Hindu-Arabic notation is strictly positional. It means if we stick an Arabic ‘٤' (4) to the left of an Arabic ‘١' (1), we would normally get a ‘٤١' (41).
$$4\times 10 + 1 = 41$$
If for one moment, suppose the glyph for ‘5' is yet to be invented and we are asked to supply one.
Excuse me, how do you write ‘five'?
One easy way is to copy the Romans: We pretend the rule is not positional but additive, i.e.
$$٤+١=4 + 1 = 5$$
However, for our new design to function properly (and to prevent mix-up) in the positional scheme, they must be fused tightly together into one single glyph. And the result is something that looks like a mirror image of B (B) or a Gujarati ‘ઇ' (i).
٤
١
Now, this gedankenexperiment can be repeated if you start with ઇ (5) and remove the vertical stroke (1) on the right to get 4 (٤). Given the fact in most early mathematical manuscripts, four is consistently written as ۴, it is more likely that ٤ is manufactured from ઇ by Mashriki scribes. The reason why ۴ is kept in Persian/Sindhu keyboard but abandoned in Mashirki keyboard is not exactly known.
۱۲۳۴
B in old mathematical tables
The Persian engineer Abū BakarMuḥammad ibn al-Ḥasanal-Karaji (953 - 1029) must have produced the first written record of binomial coefficients (arranged in the form of Pascal's triangle) but his work is lost.
See Mustapha Nadmi (2019) A Significant step toward the development of algebra: Al-Samaw'al Ibn Yahya Al-Maghribi, a twelfth century mathematician, PhD Thesis, Columbia University.Binomial coefficients in MSS Ayasofia No. 2718, 116ff, the oldest extant copy of al-Bahir fi al-Jabr (in Suleymaniye Library in Instabul, Turkey). The manuscript was discovered by Max Krause in 1936 in Istanbul. It was copied on 24 April 1325, about 150 years after the death of its author. Another slightly younger copy is available and it is Esat Efendi No. 3155.
Fortunately when al-Samaw'al ibn Yahyaal-Maghribi (1130 - 1180), a muslim Jewish mathematician, wrote his al-Bahir fi al-Jabr (The Brilliance in Algebra), he cited al-Karaji's al-Fakhri fi al-Jabr wa al-Muqabala (The Brilliance in Algebra and Equation Balancing) as one of his reference. In al-Maghribi's table of binomial coefficients \(n \choose k\), we can see first two ‘5's when the case for \(n = 5\) is written down, that is, \(5 \choose 1\) and \(5 \choose 4\). In both cases, the number is rendered as B.
B
The B-glyph is repeated throughout the table with the vertical line closing the loop. Occasionally, however, the loop is not 100% closed, as in the case of \({10 \choose 2} = 45\), and \({11 \choose 2} = 55\).
al-Maghribi's table of powers, \(2^p\) and \(3^p\). The mirrored-B five is shown in \(2^8 = 256\), \(2^9 = 512\), and \(3^8 = 6561\).
Kitab al-Amal bi al-Asturlab of al-Khawarizmi (circa 780 - circa 850). The manuscript is stored in Berlin, Staatsbibliothek, Landsberg 56, 77b-97b. Note that B was used by both al-Maghribi and al-Khawarizmi. However, al-Maghribi uses the circle zero (०) but al-Khawarizmi uses the Mashriki dot zero (٠) instead.
The slow death of B
Now once the B-glyph is registered in our brain as the symbol for ‘5', it should stayed registered even when it is rotated. In what follows, we show the gujarati ઇ rotated by \(15^\circ\), \(30^\circ\), \(60^\circ\), and \(90^\circ\). This is perhaps, how we evolve the B-glyph to the inverted heart-shaped ‘5' (۵, U+06F5. The See also the reversed open E in Latin script, ɞ, U+025Eclosed omega in Latin, ɷ, also looks like an inverted heart U+0277) in modern Persian/Sindhi keyboard. Transition from inverted heart to the modern Mashriki ‘5' (٥, U+0665) is easy, since all you need to do is just to de-emphasize the cusp.
