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Singapore Math

Friday, March 14, 2014

No Pi Day in Singapore

Number Photos

Because of the way Singaporeans write their dates (DD/MM/YYYY), celebrating Pi Day with school students in Singapore is a mere calendrical impossibility. At best, Singaporeans' celebration of Pi Day could be likened to Americans' celebration of the Abacus Day in the United States.

At a time when so much is written about the merits of Singapore math, it's just frustrating that Singapore students and teachers aren't able to commemorate the most popular constant in mathematics on March 14. However, in Singapore, we haven't let calendrical concerns prevent us from indulging ourselves in some extra-mathematical activities, peeping at some hidden pi treasures from yesteryear.

For us, the diehard supporters, let's look at a dozen irrational or transcendental musings on Pi. 

1. Pi in the Sky—The Digits of Pi Eternally Linked!

The German mathematician Ludolph van Ceulen (1540-1610) spent much of his life calculating the decimal expansion of pi. The professor at the University of Leiden determined pi to 35 decimal places. When he died in 1610, the numbers 288 were carved on his tombstone (now lost) in St. Peter's churchyard, at Leyden—in the 33rd, 34th, and 35th decimal places of pi.

Ludolph van Ceulen died of exhaustion after deriving 35 decimal places of pi,
which are engraved on his tombstone. 

2. A Pi Bill—To Pi or Not to Pi

Over the centuries, mathematical cranks, particularly the circle squarers, had submitted proofs of pi related to different values. 
To pi or not to pi: Changing pi to 3.2

The notorious Edwin Goodwin (1828-1902) who believed that a square and a circle of the same perimeter had the same area, was unique in getting so many values of pi: 4, 3.555556. 3.3333333, ..., 3.2, 2,56. His attempt to legislate a legal value of pi of 4 exemplifies that irrational forces are present, ready to shake up the ivory towers.

3. Your Digital Imprint

My birth dates are at positions 3,749,507 and 25,864,351. And my mobile numbers start at positions 308,617,971 in pi. Where do you occur in pi? It's never too late to test the power of your computer in searching for your IC number or phone numbers in the decimal expansion of pi—claim your "digital imprint," and immortalize it!

4. Tattooing Pi


Are you so much in love with the beauty of Pi that you're prepared to get a tattoo of it on some inconspicuous parts of your body?

5. The Bible Encoded in Pi

When coded in numeric form, every text in the world can be stored. Besides the biblical value of pi being equal to 3, as deduced from 1 Kings 7:23 and II Chronicles 4:2—which describes a round "sea" (large bowl) made of metal, 10 cubits in diameter and 30 cubits in circumference—could the Bible, with an estimated 10⁷ digits occur in the digits of pi? Or, at least, could some of the 66 books of the Holy Scriptures be encoded in it? Or, what about some Shakespeare's plays

For young readers, what about the frequency of your favorite digit, or "lucky number," appearing within the first million or billon places of pi?

6. Pi Digits for Sale

Pi-rate Math Geek T-shirt

If random numbers are a multi-million business, would it unethical to commercialize the digits of pi in future, although up to now they have been used primarily for scientific purposes?

7. Meta-Pi: Does π Contain π Itself?

Does pi contain itself, or at least finite sequences of it? After all, some numerologists claimed that the decimal expansion of pi contains the secrets of the universe. For instance, Dr. Matrix, the renowned numerologist and the darling of number perverts, noted that "Pi conveys the entire history of the human race."


8. Pi Quickies, Trickies, and Toughies

The beauty, ubiquity, and utility of pi has blessed recreational mathematicians with a number of mathematical quickiestrickies, and toughies.

What are the next numbers in the following sequence? 3, 1, 4, 1, 5, ...

Show that there exists precisely one set of three single-digit positive integers a, b, c such that π/4 = tan¹ 1/a + tan¹ 1/b + tan¹ 1/c[a = 2, b = 5, c = 8]

What is the probability that a coin tossed an even number of times will land an equal number of times on its head and on its tail?

What is the probability that two numbers, p and q, chosen at random be relatively prime? (6/π²)

Prove that pi is irrational.

Which is larger? eπ or πe

9. Contact with ET
Did the pi-aliens do that?

In the short story Contact by Carl Sagan, an extraterrestrial tells a woman on Earth that pi contains an important message for us in encrypted form.

