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The Weirdness in Maths

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I find it quite bizarre that there are a lot of mysteries and confusions and “weirdness” in the many wonders of mathematics. One example would be the fact that 0.999… = 1. This means that the number 0.9, followed by an infinite number of 9s, are all equal to 1. But not in the sense that this number is rounded up to 1. It is just equal to 1. But how could that be? We have all learned from middle school that the comparison of decimals involves comparing the integral part first before comparing the decimal part. And in doing so, the greater whole number, the greater its value; and then comparing the decimal part digit after digit after the decimal point. But in this case it seems that 0.999… is of a smaller value since its whole number value is 0 and 1 has an integral part of 1; this meant that 1 would have been greater number. But it seems that there exists a proof that proves that they are indeed both equal:

Let x = 0.999… Therefore, 10x = 9.999… (Multiplying by 10 means that you have to move the decimal point one place to the right). Also if you subtract these two numbers, we have (10x-x) = (9.999…-0.999…). Simplifying it, we have 9x = 9. (The decimal portion of both are both identical; hence they cancel out). Dividing both sides by 9, x = 1. And by transitive property, if x = 0.999… and x = 1, then it must be that 0.999… = 1.


There doesn’t seem to be an error in the proof; all it took was a little Algebraic Magic. However it still seems erroneous at first glance. How come a number with an integral part of 0 be equal to 1? 1.000… = 1 I may say seems more valid, but not 0.999… = 1. And that’s where the problem with infinity arises. Remember that there are an infinite number of 9s after the decimal point. This means that there is no last digit to this number. Think about it; what number would you add to 0.999… so that you can reach the value of 1? Well, that would be 0.000… where the last digit has to be 1; or is it 0.000…1 and that there’s an infinite number of 0s between the decimal point and 1? But how could this be the case since we have already talked about that there is “no last digit”. If your number contains a finite number of 0s instead of an infinite number of 0s, then adding this to 0.999… will result into a number greater than 1. So instead, we expect to add 0.000… instead (infinite number of 0s after the decimal point) to 0.999… to produce 1. But 0.000… is also equal to 0. And adding any number to 0 is always the number itself, which is 0.999… Then again, we are using the transitive property which equates that this number 0.999… is indeed equal to 1.

This is all linked to the peculiar, yet magnificent power of infinity. Here’s another one: how many types of infinity are there? Perhaps I could be more specific by saying that there are a lot of infinities out there that exist, much much greater than other infinities. Here’s an example: how many positive integers are there? The set of all positive integers contains 1, 2, 3, and so forth. And now how many positive real numbers are there? Well, this is a bit much harder since the set of positive integers is just a subset of the set of all positive real numbers. Ok so if we want to list the set of all positive real numbers, we have all these positive integers. What else then? Well, there’s 0.1, 0.2, 0.3,… 0.9. Oh but wait, I think we missed something: 0.01. It, too, is a positive real number. So 0.02, 0.03, …, 0.09 also count. But what if I tell you that you missed another positive real number, which is 0.001? With that said, our listing methods seem to be resulting into a failure. Hence, even if there is an infinite number of elements in this set, it is of a greater cardinality (the number of elements in a given set) than the cardinality of the set of positive integers. Hence, we say that the set of positive real numbers is uncountably infinite and the set of all positive integers is countably infinite. And being uncountably infinite over countably infinite provides proof that there are indeed infinities much greater than others.

There are yet a lot of “weirdness” in maths, and many of them are said to be linked with infinity. But just looking at it, it just seems so beautiful! Ever wonder why the slope of a vertical line on a Cartesian plane is undefined? Ever wonder why the sum of a series tends to converge (meaning to approach a certain definitive number), while others tends to diverge (the opposite of converge)? And have you ever wondered (elementary question) why one is not able to give a definitive number when dividing by 0? All these questions lie to some of the foundations of infinities and I find these very captivating. But yet again, there are still other “mysteries” out there in Mathematics that I also find attractive. All these are to be discussed and explored further in my future blogs.

– Royalle
Date of Post: 12.27.16


The Secret of Success: Developing Communications in the Multi-Lingual World

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We live in a diverse world. It has been designed that the Earth we live in is a multi-cultural place. And with that in mind, it is necessary to realize the importance that a lot of languages have risen from the different parts of the world, some of which include English, French, Japanese, Korean, Arabic, and Tagalog. We must also take note that a lot of these have evolved to new, modern languages. For instance, English has evolved into the modern English we knew today from the Old English language we used to know; it has been through a lot in its history – from Barbaric English to Medieval to Shakespearean to Victorian to Pre-modern to Modern. And English today is still accepting new words (such as clickbait, filmer and selfie) and even borrowing foreign words from another languages (such as boondocks, carpe diem, and per se).

