Jaghanya Parīta Asaṃkhyāta

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Jaghanya Parīta Asaṃkhyāta (Innumerable of low enhanced order), or JPA, is a large number defined in ancient Indian literature of Jainism such as the Anuyogadvāra Sūtra (ca. 1st century AD), Tiloyapaṇṇatti (ca. 6th century AD), Trilokasāra (ca. 10th century AD) and Lokaprakāśa (ca. 11th century AD). It is approximately equal to \(10\uparrow \uparrow (1.285 \times 10^{136})\) according to calculations by Radha Charan Gupta and Muni Mahendra Kumar.

Background

Jainism is an Indian religion that has existed in some form since at least the second millennium BC, but most major pieces of literature and mathematical texts from the religion were written after the time of the Buddha, in the 4th and 5th centuries BC.

The Surya Prajñapti (ca. 4th century BC) is an ancient Indian text that separates the Gaṇanā Saṃkhya (mathematical numbers) into three groups:

1. Saṃkhyāta, or “Numerable”
2. Asaṃkhyāta, or “Innumerable”
3. Ananta, or “Infinite”

Asaṃkhyāta is split into 3 groups

1. Parīta Asaṃkhyāta, or "Nearly Innumerable"
2. Yukta Asaṃkhyāta, or "Truly Innumerable"
3. Asaṃkhyāta Asaṃkhyāta, or "Innumerably Innumerable"

These categories are then split up further into

1. Jaghanya, or "Minimum"
2. Madhyama, or "Intermediate"
3. Utkṛṣṭa, or "Maximum"

Jaghanya Parīta Asaṃkhyāta is the smallest nearly innumerable number (Parīta Asaṃkhyāta), and the largest numerable number, Utkṛṣṭa Saṃkhyāta (US) is defined as equal to one less than JPA. Although the asaṃkhyāta and ananta are designed as stand-ins for any incomprehensibly large number, ancient traditions do give definitions for each of them. The value of JPA (and thus US) has a specific definition.

This number has parallels with the Buddhist number Asankhyeya (Incalculable), found in texts such as the Avatamsaka Sutra. In addition, various authors have drawn parallels between the asaṃkhyāta/JPA and aleph null (\(\aleph_0\)), the cardinal number of the infinite set of integers (also known as the "first transfinite number"). It is believed that the Jains had a very advanced view of infinity compared to contemporary civilizations, being the first to discard the idea that all infinities are equal.

Definition

Diagram of the concentric rings of land and ocean, each twice as wide as the last

The following is the definition for JPA in simplified modern language:

Imagine that there are four enormous circular cylinders (containers), each with a diameter of 100,000 yojanas and a height of 1000 yojanas. These cylinders are called śālākā, pratiśalākā, mahāśalaka and anavasthita, although they can be more simply notated as A, B, C and X(0). The first step is to fill the anvasthita container X(0) with mustard seeds until the container is full and a conical heap is formed on top so that the container holds the maximum possible number of seeds.

Now, imagine that there is a god who carries this container and starts dropping one mustard seed each in the concentric continents and oceans situated in the universe (Jain philosophers imagine the universe existing as an endless number of concentric rings, half ocean and half land). The innermost circle is Jambūdvīpa, with a diameter of 100,000 yojanas, surrounded by the Lavaṇa Ocean with a diameter of 200,000 yojanas, surrounded by another ring with a diameter of 400,000 yojanas. Each ring is twice as wide as the last.

The god continues placing a single seed on each ring until the container X(0) is completely empty, places one seed in the cylinder A, and then creates a new cylinder X(1) with the same height but a diameter equal to that of the last ring. X(1) is then filled as much as possible and the god repeats the process, taking each seed out and placing it on a new ring. Once X(1) is empty a second seed is placed in cylinder A and X(2) is created with a diameter equal to that of the last ring.

This process is repeated until cylinder A is completely full, at which point a single seed is placed in cylinder B. The seeds of cylinder A are then placed on further rings, at which point the process continues. Once cylinder B is full a single seed is placed in cylinder C, the seeds of cylinder B are placed on further rings, and the process continues. Once cylinder C is completely filled, a final anvasthita container is created with a diameter equal to the final ring, and the number of seeds that completely fills that container is Jaghanya Parīta Asaṃkhyāta.

Calculation

Let numbers of seeds in container X(i) as \(n_i\). then JPA is approximately \(n_{{n_0}^3}\).