By using zeros and ones in the decimal representation of pi, we can send meaningful messages to beings or aliens from other galaxies.

10. Pi Mnemonics and Haikus in Chinese and Malay

Circle Digits—A Self-Referential Story

For the majority of us who aren't blessed with a sharp memory, creating mnemonic devices that will allow us to more easily memorize the digits of pi is a less-stressful mental exercise. Besides, long after we've forgotten our school math, composing meaningful mnemonics could help us remember, say, the first twenty-five digits of pi until we breathe our last breath.

Other than remembering mnemonics in English, such as "May I have a large container of coffee?," why not compose some in other languages and dialects? And some pi-kus (pi-haikus) as well?

π = 3.14 and John 3:16

So close, yet so far
Rational and eternal
The union is null.

11. Pi on Mars and Venus
Is the value of pi different on other planets, or in other galaxies, if the curvature of space, or multi-dimensionality, is taken into account? At least, from a philosophical standpoint, could pi take on different values, just like the sum of the interior angles of a triangle needn't always be 180 degrees? 

12. Non-Euclidean Geometry and Pi

In Euclidean and non-Euclidean geometries, the ratio of the circumference, C, of a circle to its diameter, d, depends on the type of geometry. In other words, depending on the geometric milieu, can we then say that the value of pi varies?

Euclidean geometry: C/dπ

Lobachevskian geometry: C/d > π

Riemannian geometry: C/d < π

Pi-fully yours


Berggren, L., Borwein, J. & Borwein, P. (2000). Pi: A source book (2nd ed.). Springer.

Gardner, M. (1985). The Magic Numbers of Dr. Matrix. New York, Dorset Press.

Posamentier, A. S. & Lehmann, I. (2004). Pi: A biography of the world's most mysterious number. New York: Prometheus Books.

Yan, K. C. (2013). 17 theomatical haikus for math educators. Yan's One Minute Math Blog, May 18, 2013.

Yan, K. C. (2009). Geometrical quickies & trickiesSingapore: GLM Pte Ltd.

© Yan Kow Cheong, March 14, 2014.

Tuesday, March 11, 2014

Primes and Priests

If mathematics were a secular religion, then mathematicians could be regarded as its high priests. If Christianity were a scientific enterprise, then priests could be regarded as its prime managers.

Toying with "prime cubes"

Let’s look at some parallelisms between primes (or prime numbers, as British and Singaporeans like to call it) and priests (or pastors, depending on your denomination).

One unsolved problem in mathematics deals with the distribution of primes.
One unsolved spiritual problem in the Church involves the appointment of women-priests and women-bishops.

The prime number theorem states that the number of primes less 
than any real number x is approximately equal to x/(ln x). 
Graph from Paul Glendinning's Maths in minutes

Primes are the atoms of mathematics—all positive integers are composed of primes.
Priests are the ambassadors of Christ—God's chosen servants for His people, who are expected to exemplify a holy lifestyle.

All positive integers can be expressed as a product of primes. The fundamental theorem of arithmetic: “Every natural number greater than 1 can be written as a unique product of prime numbers.”
All born-again believers are members of a royal priesthood—they are all priests, in a spiritual sense.

Different proofs exist for the infinitude of primes—some 50 odd million known primes have been printed.        
Different proofs exist, which point to the Omnipresent, Omnipotent, and Omniscient God.

Pseudo-formulas for generating a few hundred or thousand primes exist.
Pseudo-Christs appear every now and then to deceive the believers, performing signs and wonders.

The number 1 is a pseudo-prime—its prime-like property fools the novice.
Cult leaders are pseudo-priests, out to control the lives of their followers.

A prime has four factors. For instance, the prime number 5 is divisible by 1, 5, –1, and –5.
A priest serves as a “surrogate’ to “forgive” sins, by acting as an earthly proxy for God.

Recruiting new priests is a challenging task for the Church, because of low pay and long hours, with many feeling unappreciated and undervalued.    
Finding new primes is a favorite brain-busting activity for math geeks, who spend unpaid hours on supercomputers to look for the next Mersenne prime—whose discovery is linked to the largest prime.

Different types of primes: twin primesemirpssexy primes, ....
Different types of priests: celibate, married, gay, women, ....