Given the number of various languages out there, we must keep in mind of the variety of people there are out there. And yes, I am talking about races and cultures, not in a racist sense, however. And each of these people in their races have actually become businessmen and businesswomen who actually decided to start up a small company and from there worked all the way up to the most successful multi-national corporations (MNCs). Some examples of these world-wide famous MNCs include McDonald’s, Coca-Cola, Fedex, Google, and Microsoft.

Being in a multi-national company involves a lot of great things that needs to be fixed, such as administrative decisions, risk-management planning, and organizational development. However, even if all of these administrative plans in an MNC are all fully-working well, this does not mean that the company is a 100% successful. There are still several steps that a CEO could take in order for the company to be a complete success. Maintenance, which may be one of the toughest jobs to handle, is just an example of that, but there’s more to that. I’m talking about taking a step further – being able to converse in more than one language; being fluent with a variety of languages. Knowing and being able to speak in one language is good, but not good enough. Bilingual, trilingual, perhaps? However, being multi-lingual is such a great asset in the company and should be recognized of the utmost importance in any company, may it be a small starting-up one or one such big one such as a multi-national corporation.

At this day and age, global modernization, at its core, is of its peak; and believe it or not, it continues to grow day by day. And with global modernization in mind, also keep in mind that the human race is also evolving as well. Each day, man is becoming more and more advanced – stirring away from the obsolete, going over towards the more high-technological sense of life. And with these things in mind, we must take note that the ability to speak multi-lingually is an asset, for better communication with others, for forming more connections with other people and for developing deeper bonds with others.

In this time, due to the uniqueness of each one, we are able to hold talents and skills that make and shape us. And being able to speak in different tongues is a part of this and can be taken advantage of. People who migrate and born in a new country may adapt and learn to speak new languages. The more languages you know, the better you can communicate, because there are more people you can communicate with and not just the people of your own race or culture or people who know your own language or dialect. In the context of the business world, let us take this in as an example. Say there are three businessmen, A, B, and C, who are about to meet up separately so that they could start a project collaboration. Meeting up separately meant that Businessman A meets up with Businessman B in private, Businessman B meets up with Businessman C in private, and Businessman A meets up with Businessman C in private. For instance, Businessman A is only able to speak in Mandarin, Businessman B is able to converse in three languages: English, Mandarin and British, while Businessman C can communicate in English and British only. Among them, Businessman B is said to be the most successful since the first two meetups from above (A and B, B and C) would turn out to be successful in language sense. Businessman A and B can speak in Mandarin in their meeting and Businessman B and C can speak in either English or British in their meeting. It may not be a guaranteed success in the end, but at least the fact that they are able to start conversing in a language both know is a good stepping stone. Businessman A and C, however, would have a difficulty in starting up a conversation in their meeting since each does not know a common language between them. Since they are stuck, it is highly probable that they are unsuccessful. Interpreters and translators may be good for both of them but the mere fact that one is able to understand the other is much better especially when making quick business decisions (and there could be a probability that interpreters or translators might give out the wrong translation as well, which may result into the failure of the project collaboration between the businessmen.)

In connection with the previous point, the ability to know more languages is also an asset in a sense that one is able to gain more connections; and more connections meant more trades, partnerships and collaborations. Some of these MNC partnerships and collaborations in the real world include that of Microsoft and Dropbox Inc., as well as of Apple and Beats Electronics LLC. And as a result, the greater, better, deeper connections you have with different, various companies, the more money and fundings your company will have. Hence, this will result into a better, much more successful company you will have.

As an analogy, picture how interpreters and translators these days are making a great amount of money. It is a high-paying job, which is a great in-demand in the job market. This is because they are able to speak in a variety of multiple languages. They help form a link, a connection, between two people that are not able to understand each other because of a language barrier. So imagine if a company is able to do what interpreters and translators do (not in the sense of the exact things they do, but the ability to speak in multiple languages). They are able to not only make those quick business decisions and easy conversations and meetings, but also the fact that they no longer need an interpreter or a translator to be present during their meetups. It saves them the money to pay for an expensive service of an interpreter or a translator. And you might think that a web app may help in translation of languages (such as Google Translate), but not every word, phrase or sentence can easily be translated (even to a universal language such as English). And sometimes as well, the translated version by these translator apps may not be as good and may sometimes lead into a confusion and can lead to the unsuccessful communication between businessmen, for example.

