Vedic vs Chinese vs Japanese Math: Same Problems, Three Systems

No single system is "more powerful" across the board. Vedic shortcuts win on special numbers, such as factors near 100 or squares ending in 5. Trained soroban and anzan users are the fastest at general mental arithmetic. The school algorithms and the Chinese Nine Chapters methods are the most general, and the Chinese elimination method reaches furthest into algebra.

That is the short answer. The long answer needs evidence, so this post puts the same eight problems through four systems: the 16 sutras of Vedic Mathematics, Chinese counting rods (算筹 suànchóu) with the suanpan (算盘) and the methods of the Nine Chapters (九章算術 Jiuzhang Suanshu), the Japanese soroban (算盤) with its mental form anzan (暗算), and the standard school algorithm as the baseline. For each problem we count the operations, the intermediate results you have to keep, and the carries, and we note which built-in check catches a slip.

A note on numbers. Every answer, every intermediate value and every running total in this post was computed with Python, and the script is in the walkthrough near the end so you can rerun it. Step counts for the four multiplications come from the script's simulation; the other step counts were made by hand and are marked as approximate. Dates and attributions come from MacTutor at the University of St Andrews, the Bodleian Library, the scholarly response to the Bodleian dating, and the peer-reviewed studies listed in the sources. Where sources disagree, we say so.

Key takeaways

• Origins first. The 16 sutras of "Vedic Mathematics" come from a book by Bharati Krishna Tirtha, published in 1965, five years after his death. Historians, including S. G. Dani and Kim Plofker, note that these sutras are not found in the Vedic texts. India's genuine classical mathematics (the Śulba Sūtras, Āryabhaṭa, Brahmagupta, Bhāskara II, the Kerala school) is a separate and much older history.
• Special cases: for 98 x 97, the Vedic Nikhilam method needs 4 operations and no carries. The school algorithm needs 11 operations and 6 carries. For 85², the Vedic method needs 2 operations.
• General cases: for 386 x 524, every method needs the same 9 single-digit products. Vedic crosswise multiplication combines them with 8 additions, the school method with 17, and the abacus with 15 bead additions but nothing held in your head.
• Algebra: the Nine Chapters solved linear systems by column elimination roughly two thousand years ago. Seki Takakazu studied determinants in 1683, about ten years before Leibniz. The Vedic book offers a cross multiplication rule for two unknowns.
• Evidence: a three-year randomised controlled trial with 204 children in India found that mental abacus students outperformed controls on arithmetic. The same trial found no change in basic cognitive abilities. We found no randomised trial of Vedic Mathematics in an international peer-reviewed journal.
• Practice: use all three. Learn digit sums and casting out nines as your check, the abacus for fluency, the Vedic special cases for the numbers that suit them, and elimination for anything with unknowns.

The four systems in one paragraph each

Vedic Mathematics: a 1965 book, not a Vedic text

Vedic Mathematics was written by Jagadguru Swami Sri Bharati Krishna Tirtha, the Shankaracharya of the Govardhana Matha at Puri, who died in 1960. The book was published posthumously in 1965. It sets out 16 sutras (short aphorisms such as Nikhilam Navataścaramam Daśataḥ, "all from 9 and the last from 10") and 13 sub-sutras, and applies them to arithmetic and elementary algebra. The author said the sutras came from an appendix to the Atharvaveda. Historians have looked and do not find them there. S. G. Dani, a mathematician at the Tata Institute of Fundamental Research, set out the case in "'Vedic Mathematics': Myth and Reality" (Frontline and Economic and Political Weekly, 1993). Kim Plofker's Mathematics in India (Princeton University Press, 2009) treats the system as a modern creation. Even the book's general editor, V. S. Agrawala, noted in his foreword that the claim of a Vedic origin was not supported by the known texts. None of this makes the methods wrong. Many of them are clever and correct, and some are genuinely faster for the cases they fit. It only means they are twentieth-century methods, and calling them ancient is not supported by any primary source.

India's real early mathematics, from the Śulba Sūtras (शुल्बसूत्र) and their √2 ≈ 577/408 to Āryabhaṭa, Brahmagupta and the Kerala school, is impressive on its own terms and gets its own section below.

Chinese mathematics: rods, the board and the Nine Chapters

Chinese arithmetic for most of its history was done with counting rods (算筹) laid out on a board. Numbers were written in place value, with alternate vertical and horizontal rod forms for neighbouring places so that digits did not run together, and an empty space for zero. Liu Hui's commentary on the Nine Chapters, written in 263 CE, uses red rods for positive quantities and black rods for negative ones. MacTutor summarises the dating debate over the Nine Chapters itself: the material is usually placed shortly after 200 BCE, with some scholars preferring dates between 100 BCE and 50 CE. Its eighth chapter, fangcheng (方程), solves systems of linear equations by arranging the coefficients in columns and reducing them, which MacTutor describes as the same procedure as Gaussian elimination, done with column operations. The suanpan (算盘), the Chinese bead abacus, later replaced rods for daily work. The common form has 2 upper beads worth 5 each and 5 lower beads worth 1 each on every rod.

Japanese mathematics: soroban, anzan and wasan

The soroban (算盤) came to Japan from China and was simplified. The modern form has 1 upper bead worth 5 and 4 lower beads worth 1, so each rod holds exactly one digit from 0 to 9 and nothing more. Anzan (暗算) is calculation on an imagined soroban: trained users "move" beads in a mental image. Japan's League of Abacus Associations (日本珠算連盟) still runs graded examinations, dan grades and a national anzan competition. Alongside the abacus, Edo-period Japan developed its own advanced mathematics, wasan (和算). Its best known figure, Seki Takakazu (c. 1642 to 1708), studied determinants in 1683, about ten years before Leibniz used them independently, according to MacTutor. He also found the Bernoulli numbers before Jacob Bernoulli. Wasan declined after the Meiji government built its school system on Western mathematics in the 1870s.

The baseline: the standard school algorithms

The school algorithms are the column methods most readers learned: long multiplication with one partial product row per digit of the multiplier, long division with "bring down the next digit", the long-division square root or prime factorisation, and elimination or substitution for equations. Their great virtue is that they work the same way for any numbers. Their weakness is that they do not notice when a number is special, so they spend the same effort on 98 x 97 as on 73 x 46.

How we count steps, memory and carries

These are our counting rules.

