Real Numbers

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)

Fun with Maths – Why Seven is so Awesome

  1. Most people can generally hold around seven numbers in their working memory for a short period of time (Miller’s law), which explains why our telephone numbers are mostly seven digits (excluding the country and area codes).
  2. Several sleep studies have found that seven hours is the optimal amount of sleep, not eight!
  3. There are seven colours in the rainbow: red, orange, yellow, green, blue, indigo, and violet.
  4. The ‘Seven Seas’ (as in the idiom ‘sail the Seven Seas’) is an ancient phrase for all the world’s oceans: Arctic, North Atlantic, South Atlantic, Indian, North Pacific, South Pacific, and Southern (or Antarctic).
  5. There are seven continents in the world: Africa, Europe, Asia, North America, South America, Antarctica,and Australia.
  6. Seven is used 735 times in the bible (54 times in ‘Revelation’ alone)! If we include ‘sevenfold’ and ‘seventh’, the number jumps to 860!
  7. And of course, seven is a prime number. 😉

Happy Seven! 😀

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