Mathematics

The Family of Real Numbers – Integers

Enjoy this excerpt from Tutor Leon’s Secondary 1 Math tuition class. 😊

Tutor Leon: Can anyone tell me what real numbers are? Or… what type of numbers are considered real? Or… if you can’t explain it in words, you can give me examples of what real numbers are.

Student Ethan: I know, I know… they are not fake numbers! [Class giggles]

Tutor Leon: Haha… very funny, Ethan. But… it’s actually kinda true! “Fake numbers” do exist! [Bewildered look on everyone’s faces 😮  ]

Student Ethan: Huh!?!? Sure or not, Cher?

Tutor Leon: Well… put it this way – numbers that are NOT real do exist. However, we don’t call them fake numbers. They are called imaginary numbers. For now, you do NOT need to concern yourself with imaginary numbers as they are beyond your O-level syllabus. [Sigh of relief from the class] For secondary level mathematics, numbers that you deal with are ALL real. For now, just know that besides real numbers, imaginary numbers also exists. Ok?

The Entire Class: Ok!!

Tutor Leon: So… back to my original question, what are real numbers? [Class ponders… 🤔  ]

Tutor Leon: How about this. You can think of real numbers as a family of numbers, just like how your immediate and extended families are made up of various members like your father, mother, brothers, sisters, uncles, aunties, cousins, etc. So what are the different types of numbers that belong to the real number family? Megan, what do you think?

Student Megan: Err… 1, 2, 3, 4, 5 … and so on?

Tutor Leon: Hmm… that’s not a bad start. Yes, real numbers include the numbers 1, 2, 3, 4, 5 and so on. Can you tell me what is the largest number you can think of?

Student Riley: Gazillion!! Just kidding… hahaha… err… I think it’s called infinity? 😬

Tutor Leon: That’s a decent answer, Riley! You’re kinda right. There are an infinite number of real numbers. However, infinity is NOT an actual number, but rather the idea that something is endless. Meaning that real numbers go on and on forever and ever. Does that make sense? [Class nods in unison] Ok, so the next question is… does anyone know what the numbers 1, 2, 3, 4, 5 and so on are called?

Student Riley: Cher, you mean numbers have names?

Tutor Leon: Sort of… just like how in science, in order to make sense of things around us, we categorise or classify different types of animals, plant life, etc. So the same applies to numbers, especially since there are an infinite number of them. To help us understand numbers better. Wouldn’t it make sense to categorise the different types of numbers?

Student Riley: Guess so… err… so what are the numbers called?

Tutor Leon: Good question! 😉  Numbers starting from 1 then 2, 3, 4, 5 and so on are called natural numbers.

Student Megan: Natural numbers? Why natural?

Tutor Leon: Another good question! Do you remember the time when you were much younger, as a toddler, when you were learning your ABC’s and of course learning how to count? [Class nods] Didn’t we all start counting from 1, 2, 3, 4, 5 … [Class nods again] Exactly! It’s simply the most natural way for most of us to count numbers… hence natural numbers. Sometimes natural numbers are also known as counting numbers.

The Entire Class: Oorrrhh…

Tutor Leon: So imagine that the real number family started with natural numbers but was still not a complete family yet. More numbers were added to the real number family. Any idea what comes before 1?

The Entire Class: Zero!!

For more insights on the origins of the number 0, please read “History of Zero“.

Tutor Leon: Spot on! Yes, then the number 0 joined the family. However, since zero is NOT natural, we had to think of another category to call numbers from 0, 1, 2, 3, 4, 5 and so on. Can anyone tell me what these numbers are called?

Student Ethan: Artificial numbers!! Hahaha… [Class giggles]

Tutor Leon: That’s a little corny, Ethan. Come on, seriously, what are these numbers called?

Student Ethan: 😝

Student Megan: Whole numbers!

Tutor Leon: Bingo!!

Student Ethan: Wah… steady lah… 👍

Student Megan: 😏  Just guessing.

