequence.

Divergence:

A sequence that is not converge is called Diverge.

Example: 1,2,3,4,......

In the above example the sequence leads to +∞ . So it is a Divergent Sequence.

Read the below Question Carefully for Better Understand.

QUESTION:

1. Check 1/2 , 2/3 , 3/4 ,4/5 ,5/6 ........   is Convergence or Divergence ?

ANSWER:

Step1: Find the n_th term

t_n = n/(n+1)

Step2: Find the limit of the n_th term with n↠ ∞

Similarly , if we don't get any finite number then it will be Divergent Sequence.

Limit Superior and Limit Interior Sequences:

Let x_n  be the sequence of real numbers , for any k, let S_k = { x_n : n > k} . Then

Also Limit Superior and Limit Interior is categories under Convergence and Divergence. Which we have already discussed above.

Other Types Of Sequences:

Monotonic Sequences:

A sequence of increasing(Diverging) or decreasing(Converging) is called Monotonic Sequences.

Example:

1,1/3,1/9 ,1/27 ,........ is a Converging Sequence.

0,2,4,6,8,........ is a Diverging Sequence.

Oscillatory Sequence:

The oscillation of a function or a sequence is a number that quantifies how much that sequence or function varies between its extreme values as it approaches infinity or a point.

Example: (-1) ⁿ  is an example of Oscillatory Sequence.

Cauchy's Sequence:

We say that a sequence of real numbers {a_n } is a Cauchy sequence provided that for every Є>0, there is a natural number N so that when n,m≥N, we have that|a_n -a_m |≤ Є.

Basic Properties of Cauchy's Sequence:

1. Every convergent sequence is a Cauchy sequence,
2. Every Cauchy sequence of real (or complex) numbers is bounded .
3. If in a metric space, a Cauchy sequence possessing a convergent sub sequence with limit is itself convergent and has the same limit.

ZENO'S DICHOTOMY PARADOX | ANCIENT MATH | MYTHOLOGY OF MATH

INTRODUCTION:

I am damn sure that you never heard this name before.If you know this before then you are special.

     Don't get sad we will discuss about it briefly and from basic.

ZENO:

Zeno of Elea was a pre-Socratic Greek philosopher of Magna Graecia and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known for his paradoxes.

What is Paradox?

A paradox is a statement that contradicts itself, or that must be both true and untrue at the same time. ... But a key part of paradoxes is that they at least sound reasonable.

Dichotomy Paradox:

Dichotomy Paradox means before before an object can travel a given distance , it must travel a distance . In order to travel , it must travel , etc. Since this sequence goes on forever, it therefore appears that the distance cannot be traveled.

Simply,

In Dichotomy Paradox, Zeno argued that a runner will never reach the stationary goal line on a straight racetrack. The reason is that the runner must first reach half the distance to the goal, but when there he must still cross half the remaining distance to the goal, but having done that the runner must cover half of the new remainder, and so on.

Example-1:  If the goal is one meter away, the runner must cover a distance of 1/2 meter, then 1/4 meter, then 1/8 meter, and so on ad infinitum.So the runner cannot reach the final goal.

EXAMPLE-2: you may know about the story of Rabbit and Tortoise Race.

According to zeno's paradox if the Rabbit start the race after tortoise start then rabbit will not overtake the tortoise.

Don't Confuse with this because Dichotomy Paradox means The Contradiction of itself which it true and false at same time.

   I know this is funny but it is real.

Every time the hare reaches the place where the tortoise was, the tortoise has moved on a bit further. (Image)

And if the Pictures continue to infinity then you will find that Tortoise  will be bit ahead of Rabbit with less than a nanometer or picometer of whatever..

So Now the question must arises that

1. what is the time takes to reach at the goal?

2. In real life we reach at a goal in finite time , So in Zeno,s Paradox is it finite or infinite?

3. Is Zeno's Paradox is just a illusion or works in real life?

Time Management of Zeno's Dichotomy Paradox:

Let

      Distance=2KM

      Velocity = 1km/hr

From Zeno's Point of view

• for Covering 1st half i.e 1km will take 1hour
• For covering 2nd portion i.e; 500m will take 1/2 hr.
• For Covering 3rd portion i.e; 250m will take 1/4 hr.
• And So On.....

So Total Time will be (in Hours)

   = 1+1/2+1/4+1/8+1/16+........

   = 1+2-1 +2-2 +2-3 +....

   It is a Geometric Progression

Here a=1 and r=2-1

It is a Infinite series so 1/(1-r) will be the Formula

=1/1-2-1

=2  ( Proof)

Now you can see that the answer is correct according to real life.

RANDOM VARIABLE

INTRODUCTION:

Some Thinks that Probability is very easy 

And Some think it is very tough.

      If you Think Probability is tough then Proceed to our Probability blogs to learn easily.

Let's Proceed to our Topic

✦ A random variable is often described as a variable whose values are determines  by outcomes of a random experiment.

OR, 

      We assigned some real values to each of the outcomes of a sample space ,which is called the random variable.

✦ A random variable is a function , X :S ➝ R i.e a random variable is a function whose domain is the sample space of a random experiment(S) and whose range is a real line(R).