• Core operations are the single-digit facts you use: each product such as 8 x 7, each addition or subtraction of a column, and each digit added onto an abacus rod. Placing a two-digit product onto two rods counts as two rod additions.
• Intermediate results kept are the numbers you must hold, on paper, on the board or in your head, before the final answer appears. On an abacus the beads hold the running total, so the count of numbers held in your head is close to zero. In anzan the image of the beads holds it, which is exactly the load the research studies.
• Carries are the times a column, or a rod, overflows past 9 and has to pass something left. A two-digit product written straight onto two abacus rods is not a carry. A product such as 56 in the school method, where you write the 6 and remember the 5, is a carry.
• Checks are the error checks the method gives you for free, or that its tradition teaches alongside it.

The four multiplications were counted by the simulation in the walkthrough script; the rest were counted by hand, with every step's arithmetic checked in Python.

Problem 1: 98 x 97 = 9,506

Vedic: Nikhilam with base 100

Both numbers are just below 100, which is the case Nikhilam was built for. Write each number with its deficit from the base.

98   deficit 2
97   deficit 3
--------------
Left part:  98 − 3 = 95   (or 97 − 2, the same)
Right part: 2 × 3 = 06    (two digits, because the base has two zeros)
Answer:     95 | 06 = 9,506

That is 4 operations: two deficits, one cross subtraction and one product. There are two intermediate results (95 and 06) and no carries. It works because (100 − 2)(100 − 3) = 100 × (100 − 2 − 3) + 2 × 3.

Vedic: Urdhva-Tiryagbhyam, vertically and crosswise

The general Vedic multiplication works on any pair of numbers. For two-digit numbers, you take three column values from left to right.

   9 8
 × 9 7
Left column:    9 × 9           = 81
Cross column:   9 × 7 + 8 × 9   = 63 + 72 = 135
Right column:   8 × 7           = 56
Combine from the right:
  56  → write 6, carry 5
  135 + 5 = 140 → write 0, carry 14
  81 + 14 = 95  → write 95
Answer: 9,506

That is 4 products and 3 additions, 7 operations, with 3 carries, one of them a two-digit carry (14). The method is general, but on numbers like these it is slower than Nikhilam.

Chinese: counting rods (Sunzi's layout) and the suanpan

The Sunzi Suanjing (孫子算經, usually dated between the 3rd and 5th centuries CE) describes multiplication with three rows on the board: the multiplier on top, the multiplicand at the bottom and the product building up in the middle. You multiply the leading digit of the top number by each digit of the bottom number, add each product into the middle row at the right place, then remove the used top digit and shift the bottom row one place right. A suanpan user follows the same order on bead rods.

Step   product    added at    running total on the board
1      9 × 9 = 81  hundreds    8,100
2      9 × 7 = 63  tens        8,730
3      8 × 9 = 72  tens        9,450
4      8 × 7 = 56  units       9,506

That is 4 products and 8 rod additions, with 2 bead carries. The board holds every running total, so you keep at most one two-digit product in your head at a time.

Japanese: soroban and anzan

The soroban adds the same products onto the rods from the left, with the same running totals. The difference is in the bead moves. A soroban rod holds only 0 to 9, so every overflow is a carry to the next rod, made with a complement: to add 7 to a rod showing 4, you subtract 3 (the complement of 7 in 10) and add 1 to the rod on the left. With practice these moves become one motion. In anzan the operator does the same moves on an imagined soroban.

Baseline: the school algorithm

      9 8
    × 9 7
    -----
      6 8 6     (98 × 7: 8 × 7 = 56, write 6 carry 5; 9 × 7 + 5 = 68)
    8 8 2 0     (98 × 9: 8 × 9 = 72, write 2 carry 7; 9 × 9 + 7 = 88; shifted)
    -------
    9 5 0 6

That is 4 products and 7 additions, 11 operations, with 6 carries and 2 partial product rows.

Verdict for problem 1: Nikhilam wins clearly: 4 operations and no carries, against 11 operations and 6 carries for the school method. The abacus methods are slower in moves but leave nothing for your memory except the current product.

Problem 2: 47 x 53 = 2,491

Vedic: Nikhilam with a working base of 50

These numbers sit either side of 50, so Nikhilam uses 50 as a "working base", written as 100 ÷ 2.

47   deficit −3   (3 below 50)
53   excess  +3   (3 above 50)
Left part:   47 + 3 = 50, then × 50 ÷ 100, i.e. 50 ÷ 2 = 25  → 2,500
Right part:  (−3) × (+3) = −9
Answer:      2,500 − 9 = 2,491

That is 6 operations with one borrow. Many readers will recognise the shortcut underneath: 47 x 53 = (50 − 3)(50 + 3) = 50² − 3² = 2,500 − 9. That identity is ordinary algebra, and you can use it whichever system you learned.

Vedic: Urdhva-Tiryagbhyam

Left:   4 × 5         = 20
Cross:  4 × 3 + 7 × 5 = 12 + 35 = 47
Right:  7 × 3         = 21
Combine: 21 → write 1, carry 2; 47 + 2 = 49 → write 9, carry 4; 20 + 4 = 24
Answer: 2,491

That is 7 operations and 3 carries.

Chinese counting board and Japanese soroban

Step   product      running total
1      4 × 5 = 20   2,000
2      4 × 3 = 12   2,120
3      7 × 5 = 35   2,470
4      7 × 3 = 21   2,491

That is 4 products and 7 rod additions. No rod ever overflows, so there are no bead carries. This is the easiest of the four multiplications on an abacus.

Baseline

      4 7
    × 5 3
    -----
      1 4 1     (47 × 3)
    2 3 5 0     (47 × 5, shifted)
    -------
    2 4 9 1

That is 4 products and 5 additions, 9 operations, with 4 carries.

Verdict for problem 2: the difference of squares, which the Vedic working base expresses, is fastest if you notice it. If you do not notice it, the abacus is the cleanest: no carries at all.

Problem 3: 386 x 524 = 202,264

This is the fair test, because none of the special cases apply. Neither number is near a power of 10, they are not symmetric about a round number, and nothing ends in 5.

Vedic: Urdhva-Tiryagbhyam for three digits

For 3 by 3 digits there are five columns. The middle column has three cross products.

Digits: 3 8 6 and 5 2 4
Column 1:  3 × 5                     = 15
Column 2:  3 × 2 + 8 × 5             = 6 + 40 = 46
Column 3:  3 × 4 + 8 × 2 + 6 × 5     = 12 + 16 + 30 = 58
Column 4:  8 × 4 + 6 × 2             = 32 + 12 = 44
Column 5:  6 × 4                     = 24
Combine from the right:
  24       → write 4, carry 2
  44 + 2   = 46 → write 6, carry 4
  58 + 4   = 62 → write 2, carry 6
  46 + 6   = 52 → write 2, carry 5
  15 + 5   = 20 → write 20
Answer: 202,264

That is 9 products and 8 additions, 17 operations, with 5 carries. The answer appears as one line, which is why the method suits mental work. The cost is that the middle column asks you to hold a running sum of three products while you compute them.