According to the Trilokasāra (ca. 10th century AD) a yojana (approximately 12.5 kilometers) is equal to \(24,576,000,000 = 2.458 \times 10^{10}\) sarṣapas, where a sarṣapa is the width of one mustard seed. Thus, the radius of the cylinders A, B, C and X(0) are \(r = 2.458 \times 10^{10} \times 50,000 = 1.228 \times 10^{15}\) sarṣapas, and the height of each cylinder is \(h = 2.458 \times 10^{10} \times 1000 = 2.458 \times 10^{13}\) sarṣapas. The Trilokasāra also states that the height of the conical heap is equal to the circumference of the cylinder divided by 11, which is equal to (2πr)/11. Thus, the formula for the total number of seeds in both the cylinder and conical heap is \(6r^2 h+\frac{4}{11}\pi r^3\), where \(r\) is the radius of the cylinder in sarṣapas and \(h\) is the height. This gives a value of \[n_0 = 2.342 \times 10^{45}\] total seeds in each container according to Gupta's calculation. However, ancient calculations assumed that \(\pi = 3\), and under this modification the total number of seeds is slightly lower, at \(n_0 = 1.997 \times 10^{45}\).^{Footnote 6}

Gupta showed that \[n_{i+1} = n_i 2^{p n_i}\] where Gupta used \(p=3\) for the lower bound. Although Gupta used a function defined with a sequence of \(m_{i + 1} = m_i^{m_i}\) to evaluate \(n_i\), we show evaluation with tetration here for convenience to compare with other googological numbers, where the approximation is equivalent. When \(n_0 = 2.342 \times 10^{45}\) and \(p=3\), \(n_i \approx 10 \uparrow \uparrow (i+2.219)\) for \(i>2\) with linear approximation of tetration. Therefore, JPA \(\approx n_{{n_0}^3}\) can be approximated with \[\textrm{JPA} \approx 10 \uparrow \uparrow {n_0}^3\] This approximation is valid by taking into account of some uncertainty in the values of \(n_0\) and \(p\), because when expressed as \(10 \uparrow \uparrow ({n_0}^3 + a)\), \(a\) is much smaller than \({n_0}^3\) and the value of \({n_0}^3 + a\) can be expressed as \({n_0}^3\) value.

The final value of JPA using the modern system \(\pi = 3.14159\dots\) is \[\textrm{JPA} \approx 10\uparrow \uparrow (1.285 \times 10^{136})\] Using the ancient system \(\pi = 3\), the final value is \(10\uparrow \uparrow (7.965 \times 10^{135})\).

Demonin2 showed that by assuming that a yojana is equivalent to 12.5 km, and that mustard seeds are spherical with a diameter of 2.5 mm and are packed using close-packing of equal spheres, the JPA value is considerably smaller than the accepted value in literature. The calculated JPA is: \[\textrm{JPA} \approx 10\uparrow \uparrow (4.546 \times 10^{109})\]

Origin

It is difficult to determine when the definition for JPA was first written, or if it evolved from a different form. The earliest possible point of its definition is the Surya Prajñapti (ca. 4th century BC). George G. Joseph claims it to originate from the Anuyoga Dwara Sutra, created sometime between 200 BC and 100 AD. However, this may have been a smaller or less well-defined definition of the number. The Tiloyapaṇṇatti (ca. 500 AD), written by Yativṛṣabha, is the first concrete source of the definition. Either way, it is almost certainly the largest number given a definition at the time.

Utkṛṣṭa Ananta Ananta

As mentioned earlier, the Surya Prajnapti lays out a series of numbers beyond JPA, and according to research by Muni Mahendra Kumar these numbers have various definitions laid out by sources such as the Anuyoga Dwara Sutra and the later Tiloyapaṇṇatti. The largest of these definitions is for Utkṛṣṭa Ananta Ananta (UAnAn), which could possibly be as large as \(10\uparrow \uparrow \uparrow 38\) using a rudimentary form of pentation.

Approximations

For the value of \(10\uparrow \uparrow (1.285 \times 10^{136})\)

Notation	Approximation
Hyper-E notation	\(E2.1338\#2\#2\)
Up-arrow notation	\(10 \uparrow\uparrow 10 \uparrow \uparrow 2.3291713 \approx 10 \uparrow\uparrow\uparrow 2.3672\)
Bird's array notation	{10, {10, 136}, 2}
s(n) map	\(s(1)^3f(s(1)^2f(443)), f(x)=x+1\)
Bashicu matrix system	(0)(1)(2)(0)(0)(0)(0)(0)(0)[134]
Fast-growing hierarchy	\(f_3(f_2(443))\)
Hardy hierarchy	\(H_{\omega^3 + \omega^2}(443)\)

Calculation

• \(E2.1338\#2\#2 = E2.1338\#(E2.1338\#2) \approx 10\uparrow\uparrow(10\uparrow10\uparrow2.1338) \approx 10\uparrow \uparrow (1.207 \times 10^{136})\)

See also

Googology in Asia

Indian counting system: Lakh · Crore · Padma · Tallakshana · Ogha · Ababa · Atata · Sogandhika · Uppala · Dvajagravati · Kumuda · Pundarika · Paduma · Kathana · Mahakathana · Asankhyeya · Dvajagranisamani · Vahanaprajnapti · Inga · Kuruta · Sarvanikshepa · Agrasara · Uttaraparamanurajahpravesa · Avatamsaka Sutra · Nirabhilapya nirabhilapya parivarta · Jaghanya Parīta Asaṃkhyāta
Chinese, Japanese and Korean counting system: Wan · Yi · Zhao · Jing · Gai · Zi · Rang · Gou · Jian · Zheng · Zai · Ji · Gougasha · Asougi · Nayuta · Fukashigi · Muryoutaisuu
See also: Template:Googology in Japan