An order-3 magic square made up only of prime numbers,
with the smallest possible magic constant, 177

The prime number 2 is the only even prime, and this property is often used as a catalyst to pose many contests problems and mathematical quickies.
The High Priest is the only one allowed to offer sacrifices to God on behalf of the people every year in the Holy of Holies, as related in the Old Testament.

A magic prime: 73,939,133—it makes a new prime with each digit taken from the end.
A priest can act as a magician by using the surprising property of the Möbius strip to "explain" the concept of the Trinity.

Möbius strips for your Xmas decorations
Primes are used in cryptography—for security purposes, be it in banking or on-line transactions.
Priests are often consulted by world leaders for key decisions on complex or thorny issues—they anoint them to lead their nations with divine wisdom.

Divisibility tests and computer testing are signs that some numbers could potentially be primes—there is no foolproof method for finding primes.
Spiritual gifts and leadership qualities (speaking and interpreting in tongues, prophecy, vision, charisma, ...) are signs or criteria often used to select candidates for the priesthood.  

The idea of primeness or primehood has gone into the vernacular: prime time, prime target, prime locationprime ribs.
The idea of being, or behaving like, a priest has come to be associated with honesty, power, and authority.

Primehood denotes elements such as rarity, security, oddity, or money—coming up with a general formula for generating primes may make one rich!
Priesthood suggests qualities such as holiness, morality, chastity, spirituality, or respectability. You sound like a priest!

Escher goes topological!

The lure and challenge to factorize large numbers into prime factors has led security experts to design algorithms and to write programs to encrypt and decrypt digits-long numbers—it’s a multi-million dollar business.

Health-and-wealth priests, pastors, or ministers from mega-churches often live like kings, generating much income from the sales of their books, conference preaching, and the like—their prosperity gospel appeals to many materialistic believers.

Primes are used for survival (Darwinian weapons against predators)—less competition for food.  
Priests are God's ambassadors to bring healing and deliverance to entire tribes or nations—in recent years, there have been spiritual breakthroughs in countries like South Korea, Haiti, and Uganda.

The power of the Holy Spirit sweeps across nations, delivering peoples under the bondages of occultism and curses.
Primes are used to test the power of supercomputers, and the hunt for a formula for generating primes has indirectly yielded new knowledge in many unrelated branches of mathematics.

Harry Nelson was the first person to produce a 3 x 3 matrix containing only consecutive primes.

The number 73 as a magical prime filled with numerical curiosities.
Melchizedek as the High Priest, as reported in the Old Testament—the one who blessed the patriarch Abraham.  

Art and math for young children

Clay's millennium prize (unsolved) problems, one of which is the yet-to-prove Riemann hypothesis, which is related to the distribution of primes.
Unsolved spiritual problems, such "How can Jesus be both man and God?"; "What comes before God?".

Many contests and security problems tap on the properties of primes and prime factorization
Many real-world problems have their solutions in the Bible, as priests interpret God's Word in a modern-day context—applications of His Word to solve practical problems.

The Riemann hypothesis deals with the distribution of primes. 
Riemann initially established that there are trivial zeros for the 
negative even integers, which don't contribute much to the 
overall series of the Riemann zeta function. His hypothesis was: 
The remaining zeros all include a real part equal to 1/2—they 
should lie on a line expressed as 1/2 + ix, where x is a real 
number and i is √–1. Source: Paul Glendinning's Maths in minutes

Numerologists deify prime numbers. In some superstitious milieux, as a divination tool, prime magic squares may be used as an omen to ward off evil and reduce birth pain.
Priests in some quarters often condone the worship of saints among believers, although this idolatrous practice isn't advocated in the Holy Scriptures.

Prime life cycles of insects are used as a camouflage to fool the predators. For example, there are different species of periodical cicadas, some with a 13-year life cycle and others with a 17-year life cycle. The cicadas benefit from the lengthy rotation rate, since the different-cycle cicadas compete for food less frequently.
Priests are pretty busy during these two seasons every year: Lent season leading to Easter, to commemorate the resurrection of Christ; and weeks-long caroling leading to Christmas, to celebrate the birthday of Christ.

It's now your turn to share some commonalities between primes and priests with the mathematical brethren.


Pickover, C. A. (2002). The zen of magic squares, circles, and stars. Princeton & Oxford: Princeton University Press.

Schwartz, R. E. (2010). You can count on monsters. A K Peters/CRC Press.

© Yan Kow Cheong, March 11, 2014.