Chinese counting board and Japanese soroban

Step   product      running total
1      3 × 5 = 15   150,000
2      3 × 2 = 6    156,000
3      3 × 4 = 12   157,200
4      8 × 5 = 40   197,200
5      8 × 2 = 16   198,800
6      8 × 4 = 32   199,120
7      6 × 5 = 30   202,120
8      6 × 2 = 12   202,240
9      6 × 4 = 24   202,264

That is 9 products and 15 rod additions, with 3 bead carries. Nothing is held in memory except the product you have just recalled.

Baseline

          3 8 6
        × 5 2 4
        -------
        1 5 4 4     (386 × 4)
        7 7 2 0     (386 × 2, shifted)
    1 9 3 0 0 0     (386 × 5, shifted twice)
    -----------
    2 0 2 2 6 4

That is 9 products and 17 additions, 26 operations, with 11 carries and 3 partial product rows.

Verdict for problem 3: every system needs the same 9 single-digit products. Urdhva needs the fewest additions but the most working memory. The abacus needs more moves but almost no memory. The school method needs the most additions and carries, and leaves a written record you can recheck line by line.

Problem 4: 85² = 7,225

Vedic: Ekadhikena Purvena, "by one more than the one before"

For a number ending in 5, multiply the leading part by one more than itself, then write 25.

85²: leading part 8, one more is 9
8 × 9 = 72
Append 25
Answer: 7,225

That is 2 operations and no carries. It works because (10a + 5)² = 100a(a + 1) + 25. The rule is exact for every number ending in 5: 115² = 11 × 12 then 25, which is 13,225.

Chinese and Japanese

On the board or the soroban, 85² is just 85 x 85: the products 64, 40, 40 and 25 give running totals of 6,400, 6,800, 7,200 and 7,225. That is 4 products and 6 rod additions, with 1 bead carry. Experienced soroban users learn the "ends in 5" pattern as a fact anyway, because it comes up so often in practice sheets.

Baseline

The school method gives 425 + 6,800 = 7,225: 4 products, 6 additions and 5 carries, 10 operations in all.

Verdict for problem 4: the Vedic rule is five times shorter (2 operations against 10) and is worth learning by anyone. It is also only a special case. It does nothing for 86².

Problem 5: √7,744 = 88

Vedic: inspection (for perfect squares)

If you already know 7,744 is a perfect square, the answer can be read off.

Last digit 4 → the root ends in 2 or 8 (2² = 4, 8² = 64)
First pair 77 lies between 64 (8²) and 81 (9²) → the tens digit is 8
Choose between 82 and 88: 8 × 9 = 72, and 77 is above 72 → take the larger, 88
Check: 88² = 7,744

The test in the third line works because 85² = 7,225, so a square whose first pair is above 72 must have a root above 85. That is 3 operations. It only works for perfect squares.

Vedic: the duplex (dvandva) method

The general Vedic square root works digit by digit, using the "duplex" of the root digits found so far.

Group: 77 | 44
First digit: largest square not above 77 is 64 (8²) → root digit 8, remainder 13
Divisor: 2 × 8 = 16
Bring down 4 → 134; 134 ÷ 16 = 8, remainder 134 − 128 = 6 → root digit 8
Bring down 4 → 64; subtract the duplex of 8 (8² = 64) → 0
Remainder 0, so √7,744 = 88 exactly

That is 4 main operations.

Chinese: kaifang, the Nine Chapters root extraction

The Nine Chapters extracts square roots on the counting board, place by place. The 13th-century Chinese generalisation of this procedure to higher-degree equations is what Western books later called Horner's method. MacTutor notes that Qin Jiushao's treatise of 1247 solves equations by "the standard Chinese method", today called the Ruffini-Horner method.

Shi (the number):      7,744
First root digit:      8 tens → subtract 80² = 6,400 → remainder 1,344
Double the root found: 2 × 80 = 160 (the "fixed divisor")
Trial next digit d:    (160 + d) × d should not exceed 1,344
d = 8:                 168 × 8 = 1,344 → remainder 0
Root: 88

That is about 5 operations, and it is the same procedure as the school long-division square root.

Japanese: soroban

Soroban square roots use the same place-by-place logic on the rods: the first root digit 8 removes 64, and the second digit is found by trial against the remainder 1,344. About 5 operations, as on the Chinese board.

Baseline

Prime factorisation is the common school route for small perfect squares: 7,744 = 2⁶ × 11², so √7,744 = 2³ × 11 = 88. The long-division square root is the Chinese procedure above, written with paper instead of rods.

Verdict for problem 5: for a known perfect square, Vedic inspection is fastest. For an unknown number, all four systems use what is in effect the same digit-by-digit algorithm, and the Chinese text has the oldest written version of it among the three traditions.

Problem 6: 7,429 ÷ 37 = 200 remainder 29

Vedic: Dhvajanka, the flag method

Paravartya Yojayet, the Vedic division sutra people usually quote, suits divisors just above a power of 10, such as 12 or 112. For 37, the Vedic book uses straight division, called Dhvajanka or the "flag" method. Split the divisor into a main digit (3) and a flag (7). The flag has one digit, so the last digit of the dividend is set aside for the remainder.

Dividend: 7 4 2 | 9         Divisor: 3 with flag 7
Digit 7:  7 ÷ 3 = 2 remainder 1        → quotient digit 2
Digit 4:  gross 14, minus flag × 2 = 14 − 14 = 0; 0 ÷ 3 = 0 rem 0  → quotient digit 0
Digit 2:  gross 2, minus flag × 0 = 2; 2 ÷ 3 = 0 remainder 2       → quotient digit 0
Remainder column: gross 29, minus flag × 0 = 29                    → remainder 29
Answer: 200 remainder 29

That is about 7 operations: three divisions by a single digit, three flag subtractions and the final remainder. Every division is by 3 instead of 37, which is the point of the method. If a gross value goes negative, you step the previous quotient digit down by one and continue.

A side note that the Vedic book makes much of: 37 x 27 = 999, so 1/37 = 0.027027... repeating. That is a pretty fact. It is not a faster route to this remainder.

Chinese: the counting board

The Nine Chapters lays division out with the quotient (商 shang) on top, the dividend (實 shi) in the middle and the divisor (法 fa) underneath. The divisor slides left or right under the dividend as the quotient grows.

