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The Power of Two . . . in Poetry?

We all know the importance of the powers of two in this digital age, but were they important historically? Many cultures used doubling and halving to facilitate multiplication and division (see “Travels of Two,” Amy Shell-Gellasch, Math Horizons, September 2014, 5–7). But what about the powers of two: 2, 4, 8, 16, 32, and so on?

These numbers rarely show up explicitly in the mathematical applications of earlier eras. But as we shall see, they were used in at least one surprising way: in composing and analyzing Indian Vedas—ancient Hindu literature and scriptures written during the Vedic period (roughly 1300 BCE to 300 CE).

Large numbers and ideas about infinity are common themes in Indian writing, philosophy, cosmology, and religion. Figure 1 shows symbols for numbers from the second century CE that are much larger than any numbers used in daily life.

This interest in large numbers carried over into poetry as well, as seen in the following lines from the Rig Veda, a collection of 1,000 hymns to Hindu gods, first written down around 300 BCE.

A thousand heads hath Puruhsa, a thousand eyes, a thousand feet.

On every side pervading earth he fills a space ten fingers wide.

In ancient Indian society, the writing of prose and poetry was held in very high esteem. This attitude permeated many aspects of the culture, including mathematics. Not only did Indian poets include allusions to mathematics in their writing, but Indian mathematicians also incorporated poetry into their mathematics.

At this time, mathematics was typically written rhetorically—in words only—and occasionally using syncopated notation—a mixture of words and symbols. A fully symbolic form of writing mathematics did not emerge until the Renaissance in Europe. But Indian mathematicians took rhetorical mathematics to new heights—it was written in verse. Imagine if your mathematics professor insisted you write the solutions to your homework problems as poems!

Early Indian numbers were encoded as items of the same amount, creating inherent imagery. The number 210 may have been expressed as “sky sun hands.” We have two hands, there’s one sun, and the sky is empty. Interestingly, Indian numerals were written left to right as we do today, but they were spoken right to left.

Not only was mathematics written in verse, but there were also restrictions on how the poem was composed. To maintain the aesthetic beauty, mathematicians needed to consider how their mathematics sounded when spoken. They could use many different words to represent the same mathematical idea. This flexibility allowed the practitioner to choose a word with the correct number of syllables and rhyming pattern for the poetic form being used. The final poem had to make mathematical sense, and it had to be beautifully written.

Moreover, Indian poets needed to pay attention to where the stressed syllables fell. Prosody in poetry is the art of making sure each line in a poem has the right number of metrics—syllables with the right number of stressed and unstressed syllables.

Elementary combinatorics tells us that there are 2n ways to say an n-syllable sentence, where each syllable is either heavy (stressed) or light (unstressed). For instance, there are 211 ways we can stress the sentence “There once was a mathematician from Bonn.”

But neither the ancient Indian mathematicians, nor the poets for that matter, had the notion of exponents. How did they compute this value?

The Indian mathematician Pingala (dated prior to 200 BCE), in his work Chandahsutra (Rules of Metrics) gave the following algorithm for computing 2n. “When halved, [record] two. When unity [is subtracted, record] zero. When zero, [multiply by] two; when halved, [it is] multiplied [by] so much [i.e., squared].”

Let’s follow Pingala’s cryptic prescription to compute 211. First we follow the algorithm in the first two sentences.

• 11 is odd, so subtract 1 and record 0.

• 10 is even, so halve it and record 2.

• 5 is odd, so subtract 1 and record 0.

• 4 is even, so halve it and record 2.

• 2 is even, so halve it and record 2.

• 1 is odd, so subtract 1 and record 0.

Mathematically, this procedure generates a sequence of 0s and 2s while converting the exponent, 11, into binary:

The third sentence tells us how to use the sequence of 0s and 2s to compute 211. We begin with the number 1. Then, reading the sequence in reverse order, for every 0 in the sequence, double the present result, and for every 2, square it. This gives us:

as claimed. Pretty slick, eh?

It is remarkable that the ancient Indians devised this efficient algorithm for fin ding the powers of two— especially without exponents or other symbolic notation.* We leave it as an exercise to show that Pingala’s method always computes 2n. When you’ve found the proof, write it up—in verse!

