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2=1? Is that true?

Often we see the evidence mathematically that 2=1. They show very logically. Some people who cannot know this mistake then they blindly assume that math is flawed.

I believe that mathematics is one of the perfect sciences. Because until now I have not found a defect in this science. Which I love, this science is connected to each other. Some people even consider that mathematics is the language of Deity.

Back to the topic of conversation. Ever seen evidence like this?

    \begin{eqnarray*} a & = & b\\ a^{2} & = & ab\\ a^{2}-b^{2} & = & ab-b^{2}\\ \left(a+b\right)\left(a-b\right) & = & b\left(a-b\right)\\ a+b & = & b\\ 2b & = & b\\ 2 & = & 1\end{eqnarray*}

Or something like this?

    \begin{eqnarray*} a & = & b\\ a^{2} & = & ab\\ a^{2}+a^{2} & = & a^{2}+ab\\ 2a^{2} & = & a^{2}+ab\\ 2a^{2}-2ab & = & a^{2}+ab-2ab\\ 2a^{2}-2ab & = & a^{2}-ab\\ 2\left(a^{2}-ab\right) & = & \left(a^{2}-ab\right)\\ 2 & = & 1\end{eqnarray*}

Lots of evidence like this is scattered on the internet. If we do not look closely, we will be caught in logical traps and as if they are true.

So, where is the error of evidence above?

It’s simple. The error of the above evidence is in the step that makes \frac{0}{0}=1. And we know that \frac{0}{0} is undefined.

I do not see the \frac{0}{0} . Where is it? (more…)

How To Proof Set A Is Equal To Set B

Some people are wrong ways to prove that set A is equal to set B. I also experienced it when I started learning math. Though the concept of set A is equal to set B very simple. We only have to prove that set A is a subset of set B and set B is a subset of set A. Of course the basic is very easy to understand.

Then, how to prove the set A is a subset of the set B, vice versa?

The way to prove set A is a subset of set B is by taking any element on set A and then prove that element is a element of set B. It can be easily understood that the taking of any element can be representative of all element of the set. Mathematically like this :

    \[A=B\Longleftrightarrow\left(A\subseteq B\right)\wedge\left(B\subseteq A\right)\]

with

    \[A\subseteq B\Longleftrightarrow\left(\left(x\in A\right)\Rightarrow\left(x\in B\right)\right)\]

Example : (more…)

Mindset and Basic of Mathematical Induction

Mathematical induction is one of the tools of proof in mathematics. Usually mathematical induction is used to test the hypothesis. Some events make a pattern that eventually leads to a formula. But is this formula correct? Will this pattern apply to the next? Something like that.

Mindset of mathematical induction is like a domino game that is arranged upright then the first domino is dropped. Usually we call the domino effect. I think you already know about this game.

When the first dominoes fall will result in the second domino falling. And when the second domino falls then the third domino falls. And so on. This is if the domino order is correct. If something goes wrong, of course the domino effect will stop. That’s the mindset of mathematical induction.

Mathematic induction play on the proof that he fell on the first point, meanly that formula is true at first occurrence. Next, he assumes to fall on points to k. From that assumption, our task is to prove

that it falls on points to k+1. It’s mean that on points to k+1, he gets the domino effect from falling points to k. Already understand the mindset? We repeat in more detail :

Step 1. Prove he is right for the 1’st incident

Step 2. Assume he is right for the incident to k.

Step 3. Prove he is right for the incident to k+1 using assumed assumptions. (more…)

How To Outsmart The LaTeX Symbols

Read this post until finish. You will find the easiest way to write symbols using LaTeX in your website or your paper.

When the final assignment (especially math students) like or dislike – want or not, they must use LaTeX. This is to make it easier for them when creating symbols especially when revised. LaTeX is also very helpful in the numbering of definitions, theorems, or others that can be decoded each number code without having to replace them one by one.

But in reality many people presume the use of LaTeX more difficult than using Ms. Word. It is true that we must use LaTeX codes to generate mathematical symbols or make commands. For the example :

You must write : \frac{\sqrt[4]{3x^{2}}}{x^{\frac{1}{2}}}

To show symbol :

    \[\frac{\sqrt[4]{3x^{2}}}{x^{\frac{1}{2}}}\]

This causes some people are lazy to learn the code because of time constraints, too many commands to understand, or many command code errors so that the symbols that appear do not match what he wants.

LaTeX code can actually be tricked. And some of the commands in authorship in WinEdt must be every university has a standard template. So your problem is how to outsmart the LaTeX codes.

OK. Here’s how to outsmart the writing of LaTeX symbols. (more…)

How To Put Mathematical Symbols Into Your Blog or Website

I started this web with this post. It because in writing this website most (or even all) posts must use this way.

Some people inserted mathematical symbols very complicatedly. They typed symbols in Ms. Word then saves in the form of images and then uploads images to the website. That’s very impractical. I share the best way to insert math symbols in your website / blog. OK, let’s begin to the discussion.

Step 1. Make a math symbol using the LaTeX code. For those who do not know about the code LaTeX, I will discuss this code in the next article. For understanding the meantime, please visit this link. There are translations of mathematical symbols in the LaTeX code. Next, suppose you want to bring up this symbol :

    \[\lim_{x\rightarrow a}f\left(x\right)=L\]

The symbol above has a LaTeX code : \lim_{x\rightarrow a}f\left(x\right)=L (more…)