Being able to converse in a variety of different tongues also meant that one is able to employ more people (not just from a single race or culture or people of the same language). With that said, the employer is able to employ people of other race or culture or people of different language – meaning they are able to employ more people. This helps reduce unemployment and helps stabilize the nation for a better economy.

And in terms of businesses that make the effort to localise their website content, they would gain a lot of benefits for that one. This is because a lot of companies from different parts of the world might gain attraction from seeing something that they are easily able to understand. As a result, more partnerships and collaborations can be built, which would simply just lead to the points I have raised from above.

As a concluding point, the ability to speak in a variety of different, multiple languages is an asset for developing a successful company, may it be a small company or a huge multi-national corporation. It is also necessary to recognize that it is of great importance that there are a lot of benefits of being able to communicate in several languages and being able to localise website content for others to see online so that they could gain an attraction of what they have seen is crucial in order to form easier, more partnerships and collaborations.

Scholarship Submission Link:


– Royalle

Date of Post: 12.26.16

L.T.N.P. Update #n

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L.T.N.P. stands for “Long Time No Post” and every time I would have a gap of about time t = n (could be days, weeks, months or years), I’ll put up an update like this before beginning my regular daily blog. It’s #n because I’m not sure how many times I have already made these “gaps”. So just bare with me and I’ll call it “#n”. The next time this happens again, it shall be known to be “#n+1”.

Updates so far? I’m now studying in a university, York. My last blog post was last August, and since then I was so busy doing school stuff, and now finally got my winter break (oh yeah, Merry Christmas everybody!), I can finally relax. There’s so much still to write and update about but just hang on there, hopefully I can get through.

Before the end of the break, I’ll write up a blog of all the stuff I’ve did so far over the break. Again, happy holidays to all!

– Royalle

Date of Post: 12.26.16


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It has always been a pet peeve for me when there are people around me who still do not know the BEDMAS rule. But then I always think this over the moment I hear that someone beside me in the library would argue with all their might to their friend that that 2 + 8 x 3 is 30 and not 26. But hey, I would then understand that Math is a tricky subject and that anything Math-related especially these types of problems can be quite confusing. Nevertheless, rather than being mad at that guy over at the library, it would be better if I teach them the ways of the BEDMAS.

Now BEDMAS, which stands for Brackets, Exponents, Division/Multiplication and Addition/Subtraction, represents the order of operations. It is a very handy mnemonic to easily remember the precedence among operations. There are other names for BEDMAS and one in particular would be PEMDAS, which is Parentheses, Exponents, Multiplication/Division and Addition/Subtraction. Both work the same way, as long as the order of operations is still followed.

So how does this work? The rule states that any operation found within brackets or parentheses must be worked out first. It is also important to note that the innermost grouping symbol has to be operated first before the outermost grouping. For instance, in calculating 3 x [2 – (4 + 5)], we solve first for 4 + 5 to get 9. Replace the  (4 + 5) in the original expression as 9 to make the expression much simpler by reducing the number of grouping symbols. So we then have 3 x [2 – 9]. Solving the operation inside the brackets which is 2 – 9 = -7. Replace [2 – 9] with -7, which leaves us with 3 x -7 and therefore can easily be solved to get -21.

The next precedence would be Exponents. After getting rid of the grouping brackets, then it is time to attack exponents. For instance, calculate 40 / (6 – 2^2)^3. Note here that “/” means division and “^” means an exponent raised to the power of. So here we start solving for the bracket which is 6 – 2^2. Here we see an exponent, which we should solve first. 2^2 is 4 and it should replace 2^2 here. Therefore 6 – 4 = 2. This was initially part of the bracket in the original expression and we should replace this with the number we got which is 2. Our simplified expression would thn be 40 / 2^3. Now recall that Exponents must go before any Multiplication, Division, Addition or Subtraction. Therefore 2^3, which is also equal to 2 x 2 x 2, is 8. Replace 2^3 with 8 to get 40 / 8, which is 5.

Now always keep mind to solve for bracketed operations first, beginning from the innermost operation to the outermost. Then any exponents found must be solved immediately before solving for any operation with Division, Multiplication, Addition or Subtraction. Division and Multiplication have interchangeable precedences, meaning that the first operation that appears after reading the expression from left to right has to be performed first. A simple example, for instance, would be 6 / 3 x 2. Here we first operate 6 / 3 to get 2 before multiplying it to 2 to get 4. If we have 3 x 9 / 27, therefore we first solve 3 x 9 before dividing the result by 27, which is equal to 1. Addition and subtraction, however have lower precedences than Division and multiplication; thus, division and multiplication would have to be performed first before operating addition and subtraction, while of course keeping the interchangeable precedence premise among division and multiplication. Moreover, like division and multiplication, addition and subtraction also have interchangeable precedences, performing first the operation that appears when the expression is read from left to right.