Divisor 37 under the first two digits 74 → 74 ÷ 37 = 2 → quotient 2 in the hundreds place
Subtract 2 × 37 = 74 from 74 → the dividend becomes 29 (00 2 9)
Shift the divisor right: 37 does not go into 2 → quotient digit 0
Shift again: 37 does not go into 29 → quotient digit 0
Result: quotient 200, remainder 29, written as 200 and 29/37

The Nine Chapters writes the result as a whole number with a fraction, never as a decimal. That is about 5 operations, because the board lets you subtract 2 x 37 directly without writing it down.

Japanese: soroban (shōjohō)

Modern soroban division, shōjohō (商除法), sets the dividend on the rods and places each quotient digit to its left. You subtract the products of the quotient digit and the divisor digits directly from the dividend rods: 2 × 3 = 6 from the thousands rod and 2 × 7 = 14 from the next two rods, leaving 29. The next two trials give 0, and 29 stays on the rods as the remainder. That is about 5 operations. An older method, kijohō (帰除法), used memorised division rhymes like the Chinese 九归 jiugui.

Baseline

       2 0 0
     -------
 3 7 ) 7 4 2 9
       7 4
       ---
         0 2
           0
         ---
           2 9
             0
         -----
           2 9   remainder

That is about 8 operations: three trial quotients, the product 2 × 37 = 74 (two single-digit products) and two subtractions.

Verdict for problem 6: on this problem the systems are close. The flag method trades a two-digit divisor for a one-digit one at the price of a correction step. The board and soroban avoid writing the product. The school layout is the easiest to read back later.

Problem 7: 1/19 = 0.052631578947368421 (repeating)

The decimal for 1/19 repeats every 18 digits: 0.052631578947368421, then the same 18 digits again. This is where the Vedic book is at its most elegant.

Vedic: Ekadhikena Purvena for denominators ending in 9

For 1/19, the digit before the 9 is 1, and "one more than the one before" is 2. So instead of dividing by 19 you divide by 2, carrying each remainder as a prefix to the next digit.

Start: 1 ÷ 2 = 0 remainder 1        → digit 0, next number 10
10 ÷ 2 = 5 remainder 0              → digit 5, next number 05
 5 ÷ 2 = 2 remainder 1              → digit 2, next number 12
12 ÷ 2 = 6 remainder 0              → digit 6, next number 06
 6 ÷ 2 = 3 remainder 0              → digit 3, next number 03
 3 ÷ 2 = 1 remainder 1              → digit 1, next number 11
11 ÷ 2 = 5 remainder 1              → digit 5, next number 15
15 ÷ 2 = 7 remainder 1              → digit 7 ...
Continuing gives: 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1, then it repeats

The rule works because dividing by 19 is the same as dividing 0.1 by 1.9, and the method handles the 1.9 by treating it as 2 with the error fed back through the prefixed remainders. There is a second built-in shortcut: after 9 digits, the rest are the nines complements of the first 9. The first half 052631578 plus the second half 947368421 equals 999,999,999. So you can stop after 9 halvings and write the rest by subtracting each digit from 9.

Doubling from the right (1, 2, 4, 8, then 16: write 6, carry 1, and so on) gives the same 18 digits.

That is 18 halvings, or 9 halvings plus 9 subtractions from 9, each on a number no bigger than 19.

Chinese: counting board

In the Nine Chapters style the answer would simply stay the fraction 1/19. For a decimal, the board runs long division: shift the remainder one place, find the next digit by trial, subtract.

10 ÷ 19 → 0, remainder 10
100 ÷ 19 → 5, subtract 95, remainder 5
50 ÷ 19 → 2, subtract 38, remainder 12
120 ÷ 19 → 6, subtract 114, remainder 6
60 ÷ 19 → 3, subtract 57, remainder 3
30 ÷ 19 → 1, subtract 19, remainder 11
... (18 digits in all before the remainder returns to 1)

That is 18 trial quotients, each with a two-digit subtraction: about 54 operations.

Japanese: soroban and anzan

On the soroban the same long division runs with shōjohō, subtracting q × 1 and q × 9 for each quotient digit q: about 54 fast bead moves. In anzan, 18 digits is a long chain to hold.

Baseline

The school long division is the Chinese procedure on paper: 18 trial divisions by 19, each with a multiplication and a subtraction, about 54 operations.

Verdict for problem 7: the Vedic method wins outright. Dividing by 2 instead of 19 cuts each step from a trial division plus a two-digit product to a single halving, and the nines-complement property halves the work again. This is the strongest single case for learning the Vedic sutras, and it is also a narrow one: it applies to denominators ending in 9 (with extensions in the book for some other endings).

Problem 8: the system x + y + z = 6, 2y + 5z = −4, 2x + 5y − z = 27

The solution is x = 5, y = 3, z = −2. Check: 5 + 3 − 2 = 6; 2 × 3 + 5 × (−2) = −4; 2 × 5 + 5 × 3 − (−2) = 27.

Chinese: fangcheng (方程), column elimination

The Nine Chapters writes each equation as a column of rods, right to left, with the coefficients going down and the total at the bottom. Negative numbers are black rods and positive numbers red, following Liu Hui's commentary.

             eq3   eq2   eq1
x             2     0     1
y             5     2     1
z            −1     5     1
total        27    −4     6

The text's method, often summarised as 遍乘直除 (bian cheng zhi chu, "multiply throughout, then subtract directly"), removes one coefficient at a time by multiplying a whole column and subtracting another column from it. Fractions never appear until the very end.

Step 1  Remove x from eq3: eq3 − 2 × eq1
        x: 2 − 2 = 0   y: 5 − 2 = 3   z: −1 − 2 = −3   total: 27 − 12 = 15
Step 2  Remove y from eq3: 2 × (new eq3) − 3 × eq2
        y: 6 − 6 = 0   z: −6 − 15 = −21   total: 30 − (−12) = 42
        so −21 z = 42, z = −2
Step 3  Back up: 2y + 5 × (−2) = −4 → 2y = 6 → y = 3
Step 4  Back up: x + 3 + (−2) = 6 → x = 5

That is 2 column operations, each of 4 multiply-and-subtract entries, then 2 back substitutions: about 20 operations. The negative intermediate values (−3, −21, −12) are why the Chinese text needed rules for adding and subtracting signed quantities, which the fangcheng chapter gives. The classic first problem of that chapter, three grades of grain, has the system 3x + 2y + z = 39, 2x + 3y + z = 34, x + 2y + 3z = 26, and the answer 37/4, 17/4 and 11/4 of a dou.