ઇ
ઇ
ઇ
ઇ
۵
۵
٥
Evolution of arabic numerals Seyyed Hossein Nasr (1976) Islamic science: an illustrated study, World of Islam Festival Publishing Company Ltd, London, p. 78 (Figure 22)
Persian inverted heart-shaped fives in a \(6 \times 6\) magic square Seyyed Hossein Nasr (1976) Islamic science: an illustrated study, World of Islam Festival Publishing Company Ltd, London, p. 80 (Figure 23). Note that this magic square has some mistakes in column 1 position 4, column 2 position 3, and column 4 position 4.
$$\begin{bmatrix} 2 & 31 & 33 & 34 & 10 & 1\\ 29 & 18 & 21 & 24 & 11 & 8\\ 30 & \color{brown}{23} & 12 & 17 & 22 & 7\\ \color{brown}{5} & 13 & 26 & \color{brown}{19} & 16 & 32 \\ 9 & 20 & 15 & 14 & 25 & 28\\ 36 & 6 & 4 & 3 & 27 & 35 \end{bmatrix} = \begin{bmatrix}۲ & ۳۱ & ۳۳ & ۳۴ & ۱۰ & ۱ \\ ۲۹ & ۱۸ & ۲۱ & ۲۴ & ۱۱ & ۸ \\ ۳۰ & ۲۳ & ۱۲ & ۱۷ & ۲۲ & ۷ \\ ۵ & ۱۳ & ۲۶ & ۱۹ & ۱۶ & ۳۲ \\ ۹ & ۲۰ & ۱۵ & ۱۴ & ۲۵ & ۲۸ \\ ۳۶ & ۶ & ۴ & ۳ & ۲۷ & ۳۵\end{bmatrix}$$
A 13/14-th century metal magic square with Hindu-Arabic numerals (14 by 14 cm, 1.5 cm thick), currently in Shaanxi History Museum (陜西歴史博物館). It was unearthed by Xian train station construction work in 1957, near the site of the Residence of There were three of them: Manggala (r. 1272 - 1278), the third son of Kublai Khan; Ananda (r. 1278 - 1307), the eldest son of Manggala; Yerutömör (r. 1307 - 1332), the son of Ananda.Prince Anxi of Yuan Dynasty (安西王府遺址). Note that the loop in the B glyph used in this magic square is consistently not closed. This metal artifact shows that people still use the mirrored-B glyph to denote five in 14th century Yuan China.
$$\begin{bmatrix}
28 & 4 & 3 & 31 & 35 & 10\\
36 & 18 & 21 & 24 & 11 & 1\\
7 & 23 & 12 & 17 & 22 & 30 \\
8 & 13 & 26 & 19 & 16 & 29 \\
5 & 20 & 15 & 14 & 25 & 32\\
27 & 33 & 34 & 6 & 2 & 9
\end{bmatrix} = \begin{bmatrix}
۲۸ & ۴ & ۳ & ۳۱ & ۳۵ & ۱۰
\\۳۶ & ۱۸ & ۲۱ & ۲۴ & ۱۱ & ۱
\\۷ & ۲۳ & ۱۲ & ۱۷ & ۲۲ & ۳۰
\\ ۸ & ۱۳ & ۲۶ & ۱۹ & ۱۶ & ۲۹
\\ ۵ & ۲۰ & ۱۵ & ۱۴ & ۲۵ & ۳۲
\\ ۲۷ & ۳۳ & ۳۴ & ۶ & ۲ & ۹
\end{bmatrix}$$
Persian five in Malay orthography
Here, we show four examples in which the non-standard mirrored-B/inverted heart-shaped five was employed.
This is a relatively recent artifact in Pulau Bintan, on which the year of death of Tun Abbas (Bendahara Seri Maharaja of Johor Empire) was inscribed. Note that the Persian inverted heart-shaped (♡) five is used instead the modern Mashriki five.