Tutor Leon: If it was a guess, it was an intelligent guess. Well done, Megan. With the number 0, the number family was finally made whole. In other words, numbers starting from 0, 1, 2, 3, 4, 5 and so on are called whole numbers. Is that it? Are there no other numbers in the real number family? Or perhaps are there other numbers smaller than zero? Hint… hint. 😉

Student Ethan: I know… I know… really… really… it’s negative numbers! Negative 1, negative 2, negative 3 and so on.

For more information on the origins of negative numbers, please read “History of Negative Numbers“.

Tutor Leon: Very good! That’s absolutely right! So… the question is: Are negative numbers natural?

The Entire Class: NO!!

Tutor Leon: Are negative numbers whole numbers?

Student Riley: Hell no!

Tutor Leon: So… what are numbers …-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 … and so on categorised as?

Student Megan: I can’t quite remember… but I think it’s “inter” something.

Tutor Leon: You are very close. Anyone else wants to try?

Student Riley: In… ter… jar??

Tutor Leon: Good try… almost there! They are called… integers. I-n-t-e-g-e-r-s. In-te-gers.

Student Megan: Yah… yes… yes… integers!!

Tutor Leon: So now that we know a negative whole number is an integer. Is a positive whole number an integer?

The Entire Class: Yes!!

Tutor Leon: That’s right! Numbers 1, 2, 3, 4, 5 and so on are ALL integers. How about zero? Is the number 0 an integer?

Student Ethan: Yes. Confirm! Zero is an integer.

Student Megan: I disagree! Zero is errr… errr… hmmm… actually I’m not sure. I don’t think zero is negative, or is it considered positive?

Tutor Leon: Awesome! I’m glad you guys are giving it more thought. Zero… is… an… integer! Yes, some do get confused as integers are often related to negative and positive numbers. How you can think about zero is that it is a special integer that is neither positive nor negative. Does that make a little more sense to you?

The Entire Class: Yesss!

Tutor Leon: Ok, great! Let’s summarise…

Tutor Leon then summarises with the diagram below and continues with a more in-depth Q&A session on real numbers with the math class.

To be continued…

Stay tuned for future blog posts on other real number family members, such as non-integers (i.e. fractions and decimals), rational/irrational numbers and last but not least, prime numbers.

Sponge ME, Maths Tuition (Singapore)

Geometry – Euclid, the Father of Geometry

Remember playing shape recognition games as a kid? These games often involve identifying and placing different shapes (circle, square, triangles, etc.) in the right slots. They help to develop our minds in determining different shapes, and are our first exposure to geometry! 

In broad terms, geometry is the branch of mathematics that deals with the measurements and relationships of lines, angles, surfaces and solids.

Euclid-of-AlexandriaGreek Mathematician, Euclid (fl. 300BC) is often referred to as the father of geometry. The standard geometry most of us learn in school today is also known as Euclidean Geometry.

Euclid put together all the knowledge of the earlier mathematicians and wrote Elements, a mathematical and geometric treatise consisting of 13 books. 

Known as one of the most successful and influential works in the history of mathematics, Elements served as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century.

Sponge ME, Maths Tuition (Singapore)

What are Prime Numbers?

Meet the Numeric Celebrity – Prime Numbers

That’s right, primes are quite the celebrity and not just in Hollywood movies. 😉  Here’s an excerpt of primes being featured in the movie, “Contact” where scientists discover an alien signal composed of… yes you’ve guessed it, prime numbers!

As seen in the movie, the aliens chose to send a long string of prime numbers to prove that their message was intelligent and not of natural origin. So why use prime numbers? What so special about primes?

What are prime numbers?