Vedic: elimination, then Paravartya for two unknowns

The Vedic book's chapters on simultaneous equations centre on two unknowns, with a cross multiplication rule under the Paravartya sutra. For a₁y + b₁z = c₁ and a₂y + b₂z = c₂, it gives y = (c₁b₂ − c₂b₁) ÷ (a₁b₂ − a₂b₁) and z = (a₁c₂ − a₂c₁) ÷ (a₁b₂ − a₂b₁). That is Cramer's rule for a 2 by 2 system. For three unknowns, reduce first.

Eliminate x: (2x + 5y − z) − 2 × (x + y + z) → 3y − 3z = 15 → y − z = 5
Now:  1y − 1z = 5   and   2y + 5z = −4
Denominator: 1 × 5 − 2 × (−1) = 7
y = (5 × 5 − (−4) × (−1)) ÷ 7 = (25 − 4) ÷ 7 = 3
z = (1 × (−4) − 2 × 5) ÷ 7 = −14 ÷ 7 = −2
x = 6 − 3 + 2 = 5

That is about 15 operations. The book also has special-case sutras for systems with symmetric coefficients, such as Sankalana-vyavakalanabhyam ("by addition and by subtraction") when the coefficients of x and y swap between the two equations. Our system is not one of those cases.

Japanese: Seki's determinants, with the soroban for arithmetic

Seki Takakazu's 1683 work, Kaifukudai no Hō (解伏題之法), treated elimination through what we now call determinants. In modern notation, the 3 by 3 determinant of the coefficients decides the system, and Cramer's rule gives each unknown.

D  = det [[1,1,1],[0,2,5],[2,5,−1]]   = −21
Dx = det [[6,1,1],[−4,2,5],[27,5,−1]] = −105   → x = −105 ÷ −21 = 5
Dy = det [[1,6,1],[0,−4,5],[2,27,−1]] = −63    → y = −63 ÷ −21 = 3
Dz = det [[1,1,6],[0,2,−4],[2,5,27]]  = 42     → z = 42 ÷ −21 = −2

Four 3 by 3 determinants take far more arithmetic than elimination for one system: roughly 50 operations once each determinant is expanded. The point of the determinant is not speed on one problem. It is that it gives a single formula for when a system has a unique solution (D not equal to 0) and what that solution is. The soroban is the natural tool for the multiplications and additions inside each determinant, and wasan scholars used it for exactly this kind of work.

Baseline

School elimination is the fangcheng procedure written in rows instead of columns: R3 − 2R1 gives 3y − 3z = 15, then combining with 2y + 5z = −4 gives z = −2, then back substitution. About 20 operations.

Verdict for problem 8: the Chinese method, written down two thousand years ago, is the same algorithm modern linear algebra teaches and computers run, and it scales to any number of unknowns. Seki's determinants are the deeper theory. The Vedic rule is a convenient formula for two unknowns.

The metrics table: every problem, every method

Operation counts follow the rules above. "Kept" means intermediate numbers held before the answer appears; for the abacus, the beads hold the running total and the "kept" figure is what sits in your head. Training time and generality are in the second table.

And the per-method view, for the questions that do not depend on a single problem:

Three patterns stand out once the table is laid out.

First, the shortcut methods win by a factor of about 1.5 to 5 exactly where they apply, and they apply to a minority of numbers. A randomly chosen pair of two-digit numbers is rarely near 100, rarely symmetric about a round number and rarely ends in 5.

Second, on general multiplication every system does the same digit products. The real differences are in the additions, the carries and where the intermediate results live. That is a difference of memory load, not of mathematical power.

Third, the abacus methods have the highest move counts and the lowest memory load, which is why they get very fast with training: the moves become motor habits.

Error checks compared: casting out nines and its cousins

A method is only as good as your ability to catch your own slips. The traditions differ a lot here.

Digit sums and casting out nines. Reduce each number to its digit sum, repeated until one digit remains, which is the remainder on division by 9 (with 9 standing in for 0). The digit sum of a product equals the digit sum of the product of the digit sums. The Vedic book uses this check throughout, and it predates the book by a long way in Indian, Arabic and European arithmetic.

98 x 97:     8 x 7 = 56 → 11 → 2      9,506 → 20 → 2      agrees
47 x 53:     2 x 8 = 16 → 7           2,491 → 16 → 7      agrees
386 x 524:   8 x 2 = 16 → 7           202,264 → 16 → 7    agrees
85²:         4 x 4 = 16 → 7           7,225 → 16 → 7      agrees
88²:         7 x 7 = 49 → 13 → 4      7,744 → 22 → 4      agrees
7,429 ÷ 37:  1 x 2 + 2 = 4            7,429 → 22 → 4      agrees

Casting out nines has a known blind spot: it cannot see two swapped digits (writing 9,056 for 9,506 has the same digit sum) or an error that is a multiple of 9.

The elevens check. The alternating sum of digits, taken from the right, gives the remainder on division by 11, and it does catch most swapped digits. For 386 x 524: 386 leaves 1 and 524 leaves 7, so the product should leave 7, and 202,264 does leave 7. Using nines and elevens together catches far more errors than either alone.

Reverse operations on the board. The Chinese and Japanese traditions lean on doing the inverse: multiply the quotient back by the divisor and add the remainder (200 × 37 + 29 = 7,429), or divide the product by one factor. On an abacus this is fast because the board already holds the number.

Substitution for systems. Every tradition checks a solved system the same way: put the values back into each original equation. The fangcheng procedure is also self-checking in a structural sense, because each column operation must zero exactly one coefficient.

The 1/19 check. The nines complement property is a check as well as a shortcut. If your first 9 digits plus your next 9 do not sum to 999,999,999, one of them is wrong.

Beyond arithmetic: where each tradition went further

Each tradition reached real mathematics that the arithmetic tricks do not show.

India: zero, place value and infinite series

Indian mathematicians developed decimal place value with a written zero, and Brahmagupta's Brāhmasphuṭasiddhānta (628 CE) gave rules for calculating with zero and with negative numbers. The Bakhshali manuscript uses a dot for zero. In 2017 the Bodleian Library announced radiocarbon dates placing its oldest folios in the 3rd or 4th century CE, with other folios centuries later. Kim Plofker, Agathe Keller, Takao Hayashi, Clemency Montelle and Dominik Wujastyk published a response the same year in History of Science in South Asia questioning how those dates should be read, because the manuscript's leaves gave widely different results and the text itself is usually dated later. Oxford's own later radiocarbon analysis places the manuscript between about 800 and 1100 CE, so treat the early date as disputed. From around 1350 to 1425, Madhava of Sangamagrama in Kerala found the series that Europe later credited to Gregory, Leibniz and Newton, including the arctangent series and π ÷ 4 = 1 − 1/3 + 1/5 − 1/7 and so on. That is calculus-level mathematics, more than two hundred years before Europe.