Amy Shell-Gellasch teaches mathematics at Montgomery College in Rockville, Maryland, and subjects her students to a constant barrage of historical tidbits at every turn

Email: Amy.Shell-Gellasch@montgomerycollege.edu

John Thoo teaches mathematics at Yuba College in Marysville, California. He dabbles in the history of mathematics and knows just enough to be dangerous. Email: jthoo@yccd.edu

These numbers rarely show up explicitly in the mathematical applications of earlier eras. But as we shall see, they were used in at least one surprising way: in composing and analyzing Indian Vedas—ancient Hindu literature and scriptures written during the Vedic period (roughly 1300 BCE to 300 CE).

Large numbers and ideas about infinity are common themes in Indian writing, philosophy, cosmology, and religion. Figure 1 shows symbols for numbers from the second century CE that are much larger than any numbers used in daily life.

This interest in large numbers carried over into poetry as well, as seen in the following lines from the Rig Veda, a collection of 1,000 hymns to Hindu gods, first written down around 300 BCE.

A thousand heads hath Puruhsa, a thousand eyes, a thousand feet.

On every side pervading earth he fills a space ten fingers wide.

In ancient Indian society, the writing of prose and poetry was held in very high esteem. This attitude permeated many aspects of the culture, including mathematics. Not only did Indian poets include allusions to mathematics in their writing, but Indian mathematicians also incorporated poetry into their mathematics.

At this time, mathematics was typically written rhetorically—in words only—and occasionally using syncopated notation—a mixture of words and symbols. A fully symbolic form of writing mathematics did not emerge until the Renaissance in Europe. But Indian mathematicians took rhetorical mathematics to new heights—it was written in verse. Imagine if your mathematics professor insisted you write the solutions to your homework problems as poems!

Early Indian numbers were encoded as items of the same amount, creating inherent imagery. The number 210 may have been expressed as “sky sun hands.” We have two hands, there’s one sun, and the sky is empty. Interestingly, Indian numerals were written left to right as we do today, but they were spoken right to left.

Not only was mathematics written in verse, but there were also restrictions on how the poem was composed. To maintain the aesthetic beauty, mathematicians needed to consider how their mathematics sounded when spoken. They could use many different words to represent the same mathematical idea. This flexibility allowed the practitioner to choose a word with the correct number of syllables and rhyming pattern for the poetic form being used. The final poem had to make mathematical sense, and it had to be beautifully written.

Moreover, Indian poets needed to pay attention to where the stressed syllables fell. Prosody in poetry is the art of making sure each line in a poem has the right number of metrics—syllables with the right number of stressed and unstressed syllables.

Elementary combinatorics tells us that there are 2n ways to say an n-syllable sentence, where each syllable is either heavy (stressed) or light (unstressed). For instance, there are 211 ways we can stress the sentence “There once was a mathematician from Bonn.”

But neither the ancient Indian mathematicians, nor the poets for that matter, had the notion of exponents. How did they compute this value?

The Indian mathematician Pingala (dated prior to 200 BCE), in his work Chandahsutra (Rules of Metrics) gave the following algorithm for computing 2n. “When halved, [record] two. When unity [is subtracted, record] zero. When zero, [multiply by] two; when halved, [it is] multiplied [by] so much [i.e., squared].”

Let’s follow Pingala’s cryptic prescription to compute 211. First we follow the algorithm in the first two sentences.

• 11 is odd, so subtract 1 and record 0.

• 10 is even, so halve it and record 2.

• 5 is odd, so subtract 1 and record 0.

• 4 is even, so halve it and record 2.

• 2 is even, so halve it and record 2.

• 1 is odd, so subtract 1 and record 0.

Mathematically, this procedure generates a sequence of 0s and 2s while converting the exponent, 11, into binary:

The third sentence tells us how to use the sequence of 0s and 2s to compute 211. We begin with the number 1. Then, reading the sequence in reverse order, for every 0 in the sequence, double the present result, and for every 2, square it. This gives us:

as claimed. Pretty slick, eh?

It is remarkable that the ancient Indians devised this efficient algorithm for fin ding the powers of two— especially without exponents or other symbolic notation.* We leave it as an exercise to show that Pingala’s method always computes 2n. When you’ve found the proof, write it up—in verse!

Amy Shell-Gellasch teaches mathematics at Montgomery College in Rockville, Maryland, and subjects her students to a constant barrage of historical tidbits at every turn

Email: Amy.Shell-Gellasch@montgomerycollege.edu

John Thoo teaches mathematics at Yuba College in Marysville, California. He dabbles in the history of mathematics and knows just enough to be dangerous. Email: jthoo@yccd.edu