Primes are the building blocks of all numbers. Think of prime numbers as atoms, just like in chemistry where we say that a water molecule is formed from two hydrogen atoms and one oxygen atom (notated as H2O). Likewise, the number 12 is the product of the prime factors 2×2×3 (notated as 22 3). So just like water can be decomposed into hydrogen and oxygen, all numbers can be decomposed into primes. Here are a few more examples:

  • 8 = 2×2×2 = 23
  • 20 = 2×2×5 =22 5
  • 180 = 2×2×3×3×5 = 22 32 5

This process of decomposing a number into its prime factors is called prime factorisation (a topic to be left for another time). Like atoms, prime numbers can’t be decomposed further or rather can’t be divided further, like 2, 3, 5, 7, etc. In other words, prime numbers are only divisible by 1 and itself, and a number that has more than 2 factors is known as a composite number. For example:

  • 2 = 1×2 (2 factors only) → Prime
  • 3 = 1×3 (2 factors only) → Prime
  • 4 = 1×4 or 2×2 (3 factors) → Composite
  • 5 = 1×5 (2 factors only) → Prime
  • 6 = 1×6 or 2×3 (4 factors) → Composite

At this point, you may be wondering – what about the number 1? Is it prime? Well, 1 is somewhat of a special case. If you think about it, 1 = 1×1×1×1×1… and this is where things get a little crazy. If you were to just consider the number of factors 1 has, it’s 1, which is also itself! So… is it prime? There is certainly a little more than meets the eye. 1 used to be prime, but it’s no longer prime. Haha… and the story of primes continue to unravel. To find out more, watch the short video below, where James Grime the Numberphile, concisely explains the Fundamental Theorem of Arithmetic (don’t worry, it’s just a fancy name – watch the video and all will become clear) and how it applies to 1 and prime numbers.

Cool right? Okay, so we now know that 1 is neither prime nor composite. It’s just the lonely one. Awww… poor 1. 😢

Well, is that all there is to prime numbers? Far from it! Here are a few more observations and interesting facts about prime numbers:

  • I’m sure you’ve noticed this. 2 is the only even number that is prime. The rest of the prime numbers are odd.
  • As numbers get larger, primes become less frequent and twin primes (see below) get even more rare.
  • In any case, we’ll never run out of prime numbers, as they are infinite. Any idea what’s the largest prime number ever discovered to date? Watch the final video below to find out.
  • Twin primes are pairs of primes that differ by two. The first twin primes are {3,5}, followed by {5,7}, {11, 13} and so on. It has been conjectured (meaning it’s never been proven) that there are infinitely many twin primes. This is known as the twin prime conjecture, a.k.a. Euclid’s twin prime conjecture.
  • Prime factorisation is hard work and when numbers get extremely large, you can imagine how tedious and slow it’ll be. ?
  • On top of this, primes do not have a pattern we can easily decipher, meaning there is no easy way to tell when the next prime number will appear. But that’s actually a blessing in disguise. Why? Read on to find out more.

Why are prime numbers so important?

Did you know that prime numbers are worth billions of dollars? 😲  Why are prime numbers so valuable to organisations, government agencies and companies like Apple, Google, eBay or Visa? Wondering how numbers can be worth even a single cent? Well, though prime numbers have little value in themselves, they are used in every credit/debit card transaction, including ATMs, online payments and even trading (e.g. stocks and shares) transactions totalling billions of dollars every day. In fact, prime numbers power the mathematics behind the cryptography (used for cyber security) of your WIFI connections, email accounts, blogs, Facebook, Twitter, etc.

To find out how primes combined with the difficulty of factoring large numbers are used to protect and secure our emails and payment transactions, please watch the short video below.

Aren’t prime numbers just fascinating? As Carl Sagan, author of the science fiction novel, “Contact” so eloquently pointed out – there is a certain importance to the status of prime numbers as the most fundamental building block of all numbers, which are in turn themselves the building blocks that help us understand our universe. 🤔  Regardless of how an advance alien life form may think or look like, one thing is for certain, if it understands the world around it, it most certainly understands the concept of primes.