China: linear systems, remainders and root extraction

Fangcheng is Gaussian elimination. The Sunzi Suanjing poses the remainder problem whose smallest answer is 23 (a number that leaves 2 when divided by 3, 3 when divided by 5 and 2 when divided by 7). Qin Jiushao's Shushu Jiuzhang (数书九章) of 1247 gives the general dayan (大衍) method for such problems, and solves polynomial equations by the method later named after Horner. Zu Chongzhi (429 to 501) gave 355/113 for π, correct to 6 decimal places.

Japan: determinants, Bernoulli numbers and temple geometry

Seki's determinants (1683) and his Bernoulli numbers are the headline results of wasan. The sangaku (算額), wooden tablets of geometry problems hung in shrines and temples during the Edo period and collected by Fukagawa Hidetoshi and Tony Rothman in Sacred Mathematics (Princeton University Press, 2008), show a broad public culture of problem solving. Wasan was a living research tradition until the Meiji reforms of the 1870s moved schools to Western mathematics.

What reached modern mathematics and computing

A short word on that last row. Urdhva-Tiryagbhyam's columns are exactly the coefficients of the product of two polynomials: (3t² + 8t + 6)(5t² + 2t + 4) has coefficients 15, 46, 58, 44, 24, and setting t = 10 gives 202,264. That is a genuine insight into why the method works, and it connects to how computers multiply very large numbers. It does not make the method ancient.

Which is more powerful? An honest verdict

The question has five different answers depending on what "powerful" means.

Fastest for mental arithmetic, with training: soroban and anzan

The strongest evidence of any system here is for the mental abacus. Hatano, Miyake and Binks published "Performance of expert abacus operators" in Cognition in 1977, and Stigler studied "the effect of abacus training on Chinese children's mental calculation" in Cognitive Psychology in 1984. Frank and Barner (Journal of Experimental Psychology: General, 2012) studied mental abacus users in India and found that the mental image seems to be stored column by column, each column as a separate unit, and that practised users' calculations were relatively insensitive to verbal interference, which fits the idea that the representation is visual and not verbal. The largest study is the randomised controlled trial by Barner, Alvarez, Sullivan, Brooks, Srinivasan and Frank (Child Development, 2016): over 3 years, 204 children aged 5 to 7 at the start, in ordinary classrooms. Mental abacus students outperformed controls on arithmetic tasks.

The limits matter as much as the result. The same trial found that the training did not change basic cognitive abilities such as working memory. Instead, children's spatial working memory at the start of the study predicted how well they learned. So the evidence supports "mental abacus training can make children better at arithmetic in a classroom setting", and does not support claims about general intelligence or broad cognitive benefits. It is one trial in one country, and the outcomes are arithmetic tests, not long-term mathematics achievement.

Best shortcuts for special cases: Vedic sutras

For numbers near a power of 10, squares ending in 5, fractions with denominators ending in 9 and known perfect squares, the Vedic methods take between about a third and two thirds fewer operations in our counts, and up to five times fewer for 85², and they need no tool. The evidence base is thin. We searched for randomised trials of Vedic Mathematics instruction in international peer-reviewed journals and found none. The studies we did find were small, local and often had no control group, so their results cannot be attributed to the method. The fair claim is that the special-case methods are faster when they apply, which our counts show directly, and that there is no good evidence yet for wider learning benefits.

Most general: the school algorithms and the Nine Chapters methods

Long multiplication, long division, the digit-by-digit square root and elimination work on any numbers and any size of system with no recognition step. The Chinese methods are the oldest written versions of several of them (root extraction, elimination, signed numbers). Generality is what makes them the foundation of algebra and of computer arithmetic.

Furthest into algebra: Chinese elimination, then Japanese determinants

If "powerful" means "solves the hardest problems", fangcheng and Seki's determinants go furthest of the methods in this post. The Vedic book reaches two unknowns with its cross multiplication rule and a set of special cases.

Best for teaching children: it depends on the goal

For arithmetic fluency, the abacus has the best evidence, with the limits above. For place value and algebra, the school and Chinese elimination methods connect most directly to what comes next. The Vedic special cases teach children to look at a problem before they start. None of the studies we found supports claims about IQ or brain development from any of the three systems, and we make none.

The overall answer

If you must pick one word for each: soroban is the fastest engine, Vedic is the best bag of tricks, and the Chinese and school methods are the most general. The strongest practitioner uses all three: notice the special case, compute fast with a low memory load when there is none, and check with digit sums.

A combined practice plan: 12 weeks, 20 minutes a day

This plan borrows the best of each system. It assumes an adult or older child who already knows the multiplication table.

The most useful habit in the whole plan is week 12: look first, then choose. Special cases call for Vedic methods, large plain products for the soroban or the crosswise method, and anything with unknowns for elimination.

Walkthrough: reproduce every number in Python

Every figure in this post comes from the script below. Save it as traditions.py and run python3 traditions.py. It uses only the standard library.

from fractions import Fraction

def digit_sum(n):
    n = abs(n)
    while n > 9:
        n = sum(int(d) for d in str(n))
    return n

def school(a, b):
    """Long multiplication: products, additions, carries, rows."""
    A = [int(d) for d in str(a)][::-1]
    B = [int(d) for d in str(b)][::-1]
    prods = adds = carries = 0
    rows = []
    for j, bd in enumerate(B):
        c, row = 0, []
        for ad in A:
            prods += 1
            if c:
                adds += 1
            t = ad * bd + c
            row.append(t % 10)
            c = t // 10
            if c:
                carries += 1
        if c:
            row.append(c)
        rows.append(int("".join(map(str, row[::-1]))) * 10 ** j)
    c = 0
    for col in range(len(str(sum(rows)))):
        digs = [(r // 10 ** col) % 10 for r in rows if r >= 10 ** col]
        n = len(digs) + (1 if c else 0)
        if n > 1:
            adds += n - 1
        s = sum(digs) + c
        c = s // 10
        if c:
            carries += 1
    return prods, adds, carries, rows

def urdhva(a, b):
    """Vertically and crosswise: column sums, then one combining pass."""
    A = [int(d) for d in str(a)]
    B = [int(d) for d in str(b)]
    cols, prods, adds = [], 0, 0
    for k in range(len(A) + len(B) - 1):
        terms = [A[i] * B[k - i] for i in range(len(A)) if 0 <= k - i < len(B)]
        prods += len(terms)
        adds += len(terms) - 1
        cols.append(sum(terms))
    c = carries = 0
    for v in cols[::-1]:
        if c:
            adds += 1
        c = (v + c) // 10
        if c:
            carries += 1
    return cols, prods, adds, carries