Here’s a recap of the rules of BEDMAS:

  1. Begin performing bracketed operations in an expression, starting from the innermost going to the outermost brackets (or parentheses or braces or any other grouping symbol for that matter).
  2. Simplify exponents whenever possible.
  3. Solve for division and multiplication operations. Precedences are interchangeable.
  4. Finish up with addition and subtraction. Precedences are also interchangeable.
  5. Repeat back to step 1 if there are any other further brackets and continue performing the cycle.

Now go and test yourself if you can calculate the final result of the following expressions using BEDMAS rule, and then check your answers:

  1. (2^7 – 1 x 5) + 3^4
  2. 1 + 1 + 1 + 1 + 1 x 0
  3. 180 / [2 + (8 x 1/2)]^2


  1. 214
  2. 4
  3. 5

– Royalle
Date of Post: 08.20.16

A Dream Map

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One of the things that got me fascinated about here in Toronto are bus and subway map routes. It’s never a surprise that I could navigate my way from one spot to the next by just riding and hopping through buses and trains. That’s why my friends here always count on me to be the “navigator” whenever we go on our “little adventures”.

As I was saying, my fascination for these things has led me to come to get to know the bus routes and subway station names much easier especially when I am in a foreign place. For instance, last summer my family and I went for a little vacation to New York and one knows how complex subway lines are there upon first glance. But trust me, once you get to know it, you can easily navigate where you are and where you are headed to, not to mention which subway lines to ride and what not.

Just recently last week, I finished building up my own subway dream map for Toronto. I included in the map the current existing one, while extending some subway lines and adding some new ones, with their corresponding subway stations. I would think that it would be great if Toronto would get rid of all its buses and focus on building more subways instead.

Environment-friendly? Well, not necessarily. The construction of a subway station and trains can sometimes be considered factors for the destruction of our environment, and can be very costly too when it comes to expenses. But hey, these are just some of my opinions. Tell me what you think about it, as well as the subway dream map I have made. Note that the subway map below is only fictional and that not everything below is real; some subway lines and stations do exist, some are planned, however most are just entirely made up!


View the map up close by selecting one of the shared files below:



– Royalle

Date of Post: 08.18.16

Operation: RE-LIVE

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There’s just been a lot of things that had happened here while I was away (check my blog below entitled “So, What happened?” for the full story of what had happened to my WordPress account). Many changes in the interfaces, how the profiles were set up, as well as the daily and weekly challenges (which I have decided to begin participating).

Apart from that, most probably I would be posting stuff daily (more than usual), just to catch up for the time that I have “lost” all throughout those years without my WordPress account. Like I said from my previous post, whenever I feel like writing, I have Microsoft Word with me when I was writing (if you came to ask why I didn’t just make a new WordPress account, then the answer is that this account is way too precious to be replaced; she’s irreplaceable).

So I would continue to publish blog posts (or anything extraordinary that have happened to me or anything along those lines, like a public diary per se) and any creative pieces (such as poems, fiction stories, novel excerpts, photographs, artworks, etc.)

My goal is for this WordPress account to re-live and not die out of oblivion. Again, for all the followers I have and the likes, comments thank you so much. You are all awesome!!!

– Royalle

Date of Post: 08.18.16

Red, Red Moon

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via Daily Prompt: Moon

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“Come, look!” she cried as she came rushing through me that night while I was reviewing my notes. I didn’t even know what the hell was going on, but to my expense, alright.

She was actually referring to the beautiful sight outside the window view.

“It’s just a moon, my dear. What’s all the excitement about?” I told her as I felt the loss of excitement once I came near the window pane.

Little did she know that I felt more excited than ever, more excited than her, for seeing such a marvelous sight. That round thing in the night sky. It was a full moon. And it had the most unusual color, a color that I have never seen. Red.

But that feeling of excitement lasted only for a couple of seconds. I cherished every moment of it. It was the only thing I had that time… until I came to a full realization, that the “she” that had dragged me all the way to the window never even came to exist in the first place. It was as if she turned into air and dust upon seeing the marvelous moon. She was nothing more but a figment of my imagination.

Upon realization, tears began rolling down my face. That was one cruel night…


– Royalle

Date of Post: 08.18.16