Hope you found this article insightful and educational. Happy maths! 😁

Sponge ME, Maths Tuition (Singapore)

Oh… and if you are interested to find out what is the largest prime number ever discovered to date (Jan 2016), here is Matt Parker on the latest Mersenne Prime that holds the envious world record. Who knows? Maybe you might be the next record breaker for finding the “world’s largest prime”. Find out how it’s done and more in this video. Enjoy! 😉

Real Numbers – History of Negative Numbers

A page of The Nine Chapters on the Mathematical Art

A page of The Nine Chapters on the Mathematical Art

The first mention of negative numbers can be traced to the Han dynasty (206 BCE–220 CE), the second imperial dynasty of China.

Three Han mathematical treatises — the Book on Numbers and Computation, the Arithmetical Classic of the Gnomon and the Circular Paths of Heaven, and the Nine Chapters on the Mathematical Art — still exist.

Negative numbers first appeared in the Nine Chapters on the Mathematical Art as black counting rods, while positive numbers were represented by red counting rods.

The Chinese were able to solve simultaneous equations involving negative numbers.

Amazing, isn’t it? 🙂

Sponge ME, Maths Tuition (Singapore)

Real Numbers – History of Zero

The History of Zero

Indian mathematician and astronomer, Brahmagupta (598–668 CE) was the first to formalise arithmetic operations using zero.

He used dots underneath numbers to indicate a zero. He also wrote rules for reaching zero through addition and subtraction, as well as the results of arithmetic operations with zero.

This was the first time in the world that zero was recognised as a number of its own, as both an idea and a symbol.

The Discovery of Zero – Excerpt from BBC’s the Story of Maths

Are the numbers ‘0,1,2,3,4,5,6,7,8,9’ Indian or Arabic? Why was the number zero initially despised by the western world? How did the partnership of ‘zero’ and ‘one’ change the world, eventually giving rise to the Internet age?

If your interest has been piqued, please continue to watch the video below (a BBC documentary) to find out more about the amazing story of the numbers zero and one, taking us across the world, from east to west. We love this story and hope you do too. Enjoy! 🙂

Sponge ME, Maths Tuition (Singapore)

The Story of the Numbers Zero and One – Part 1

The Story of the Numbers Zero and One – Part 2

5 Tips to Avoid Careless Mistakes for Maths (Tip #5): Check Your Answers

Maths Tip - Check Your Answers

By Leon, Private Tutor, Sponge ME, Maths Tuition Singapore

As the saying goes, “Prevention is better than cure.” I cannot agree with it more. However, avoiding CSMs (Careless Stupid Mistakes) can and will never be 100% full proof – you know and accept this. That doesn’t mean you resign to it without first putting up a good fight. If Plan A fails, there is always Plan B. A backup plan will ensure you keep CSMs to a bare minimal – it’s called checking your answers!

You may be thinking that you barely have time to check your answers, let alone finish your exam papers. Yes, that may be true especially if you lack practice. However, if you make it a habit to carry out intermediate checks as you work through your maths problems, you will realise that it is a lot more time efficient compared to checking only at the end.

For instance, if you had to navigate your way across the great oceans using nothing more than a compass – would you only check your compass once at the end of the journey (provided you get there in the first place), or would you be checking your compass intermittently throughout the journey to make sure you are on track? Checking your answers is based on the same logic. By placing so-called mental check points throughout your working steps, you will be able detect an error early, rather than when it’s too late.

Of course, exactly how to check and where to check is a topic best left for another day. But in a nutshell, you should be equipped with a set of ‘check tools’, namely the Sanity Check (good for making quick sense of numerical answers), the Reverse Check (good for checking algebraic manipulative errors) and the Loop Check (good for quickly checking solutions by means of substitution) and learn how to use them effectively. There are also ways to make use of certain features available on the latest approved scientific calculators to double-check not just arithmetic, but also algebraic and statistical calculations.

