Maths 101

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)

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