Beautiful Numbers

While reading the The Da Vinci Code a few years back, I came across this passage:

He felt himself suddenly reeling back to Harvard, standing in front of his “Symbolism in Art” class, writing his favorite number on the chalkboard.

1.618

Langdon turned to face his sea of eager students. “Who can tell me what this number is?”

A long-legged math major in back raised his hand. “That’s the number PHI.” He pronounced it fee.

“Nice job, Stettner,” Langdon said. “Everyone, meet PHI.”

“Not to be confused with PI,” Stettner added, grinning. “As we mathematicians like to say: PHI is one H of a lot cooler than PI!”

Langdon laughed, but nobody else seemed to get the joke.

Stettner slumped.

“This number PHI,” Langdon continued, “one-point-six-one-eight, is a very important number in art. Who can tell me why?”

Stettner tried to redeem himself. “Because it’s so pretty?”

Everyone laughed.

“Actually,” Langdon said, “Stettner’s right again. PHI is generally considered the most beautiful number in the universe.”

The laughter abruptly stopped, and Stettner gloated.

As Langdon loaded his slide projector, he explained that the number PHI was derived from the Fibonacci sequence – a progression famous not only because the sum of adjacent terms equaled the next term, but because the quotients of adjacent terms possessed the astonishing property of approaching the number 1.618 – PHI!

Despite PHI’s seemingly mystical mathematical origins, Langdon explained, the truly mind-boggling aspect of PHI was its role as a fundamental building block in nature. Plants, animals, and even human beings all possessed dimensional properties that adhered with eerie exactitude to the ratio of PHI to 1.

– From the Da Vinci Code, by Dan Brown

That passage set me thinking about other numbers considered pretty or at least very interesting. As an Indian I can point to quite a few numbers that we can be proud of. At the top of the list is Aryabhata’s invention of the most famous number of them all – 0, which by helping establish the place value system and the decimal number system made innumerable mathematical and scientific discoveries possible and practical. Just imagine having to write a number like 999,999 in the Roman Numeral system. You would need to write CM XC IX CMXCIX. And if you weren’t aware that adding the “overline” multiplies a number by 1000, then you would have to struggle significantly more to represent a number such as 999,999.

As an ex-IIT’ian I have had a fascination for numbers and so have many of my classmates. Both during and after life at IIT I have seen my friends use one particular number quite often – 1729. I myself used 1729 as my page id when I was building my hostel’s website back in the days when you needed to sign up for a free web-page at sites like GeoCities. A few years after graduation my friend asked me to unlock his bicycle. The code – 1729. A few more years later another friend sent out an email saying that his previous email id had been handed out to several mailing lists and he was receiving a lot of spam. So he changed his email id to something that had the number 1729 in it. If you are not very mathematically inclined you might think of 1729 as a very weird number to be fascinated with. But there is history behind it. 1729 is in fact called the Hardy-Ramanujan number, following a very famous conversation between G. H. Hardy and Srinivasa Ramanujan.

Hardy used to visit him, as he lay dying in hospital at Putney. It was on one of those visits that there happened the incident of the taxi-cab number. Hardy had gone out to Putney by taxi, as usual his chosen method of conveyance. He went into the room where Ramanujan was lying. Hardy, always inept about introducing a conversation, said, probably without a greeting, and certainly as his first remark: “I thought the number of my taxi-cab was 1729. It seemed to me rather a dull number.” To which Ramanujan replied: “No, Hardy! No, Hardy! It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways.”

That is the exchange as Hardy recorded it. It must be substantially accurate. He was the most honest of men; and further, no one could possibly have invented it.

– Foreword by C. P. Snow, to G. H. Hardy’s “A Mathematician’s Apology (Canto)”

For those trying to figure out what Ramanujan meant, 1729 = 12³ + 1³ = 10³ + 9³. Not only is this number an Indian favorite (Ramanujan was Indian), but mathematicians worldwide recognize it for the brilliance and simplicity of the discovery. This number is also referred to as a Taxicab number due to the associated incident, though the unique property of this number was actually discovered by Bernard Frénicle de Bessy.

There is another number that piqued my interest, however, when I was preparing for the Indian National Mathematics Olympiad in 1994. I came across a number that was referred to as the Kaprekar Number – 6174. In later years I came to know that the information was inaccurate, because this number was called Kaprekar Constant, and Kaprekar Numbers referred to a separate category of numbers. A Kaprekar Number is a number that is thus defined:

A Kaprekar number for a given base is a non-negative integer, the representation of whose square in that base can be split into two parts that add up to the original number again.

As the Wikipedia article states, 45 is a Kaprekar number because 45 = 20 + 25 and 45² = 2025. These numbers were discovered by another Indian mathematician Dattaraya Ramchandra Kaprekar, who had a penchant for discovering several results in number theory and was very well known as a recreational mathematician. Funny what people come up with during their free time!

But back to the Kaprekar constant – 6174. Again, this falls wholly into the category of an unremarkable-looking number. But there is a lot more to it. Arrange the digits of the number in descending order: 7641. Arrange its digits in ascending order: 1467. Subtract the two: 7641 – 1467 = 6174. This happens to be the only 4-digit number that exhibits this property. If you think that is surprising, there is more. Take any 4 digit number with at least 1 digit different from the rest. Repeat the operation of subtracting the ascending order of digits from the descending order. After a finite number of iterations you will hit 6174!! I was so impressed with this number that I couldn’t rest till I had established the proof of this. Yutaka Nishiyama has a well-documented proof, which is much more rigorous than what I came up with (plus I am too lazy to type out my proof in HTML here).

There are other Kaprekar constants when you change the number of digits to 3 (495) or something else.

I am sure there are several other numbers that have even more quirky properties. Having had an affinity towards mathematics in general since a young age and towards number theory in particular since I was 15, I know that I am missing out on such a huge treasure by pursuing a career in something so far removed from mathematics.