Once again here are the 5 tips to avoid careless mistakes for your GCE O-Level Mathematics exams:

  1. Do NOT Skip Steps
  2. Know Your Tendencies
  3. Watch Your Units
  4. Keep It Neat
  5. Check Your Answers

I hope that you have found this series of posts educational and have realised that though CSMs can’t be entirely eliminated, they can definitely be suppressed with a high level of success and anyone can learn how to do this. All the best for your exams! 😀

5 Tips to Avoid Careless Mistakes for Maths (Tip #4): Keep It Neat

Maths Tips - Keep it Neat

By Leon, Private Tutor, Sponge ME, Maths Tuition

I know, some of you may be thinking: “This is not a tip! Everyone knows that if you are neat, it’ll help reduce careless mistakes. I just write the way I do. I know it sucks, but I can’t help it!” I hear you. Firstly, I’m not asking you to change your handwriting. That will be somewhat impractical (especially for math) and too steep a mountain to climb. What I’m suggesting is that you make a few adjustments to the way you present your mathematical solution, i.e. in a more organised and consistent manner.

So, besides making sure your ‘a’ doesn’t look like a ‘9’, or your ‘z’ like a ‘2’, the key is to find a standard format that you can easily apply over and over again for the various topics in mathematics. A standard format typically consists of the following 3 steps – (1) state your equation, (2) substitute all known values and (3) solve for the unknown.

At the end of the day, the main purpose of adopting a standard format is so that you have a familiar and reliable set up which you can consistently repeat with little effort. This will allow your mind to fully focus on solving the actual mathematical problem at hand. If done well, it will definitely improve your overall neatness and reduce the likelihood of careless mistakes.  Hooray! Yet another one bites the dust! 🙂

And last but not least, Tip #5…

5 Tips to Avoid Careless Mistakes for Maths (Tip #3): Watch Your Units

Maths Tips - Watch Your Units

By Leon, Private Tutor, Sponge ME, Maths Tuition Singapore

Another culprit is using the wrong units, i.e. the Unit of Measurement (UOM) in calculations. This frequently occurs in questions or problems involving rates or quantities such as speed, distance, time, money, and measurements of weight, length, etc.

The likelihood of a UOM-related CSM (Careless Stupid Mistake) increases when you do not use units in your working or statements. In most cases, students trivialise the importance of UOMs and in some cases totally ignore it. Hence, it leads to mistakes. The best way to become more acquainted with UOMs is simply to use them in your calculations or mathematical statements.

In my maths tuition classes, I break down common GCE O-level maths questions into specific types or categories in order to sensitise my students to the ‘warning signs’ (among other reasons). And when they detect ‘trouble’, they immediately become prudent and convert all rates and quantities to the same units before attempting to solve the problem. Yet another CSM crushed! Woohoo! 😀

Tip #4 is on its way…

5 Tips to Avoid Careless Mistakes for Maths (Tip #2): Know Your Tendencies

Maths Algebra Pattern

By Leon, Private Tutor, Sponge ME, Maths Tuition Singapore

Having tutored many students over the years, and helping them to prepare for their GCE O-level Mathematics and Additional Mathematics exams in Singapore, I can’t help but notice certain patterns of occurrences, i.e. that everyone (myself included) has a tendency or inclination to make a specific type or types of CSMs (Careless Stupid Mistakes).

For example, some students tend to make what I call copy or transfer errors, i.e. they copy down the question wrongly, miss out a variable or index here and there, or transfer a sign wrongly from one step to the next. Others tend to make simple operational errors like adding instead of multiplying and the list goes on. The point is –  it’s likely that you will be more prone to making a specific type of CSM, and honestly sometimes all that is needed is a conscious effort and think “Aha! I’ve made a CSM doing this before, I better be more careful this time round.”

But of course, this only works if you are first aware of your own tendencies. In my maths tuition classes, I’ve developed specific exercises to help hasten this process of self-awareness. It’s not rocket science. It’s just a comprehensive collection of typical GCE O-Level maths questions intentionally littered with the most common CSMs. The goal of the exercise is to spot the CSM and make the correction. It’s a simple yet effective way to discover and weed out common CSMs in a more proactive manner.