 Example: Let X is a random variable , the no. of heads in the experiment of tossing a coin twice , then in this case  S ={ HH,HT,TH,TT} and X ={2,1,1,0} i.e X(HH)=2, X(HT)=1, X(TH)=1,X(TT)=0. Thus the domain of  X is S and range is {0,1,2}.

✦ For a mathematical and rigorous definition of the random variable , let us consider the triplet (S,B,P), where S is the sample space B is the 𝞼-field of subsets in S and P is the probability function on B.

   So mathematically , a random variable is a function X(𝝎) with domain S and range (-∞ , ∞) ,such that for every real number 'a' , the event [ 𝝎 :  X(𝝎) ≤ a ] ∈ B.

✦ Random variable X can be written as P( X ≤ a ) to make probability statements about 'a'.

     For simple example given above , we should write P( X ≤ 1 ) = P{ HH,HT,TH,TT } = 3/4.

NOTATION:

✦ One dimensional random variable will be denoted by capital letters X,Y,Z,... etc. A typical outcome of the experiment ( i.e a typical elements of the sample space ) will be denoted by 𝝎 or е. Thus X(𝝎) represents the real number which the random variable X associates with the outcome 𝝎 .

✦ The value of the random variable will be denoted by small letters x,y,z,... etc. 

SOME PROPERTIES OF RANDOM VARIABLE :

✦ If  X₁ , X₂  are random variables and C is a constant , then CX₁ , X₁ + X₂ , X₁ - X₂ , X₁ * X₂  also random variable.

✦ If X is a random variable , then

• 1/X , where (1/X ) ( 𝝎 ) = ∞ if X(𝝎) = 0.
• X + 𝝎 = max[ 0 , X(𝝎) ]
• X - 𝝎 = min [ 0 , X(𝝎) ]
• | X |

are also random variables.

✦ If  X₁ , X₂  are random variables ,then (i) max[ X₁ , X₂ ] & (ii) min[ X₁ , X₂ ] are also random variables .

✦ If X is a random variable and ∱(.) is a continuous function , then ∱(X) is a random variable.

✦ If X is a random variable and ∱(.) is a increasing function , then ∱(X) is a random variable.

✦ If ∱ is a function of bounded variations on every finite interval [a,b] and X is a random variable, then ∱(X) is a random variable.

REGULA - FALSI METHOD | ENGINEERING MATHEMATICS | FINDIND ROOTS OF AN EQUATION

INTRODUCTION:

Regula-Falsi method is also used for finding approximate roots of an equation .It is very similar to bisection method.

                If you don't read about bisection method OR forgot the bisection method just click here to read now.

This is also called False Position Method.

THEORY:

As before (in Bisection Method), for a given continuous function f(x) we assume that f (a) and f (b) have opposite signs ( a = lower guess, b = upper guess). This condition satisfies the Balzano's Theorem for continuous function.

Note::Theorem (Bolzano) : If the function f(x) is continuous in [a, b] and f(a)f(b) < 0 (i.e. f(x) has opposite signs signs at a and b) then a value c ∈ (a, b) exists such that f(c) = 0.

Now after this bisection method used the midpoint of the interval [a, b] as the next iterate to converge towards the root of f(x).

A better approximation is obtained if we find the point (c, 0) where the secant line L joining the points (a, f (a)) and (b, f (b)) crosses the x-axis (see the image below). To find the value c, we write down two versions of the

slope m of the line L:

We first use points (a, f (a)) and (b, f (b)) to get equation 1 (below), and then use the points (c, 0) and (b, f (b)) to get equation 2 (below). Equating these two equation we get equation 3 (below) which is easily solved for c to get equation 4 (below):

The three possibilities are the same as before in the bisection method:

• If f (a) and f (c) have opposite signs, a zero lies in [a, c].
• If f (c) and f (b) have opposite signs, a zero lies in [c, b].
• If f (c) = 0, then the zero is c.

A short GIF for how this method works:

Working Rules:

1. Find Points x0 and x1such that x0< x1and f(x0).f( x1) < 0
2. Take the interval [x0, x1] and find next value x2= x0-f( x0).(( x1- x0)/f( x1)-f( x0))
3. If f(x2)=0 then x2is an exact root.
4. If f(x1).f(x2) < 0 then x1= x2
5. If f(x0).f(x2) <0 then x0= x2
6. Repeat the step 2-5 until f(xi)=0 or f(x0)≤ Accuracy

Solve: Find a root of an equation f(x)=x3-x-1 using False Position method

Solution:
Here x3-x-1=0

Let f(x)=x3-x-1

x	0	1	2
f(x)	-1	-1	5

1st iteration :

Here f(1)=-1<0 and f(2)=5>0

∴ Now, Root lies between x0=1 and x1=2

x2=1-(-1)⋅(2-1)/(5-(-1))

x2=1.16667

f(x2)=f(1.16667)=-0.5787<0

2nd iteration :[ FROM HERE THIS LEAVES FOR READERS AND HOPE ALL ITERATION WILL BE SOLVED ]

LASTLY WE GOT

[HERE n= No. of Iteration , x0 = value of 1st Guess point and f( x0 )= value of Fuction at x0 ]
Approximate root of the equation x3-x-1=0 using False Position method is 1.32464