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 H_{2}O). Likewise, the number 12 is the product of the prime factors 2×2×3 (notated as 2^{2} 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 = 2^{3}

20 = 2×2×5 =2^{2} 5

180 = 2×2×3×3×5 = 2^{2} 3^{2} 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! 😁

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! 😉

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.

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! 🙂

Fear is the bad feeling that one has when he is in danger or when a particular thing frightens him. A German proverb goes, “Fear makes the wolf bigger than he is.” This is absolutely true as fear will often cause people to imagine the worst and act irrationally. In that case, can fear be any good? Personally, I think a small amount of fear is good and even necessary as it not only acts as a form of control and deterrence but also serves to motivate oneself. Nonetheless, being overly fearful is bad as it will severely hamper man’s progress. In this essay, I will discuss how fear can be a double-edged sword, bringing both advantages and disadvantages to man.

Fear is good as it deters people from doing dangerous acts and prompt them to control and regulate their behaviour. For instance, despite the numerous wars since World War Two, the atomic bombings of Hiroshima and Nagasaki in Japan remain the only use of nuclear weapons in warfare. This is because the world is fearful of the widespread devastation that such weapons will bring about. Therefore, the fear of total annihilation has prevented world leaders from acting irresponsibly and going down the path of self-destruction.

Next, fear is good as it is a powerful motivator. For individuals such as students and entrepreneurs, the fear of failure will prompt them to work hard and put in their best effort in their studies and business undertakings. This will lead to results and progress. Similarly for nations, the fear of losing their competitive edge will spur them to constantly improve and reinvent themselves to keep pace with the fast-changing world. For example, Singapore is taking active steps to maintain and improve her skilled and flexible workforce to ensure that she remains competitive and does not fall behind major economies like China. Retraining schemes and upgrading courses have been provided for the workforce to ensure that it stays relevant. Hence, we can see that the fear of losing out to others is one of the reasons that has motivated nations to take active steps in improving their economies. Without fear, nations will become complacent and they will eventually fall into a decline.

However, although fear is good, man must keep in mind that too much fear may be detrimental to his development. Being overly fearful of the unknown and intangible will prevent people from venturing into areas previously unexplored. For instance, in the area of space exploration, Apollo 11 would have never landed the first humans on the moon if the Americans had let fear get in the way of their dream. As the late John F. Kennedy once said, “We choose to go to the moon in this decade and do the other things not because they are easy, but because they are hard.” To achieve great feats, man must learn to conquer his fear and find the courage to overcome the obstacles that life presents. Only then can the human race continue to make progress and enjoy the sweet smell of success.

In sum, fear is good as it will ultimately lead to a well controlled and motivated society. Nevertheless, people must keep in mind that they should not be clouded by fear as it will hinder their progress. I believe that a small dose of fear and a good deal of courage will make a great man as such a man will have the spirit to pursue his goals and the sense to act responsibly in the process.

A wise man once said, “Stress should be a powerful driving force, not an obstacle.” To all Spongers who are taking their GCE O-level exams in Singapore this year, we know you will take everything in your stride and exceed all expectations. To your success! 🙂

Write about a stranger who left a deep and lasting impression on you because of his or her actions.

He came, he saw, he helped. Then, he left without even telling us his name. To this day, I still remember his face and mannerism vividly. How could I ever forget him and the kindness he showed us?

It was during the March holidays and my mother and I were on our way to Malaysia to visit my aunt. We were in high spirits and the mood was set for an enjoyable day. Unfortunately, halfway through our road trip, one of our car tyres was punctured in the middle of the highway. As my mother did not know how to change a car tyre, we had no choice but to seek help. For almost an hour, we waved at every passing vehicle but no one slowed down, much less stopped. As if the situation could not get any worse, the weather changed suddenly. Thunder rumbled and lightning cracked open the ashen sky. Dark ominous clouds gathered overhead as rain threatened to fall.

“Where are all the helpful people! I can’t imagine being stranded here for hours!” I began to whine.

Just then, a screech of brakes was heard.

A battered old truck stopped a few centimetres ahead of me and out came a towering man with broad muscular shoulders and strong heavily-tattooed arms. His weather-beaten skin was as coarse as an alligator’s and he had a pock-marked face that looked like a pimple plantation. His eyes were so tiny that they were almost non-existent and his bulbous nose had the shape of a large garlic clove. To put it plainly, he was ugly and formidable looking.

“Do you need help?” the intimidating stranger asked in a gruff voice.

An irrational fear overwhelmed me, causing my heart to palpitate so fast that it might just leap out of my mouth. Instinctively, I moved behind my mother for protection.

“Ermh… yes please. Our tyre is punctured,” my mother muttered hesitatingly after what seemed like eternity.

Without a word, the stranger walked back to his beat-up truck and took out a toolbox. Taking the spare tyre from my mother, he flashed us an enigmatic smile and started work. We stood near him uneasily, half thankful and half suspicious of his motives.

Minutes passed and a gentle drizzle began to drift down from the darkened sky. Heat was instantly radiated from the ground as the light drizzle cooled the surroundings. My mother immediately told me to get into the car while she took an umbrella to place over the stranger to prevent him from getting wet. However, as the rain got heavier, my mother also came into the car at the stranger’s bidding.

“It’s okay. There’s no point in you standing here and getting wet too,” he said, his coarse voice muffled by the pelting rain.

For the next ten minutes or so, we sat silently in the car and watched the good Samaritan fix our tyre. He was focused on the task even though the wind had grabbed the umbrella and he was drenched to the skin.

When he was finally done, he simply knocked on the car window and said casually, “Your car is good to go.”

Before we could utter a word of thanks, he turned around and hopped into his truck. Then he left as quickly as he came. Just like that.

This mysterious man has left a deep and lasting impression on me because he has taught me two valuable life lessons. Firstly, he has taught me not to judge a book by its cover. When I first saw him, I instantly associated him with criminals. Nevertheless, he turned out to be the kindest person I have ever met. Thanks to him, I no longer form an opinion about others just because of the way they look. More importantly, he has taught me what it really means to help others. There are people out there who are willing to help those in need without asking for anything in return. These are the people who make the world a better place with their kindness and consideration. Although our encounter was brief, I will never forget this memorable character who gave my mother and me a helping hand and so much more.

To learn something well, teach it to someone else. It’s like looking into the mirror of your mind. If your teachings are well understood, it means you have clarity. If not, deeper understanding is required. Einstein once said, “If you can’t explain it simply, you don’t understand it well enough.”

Hence, teaching to learn and learning to teach are sides of the same coin. A great teacher is a greater learner.