Tip #3 coming right up…

5 Tips to Avoid Careless Mistakes for Maths (Tip #1): Do NOT Skip Steps!

Maths Tips - Do not skip steps

By Leon, Private Tutor, Sponge ME, Maths Tuition Singapore

I believe many have heard maths teachers and tutors say over and over again: Do NOT skip steps! As a full time maths tutor myself, I say it too (guilty as charged), but only after convincing my students that it is absolutely critical that they do not skip steps in their working, especially if they are aiming for a distinction in both their E and A Maths papers, especially for A Maths.

There are in fact several reasons for not skipping steps, but pertaining to CSMs (Careless Stupid Mistakes) it is perhaps the best way to avoid arithmetic errors (i.e. operational errors when adding, subtracting, dividing, multiplying, etc.) and algebraic manipulation errors (i.e. expanding, simplifying and factorising) which are perhaps two of the most common types of CSMs out there. The probability of making a CSM goes up significantly when doing arithmetic operations or algebraic moves in your head, rather then penning down your steps. For those who don’t practice enough and lack mechanical fluency, the likelihood of making a CSM increases even more.

You may think you are saving time by omitting working steps, i.e. you equate fewer steps to less time, when in actual fact this is a huge misconception and worse still you do so at the expense of accuracy. Why? Well put it this way, writing down fewer steps does NOT equate to thinking in fewer steps.

This brings me to my main point: the reason why CSMs arise from skipping steps is simple – you’re trying to do too much mentally at one time. It actually takes a lot less effort and yes less time to just pen down each thought, i.e. one methodical step at a time. In other words, you can achieve both speed and accuracy by taking small quick steps, rather than taking large slow ones. The trick is to breakdown your mental thoughts into smaller, easier to manage pieces by penning them down fluently. Developing this one good habit alone can do wonders! 🙂

Stay tuned for Tip#2…

5 Tips to Avoid Careless Mistakes for GCE O-Level Mathematics (Intro)

Maths Tips

By Leon, Private Tutor, Sponge ME, Maths Tuition Singapore

So what is the crux of it? Frankly, careless mistakes can be a real pain in the butt for every student dealing with mathematics. I know I was plagued by it in my early secondary school life (more about that another time), but fortunately I found ways to suppress this infamous silent killer I call the CSM (Careless Stupid Mistake). For some, it may cost them to miss out on a distinction. For others, it may be tipping them from a pass to a fail. Whatever the case, there is indeed hope for the inflicted! 🙂

The truth is we are NOT perfect. Nobody is (as long as you’re human). Everyone, including teachers, academics and yes, even mathematicians all make mistakes. A book written by Alfred Posamentier and Ingmar Lehmann, entitled “Magnificent Mistakes in Mathematics” captures and characterises this imperfection that exists in even the best of us. World renowned mathematicians such as Pythagoras, Galileo, Fermat, Leibniz, Euler and several others have had their fair share of blunders, falling prey to making mathematical mistakes. So don’t beat yourself up about it too much. Neither should you shrug it off and say that is can’t be helped. There is a way, but like any worthy cause, it requires effort.

The first step to dealing with CSMs is to realise and accept that they can never be entirely eliminated, but in most cases can be avoided or minimised. The next step is to gain awareness of the most common types of CSMs out there and what your own tendencies are. Don’t be surprised that just from this level of awareness, some initial improvement can be made.  However, to successfully keep the most common CSMs at bay requires more than theory (for those out there looking for a quick fix, I’m sorry to say that there is none). Having said that, it is not difficult! I repeat – it is not difficult! It just requires you to put into practice what is taught in class so that you can develop good habits that will replace your bad ones.

Stay tuned as I reveal 5 useful tips to help you conquer the dreaded CSMs. To be continued very soon…