def board(a, b):
    """Counting board or abacus: add each product onto rods, left to right."""
    A, B = str(a), str(b)
    rods = [0] * (len(A) + len(B) + 1)
    prods = rod_adds = carries = 0
    totals = []
    for i, x in enumerate(A):
        for j, y in enumerate(B):
            p = int(x) * int(y)
            prods += 1
            pos = (len(A) - 1 - i) + (len(B) - 1 - j)
            for k, dig in enumerate((p % 10, p // 10)):
                if dig:
                    q = pos + k
                    rods[q] += dig
                    rod_adds += 1
                    while rods[q] >= 10:
                        rods[q] -= 10
                        rods[q + 1] += 1
                        carries += 1
                        q += 1
            totals.append(sum(d * 10 ** k for k, d in enumerate(rods)))
    return prods, rod_adds, carries, totals

for a, b in [(98, 97), (47, 53), (386, 524), (85, 85)]:
    print(a, "x", b, "=", a * b)
    print("  school", school(a, b))
    print("  urdhva", urdhva(a, b))
    print("  board ", board(a, b))
    print("  nines check", digit_sum(digit_sum(a) * digit_sum(b)), digit_sum(a * b))

print("sqrt 7744 =", 88, 88 * 88 == 7744, 7744 == 2**6 * 11**2)
print("7429 / 37 =", divmod(7429, 37))

# 1/19 by the Ekadhikena halving method
digits, x = [], 1
for _ in range(18):
    q, r = divmod(x, 2)
    digits.append(q)
    x = r * 10 + q
print("1/19 = 0." + "".join(map(str, digits)))
print("halves sum", int("052631578") + int("947368421"))

# The 3 by 3 system by Cramer's rule
def det3(m):
    return (m[0][0] * (m[1][1] * m[2][2] - m[1][2] * m[2][1])
            - m[0][1] * (m[1][0] * m[2][2] - m[1][2] * m[2][0])
            + m[0][2] * (m[1][0] * m[2][1] - m[1][1] * m[2][0]))

A = [[1, 1, 1], [0, 2, 5], [2, 5, -1]]
b = [6, -4, 27]
D = det3(A)
for k, name in enumerate("xyz"):
    Ak = [[b[i] if j == k else A[i][j] for j in range(3)] for i in range(3)]
    print(name, "=", Fraction(det3(Ak), D), "(D =", D, ", D" + name, "=", det3(Ak), ")")

The output reproduces the multiplication counts in the metrics table, the running totals, the 1/19 digits and the determinants.

Common mistakes

• Calling the 16 sutras ancient or Vedic in origin. They come from a book published in 1965. Say "the methods in the book Vedic Mathematics", and give India's genuine classical mathematics its own credit.
• Using Nikhilam far from the base. For 61 x 58 with base 100, the deficits are 39 and 42, and their product (1,638) is harder than the original problem. Pick a nearer working base (here 60) or use a general method.
• Forgetting the digit count on the right side of Nikhilam. With base 100 the right part has two digits: 2 × 3 is written 06, not 6. With base 1000 it has three: 998 x 997 = 995 | 006 = 995,006.
• Treating casting out nines as proof. It misses swapped digits and errors that are multiples of 9. Add the elevens check, or multiply back.
• Holding too much in your head with Urdhva. On 3 by 3 digits the middle column has three products to sum. Write the column sums down until the method is automatic.
• Learning anzan before the physical soroban is fluent. The mental image is a copy of physical moves. Without the moves, there is nothing to imagine.
• Losing signs in elimination. Fangcheng needed explicit rules for positive and negative quantities for a reason. Write signs on every intermediate value, as the red and black rods did.
• Expecting a special-case method to generalise. The squares-ending-in-5 rule does not help with 86², and the 1/19 rule does not help with 1/17.

FAQ

Is Vedic Maths really from the Vedas?

No primary source shows that. The 16 sutras and 13 sub-sutras come from Bharati Krishna Tirtha's book Vedic Mathematics, published in 1965. Historians, including S. G. Dani and Kim Plofker, note that the sutras do not appear in the Vedic texts. The methods can still be useful; they are twentieth-century methods.

Which is faster, Vedic maths or abacus?

For special numbers, such as factors near 100 or squares ending in 5, Vedic shortcuts take fewer steps. For general arithmetic, trained soroban and anzan users are faster, and the mental abacus has the stronger evidence, including a three-year randomised controlled trial.

Is the Chinese abacus different from the Japanese abacus?

Yes. The common suanpan has 2 beads above the bar and 5 below, so a rod can show up to 15. The modern soroban has 1 above and 4 below, so each rod shows one digit from 0 to 9.

What is anzan?

Anzan (暗算) is mental calculation using an imagined soroban. Research on practised users suggests the image is stored column by column and is relatively unaffected by verbal interference, which fits a visual rather than verbal process.

Did the Chinese really invent Gaussian elimination?

The Nine Chapters, dated by most scholars to between about 200 BCE and 50 CE, solves linear systems by column elimination: the same procedure as Gaussian elimination, roughly two thousand years before Gauss.

Did Seki Takakazu discover determinants before Leibniz?

According to MacTutor, Seki studied determinants in 1683, and Leibniz independently used them about ten years later. MacTutor describes Seki's version as the more general.

Which system is best for kids?

For arithmetic fluency, mental abacus training has the best evidence, from a randomised classroom trial that also found no change in basic cognitive abilities. Vedic special cases help number sense.

Can Vedic maths solve equations?

The book gives a cross multiplication rule for two linear equations in two unknowns, which is Cramer's rule, and special cases for symmetric coefficients. For three or more unknowns you eliminate first, as in the Chinese and school methods.

How long does it take to learn soroban?

Basic bead work takes months of regular practice, and strong anzan usually takes years. No rigorous published standard exists; graded examinations are the practical measure.

What is casting out nines?

A digit-sum check: the digit sum of a product must match the digit sum of the product of the factors' digit sums. It misses swapped digits and errors that are multiples of 9.

Deep dives

Each tradition has its own long post in this series:

• Indian and Vedic Mathematics Explained: the 16 Sutras, Śulba Sūtras, Zero, Āryabhaṭa, Brahmagupta and the Kerala School
• Chinese Mathematics Explained: Counting Rods, the Nine Chapters, Fangcheng, the Remainder Theorem and Horner before Horner
• Japanese Mathematics Explained: the Soroban, Anzan, Wasan, Seki Takakazu and Sangaku Temple Geometry

Checksums, remainders and checkable steps: an honest bridge

Safeguard does not do mathematics, and nothing in this post was computed by a Safeguard model. The parallel to our work is in the habits these traditions teach. Casting out nines is a checksum: a small summary that catches a corrupted result without redoing the work, the same idea as the hashes that confirm a downloaded package is the one that was published. The Chinese remainder theorem is used today to speed up RSA decryption. A counting board shows every intermediate step, and a software supply chain is safer when every step from source to build to release is recorded and can be checked again.