Tuhfat al-Nafis, Terengganu Manuscript, p. 191: . . . yang disuruh Yang Dipertuan Muda itu lalu diikat dan didapatnya tangan. Kemudian lalu disuruh bunuh dikerat kepalanya dikirimkan Lingga kepada Baginda Sultan Mahmud di Lingga. Maka habislah kisahnya Lebai Tamat itu . . . al-hikayat kata orang . . . ceriteranya pada hijrah sanah 1225, pada lima belas hari bulan Zulhijah pada hari Khamis pukul dua belas maka Baginda Sultan Mahmud pun memanggil paduka adinda bagindanya Yang Dipertuan Muda Jaafar . . . Note that the ‘5' in 1225 is orthographically rendered as B, instead of the Mashriki single-loop five (٥). Given the fact the manuscript was paginated with the single-loop five, it is likely that the script did not appreciate that the numeral he copied was a five and the glyph in the parent manuscript was merely faithfully replicated.
The seal of Sultan Abd al-Rahman of Lingga (see A. T. Gallop 2019, Seal #910, p. 316). The seal reads: al-Wathiq bi Rabb al-‘Arsh al-Wadud al-Sultan Abd al-Rahman bin al-Sultan Mahmud | Sanah 1225 min Hijrah al-Nabi salla Allah 'alayhi wa-salam seribu dua ratus dua puluh lima tahun yawm al-Jumaat. Note that the mirrored-B five in this seal is rendered like ‘৪' (‘4' in Bengali script, U+09EA) or our modern 8 with two loops, rather than the traditional Persian mirrored-B.
Pengasas Kuala Lumpur bukan Sutan Puasa,bukan Haji Abdullah Hukum,apatah lg Yap Ah Loy.Pengasas sebenar adlh Raja Abdullah ibni Raja Jaafar.Beliaulah yang dianugerahkn Sungai Kelang dan seluruh jajahannya hingga ke Sg Lumpur oleh Sultan Muhammad.Buktinya adlh 2 pucuk warkah ini. pic.twitter.com/riYjaoQc0l
Sultan Abd al-Samad renewed/re-assertained Raja Abdullah's position as the Governor of Klang Valley on 18 August 1864 (the calendrical data in the enfeoffment letter says Thursday 1 pm, 16 Rabi al-Awal 1281). The year in the seal reads: 1275 (11 August 1858 < 1275 < 30 July 1859). Note that the number ‘5' in 1275 is written as a mirrored B (ઇ). When Raja Sulaiman (the father of Raja Mahdi) died, the land around the Klang River was earlier assigned to Raja Abdullah by Sultan Abd al-Samad's predecessor in 1266 AH (17 November 1849 < 1266 < 5 November 1850), partly because Sultan Muhammad was indebted to Raja Jummat (Raja Jumaat was the elder brother of Raja Abdullah, he was also the brother-in-law of Raja Abd al-Samad). Note that the ‘4' in Sultan Muhammad's seal is written as ۴ (instead of our usual Mashriki ‘٤'). The year 1241 is the first regnal year of Sultan Muhammad (16 August 1825 < 1241 < 4 August 1826). Sultan Muhammad's letter shows that the appointment of Raja Abdullah took place in 1849/1850 and not 1853 (the year traditionally assigned as the year of death of Raja Sulaiman). Sultan Abd Samad's letter shows that the succession kerfuffle was quieted down only in 1858/1859 although Sultan Muhammad vacated his seat in 1857.
1957/0118652W In the Court of the Senior Magistrate at Kuala Lumpur Civil Suit No. 138 of 1898. In the matter of the Estate of Yap Ah Loy, deceased, between Yap Hon Chin (28) Yap Loong Shin (23) Yap Leong Soon (18) Yap Kim Neo Yap Leong Sem by his next friend Ong Chi Siew Yap Leong Fong Plaintiffs and Kok Kang Keow (48), otherwise called Kok Ngeo Nga who is sued as Administratrix of the Estate and Effects of Yap Ah Loy, deceased. On the tombstone of Yap Ah Loy, we were given the following list: 1 隆興 (b. 29 December 1869, d. 5 January 1933), 2 隆盛 (Loong Shin, b. 4 April 1875, d. <1925?), 3 隆顺 (Leong Soon, b. 6 March 1880, d. 8 December 1907, died when he was only 27.8), 4 隆發 (b. 10 August 1882, d. 21 September 1900, died when he was only 18...
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