If you build anything around this material, from a practice app for soroban drills to a classroom quiz site or a Python notebook, it runs on open source dependencies like any other software. Those dependencies need the same checking and timely fixes as the code in a production service.

Fix it without waiting for upstream

Whatever the vulnerability, most of the risk sits in two gaps. The first is the time between a fix being published and that fix actually running in your production builds. The second, and worse, is the time before any fix exists at all, while a maintainer works out a patch and a release schedule. Waiting through both gaps is a habit, not a requirement. Safeguard gives teams three ways to close them.

1. Autonomous remediation: pull requests, not tickets

When a fixed version exists, Safeguard's Griffin models open a tested, policy gated pull request that moves the affected dependency to the safe version. You can run it on one repository or in bulk across every project that carries the vulnerable version. Remediation covers eight categories, including dependencies, licenses, security posture and findings. Reachability analysis ranks the vulnerable code paths your application actually calls first, so the first pull requests you merge are the ones that matter most. Autonomous remediation ships switched off, with every category set to Manual, so your team decides how much runs on its own and what still needs a human approval.

2. Gold Open Source: the safe version in seconds, free

gold.safeguard.sh is free and needs no login. Search any CVE or package to see affected and fixed versions, CVSS, EPSS, CISA KEV status and a recommended safe version, or use the "Am I affected?" checker. The same data is available through a free read API, README status badges and RSS threat feeds. The safeguard-sh/gold-scan-action GitHub Action can fail a build on a known exploited vulnerability, on any known vulnerability, or never, so a version you have just removed cannot quietly come back in next week's merge.

3. Gold Registry: start from components that are already fixed

registry.safeguard.sh provides hardened, drop in replacements for upstream packages and container images. Every Gold artifact is held continuously to zero exploitable critical or high CVEs, zero malware and a completed license check, with SLSA L3 or higher provenance, a Sigstore signature and a public evidence page you can hand to an auditor. An artifact loses its Gold badge if it regresses. Teams can point their own package directory at the registry so hardened equivalents resolve without changing how developers install anything.

When upstream has no fix yet, or will not ship one, Safeguard first works to get the fix accepted upstream. If the issue is severe or upstream will not move, Safeguard builds a fork, fixes the vulnerability there and publishes a hardened Gold build, so you are not stuck on a maintainer's release schedule. You can also request a hardened, compatibility tested build of a specific package. This fork and patch flow is in early access today, so ask which packages it covers before you plan around it.

One honest limit: when the vulnerable code lives in a vendor appliance, a hosted service or an operating system runtime you do not build yourself, the fix still has to come from that vendor or distribution. The steps above apply to the open source dependencies and container images you build and ship, which is where most teams carry the largest and least tracked share of their exposure.

See how Safeguard would help

None of these demos involve any arithmetic; they show the software side described in the bridge above.

The first demo is a real Safeguard CLI session. It installs the CLI, opens the interactive terminal and asks it to fix a lodash CVE in a repository. It reads the finding and the advisory, asks before any command that changes something, edits package.json to lodash 4.17.21, runs npm install and npm test with approval, prints a summary and pushes nothing. The same flow applies to a web app for abacus or mental maths drills: scan the repository, then let the assistant propose the dependency upgrade under your approval.

Click through it at your own pace, or open it full screen.

The second demo sets up the Safeguard runner on your own machine, approves it with its device code and then scans from the terminal. That suits teachers and researchers who keep notebooks and scripts like the walkthrough above on their own laptops and want the scan to run where the code lives.

Click through it at your own pace, or open it full screen.

The third demo connects Claude to the Safeguard MCP server with OAuth and lets you choose which tools Claude can use. If you already work with Claude while building an education app, it brings dependency findings into the same conversation, limited to the tools you allow.

Click through it at your own pace, or open it full screen.

To try it on your own repositories, download Safeguard and follow the CLI documentation.

Quick check

Sources

• MacTutor History of Mathematics, An overview of Indian mathematics
• MacTutor History of Mathematics, Nine chapters on the mathematical art
• MacTutor History of Mathematics, Chinese numerals
• MacTutor History of Mathematics, Takakazu Seki
• MacTutor History of Mathematics, Madhava of Sangamagrama
• MacTutor History of Mathematics, Qin Jiushao
• MacTutor History of Mathematics, Zu Chongzhi
• MacTutor History of Mathematics, Brahmagupta
• University of Oxford GLAM, Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol zero (2017)
• K. Plofker, A. Keller, T. Hayashi, C. Montelle and D. Wujastyk, The Bakhshālī Manuscript: A Response to the Bodleian Library's Radiocarbon Dating, History of Science in South Asia 5 (2017)
• D. Barner, G. Alvarez, J. Sullivan, N. Brooks, M. Srinivasan and M. C. Frank, Learning Mathematics in a Visuospatial Format: A Randomized, Controlled Trial of Mental Abacus Instruction, Child Development 87(4), 2016
• M. C. Frank and D. Barner, Representing exact number visually using mental abacus, Journal of Experimental Psychology: General 141(1), 2012
• J. W. Stigler, "Mental abacus": The effect of abacus training on Chinese children's mental calculation, Cognitive Psychology 16(2), 1984
• G. Hatano, Y. Miyake and M. G. Binks, Performance of expert abacus operators, Cognition 5(1), 1977
• League of Japan Abacus Associations (日本珠算連盟)
• RFC 8017, PKCS #1: RSA Cryptography Specifications Version 2.2

Further reading, cited in the text but not fetched for this post: Bharati Krishna Tirtha, Vedic Mathematics (Motilal Banarsidass, 1965); S. G. Dani, "'Vedic Mathematics': Myth and Reality", Economic and Political Weekly 28(31), 1993; Kim Plofker, Mathematics in India (Princeton University Press, 2009); Kangshen Shen, John N. Crossley and Anthony W.-C. Lun, The Nine Chapters on the Mathematical Art: Companion and Commentary (Oxford University Press, 1999); Joseph Needham, Science and Civilisation in China, volume 3 (Cambridge University Press, 1959); Fukagawa Hidetoshi and Tony Rothman, Sacred Mathematics: Japanese Temple Geometry (Princeton University Press, 2008).

Last reviewed: 25 October 2026.