Lattices, Dust Boards, and Galleys

J. B. ThooYuba College

2012 CMC3-South Conference, Orange, CA

References• Jean-Luc Chabert (editor) et al. A History of Algorithms: From the Pebble to the

Microchip. Springer-Verlag, Berlin, 1999. Translator of the English Edition: Chris Weeks.

• Sir Thomas Heath. A History of Greek Mathematics Volume II: From Aristarchus to Diophantus. Dover Publications, Inc., New York, 1981.

• Annette Imhausen. Egyptian mathematics. In Victor J. Katz, editor, The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press, Princeton, 2007.

• Victor J. Katz. A History of Mathematics: An Introduction. Pearson Education, Inc., Boston, 3rd edition, 2009.

• Leonardo of Pisa (Fibonacci). Fibonacci’s Liber Abaci: A Translation into Modern English of Leonardo Pisano’s Book of Calculation. Springer, New York, 2002. Translated L. E. Sigler.

Multiplication

• Also called grating method, gelosia, and multiplication using a tableau, or grid, or net, or the jalousie.

• India: Ganesa’s commentary (16th c.) on Lilavati by Bhaskara II (12th c.).

• Arabic: arithmetic book Talkhis by Ibn al-Banna (13th c.).

• China: Jiuzhang suanfa by Wu Jing (1450).

• Europe: earliest known in a Latin ms. ca. 1300 in England. Also later in Renaissance period arithmetic books like the Treviso (anonymous; 1478) and the Summa by Pacioli (1494).

Lattice

34 × 25

3 4

2

5

Draw a lattice

34 × 25

3 4

2

5

6

5 0

8

21

Draw a lattice

Find the partial products

34 × 25

3 4

2

5

6

5 0

8

21

8

5 0

1Draw a lattice

Find the partial products

Add down that diagonals

You Try

Try 135 × 12 found in the 16th century Indian astronomer Ganesa’s commentary on the 12th century Indian book Lilavati by Bhaskara II.

1 3 5

1

2

You Try

Try 135 × 12 found in the 16th century Indian astronomer Ganesa’s commentary on the 12th century Indian book Lilavati by Bhaskara II.

1 3 5

1

2

1 3 5

2 6 01

6 2 01

Chabert et al., A History of Algorithms, p. 24.

Chabert et al., A History of Algorithms, p. 26.

Chabert et al., A History of Algorithms, p. 26.

Egyptian

•Hieratic •Hieroglyphic

Rhind Papyrus

• Base 10.

• Not positional.

• Additive: each symbol repeated as many times as needed, with ten of one symbol replaced by one of next higher value.

• No more than four of the same symbol grouped together, and when more than four were needed the larger group would be written to the left of above the smaller group.

1.3. EGYPTIAN AND ROMAN NUMBER SYSTEMS 43

one | two ||three ||| four ||||five ||| || six |||

|||

seven ||||||| eight ||||

||||

nine ||| |||||| ten 2

twelve ||2 twenty-one |22twenty-seven ||||

||| 22 sixty 222222

one hundred 3 one hundred one |3eight hundred fifty 222

2233333333 nine hundred ninety-nine ||| ||

||||222 222222

333 333333

And four thousand, three hundred seventy-nine is||| ||||||2222222 3334444. !

Now You Try 1.12. Write the decimal number and the word name for

the following Egyptian hieroglyphic numerals.

(1) ||||22333(2) |||2246777!

"

Now You Try 1.13. Write the following numbers using Egyptian hi-

eroglyphic numerals.

(1) Eighty-three.

(2) Two hundred nineteen.

(3) Six thousand, seven hundred nine.

(4) Three million, four hundred seventy-two thousand, six hundred thir-

teen.

"

According to Cajori, a multiplicative principle also came into use around

1600–1200 bc. He gives the following two examples.

one hundred and twenty thousand 4 223 120" 1000

two million, eight hundred thousand6

||||||||

!! 28" 100,000

In the first example, we see that a nonstandard (to the left) placement of the

glyph 4 for thousand implies multiplication by one thousand. In the second

Four thousand, three hundred seventy-nine

1.3. EGYPTIAN AND ROMAN NUMBER SYSTEMS 41

A’hmosè Papyrus (or Ahmes Papyrus) after the Egyptian scribe who copied

it about 1650 bc. A’hmosè attributed the material to some document from

the Middle Kingdom about two hundred years prior. The papyrus begins

“Correct method of reckoning, for grasping the meaning of things and know-

ing everything that is, obscurities and all secrets,” which is often used as

its title. The Ahmes contains 87 mathematics problems, mostly of a practi-

cal nature, but some, for instance Problems 26 and 27, belong to the small

group of “aha” problems (“aha” in Egyptian refers to the unknown quantity)

that are purely mathematical. (See examples 4.5 and 4.6.) Only part of the

Ahmes has been found. The part bought by Rhind has been at the British

Museum since 1865, while another section was discovered in the Egyptian

collection of the New York Historical Society in 1922 and is currently at the

Brooklyn Museum of Art [40].

Another of the more prominent extant Egyptian texts is the Moscow

Mathematical Papyrus which contains 25 problems. It can be found at the

Moscow Museum of Fine Arts. We will see examples from the Ahmes and

the Moscow papyri later on.

1.3.1.3. Number system. The Egyptian number system is a base 10 sys-

tem, but Cajori [30, p. 11] tells us that “traces of other systems, based on the

scales of 5, 12, 20, and 60, are believed to have been discovered.” Here are

the Egyptian hieroglyphs used for the powers of 10 from one to ten million.

| 2 3 4(sta! or stroke) (heel bone or loaf) (coiled rope or snake) (lotus flower)

one ten hundred thousand

5 6(bent or pointed finger) (burbot or tadpole)

ten thousand hundred thousand

7 !(astonished man) (rising sun)

million ten million

1.3. EGYPTIAN AND ROMAN NUMBER SYSTEMS 43

one | two ||three ||| four ||||five ||| || six |||

|||

seven ||||||| eight ||||

||||

nine ||| |||||| ten 2

twelve ||2 twenty-one |22twenty-seven ||||

||| 22 sixty 222222

one hundred 3 one hundred one |3eight hundred fifty 222

2233333333 nine hundred ninety-nine ||| ||

||||222 222222

333 333333

And four thousand, three hundred seventy-nine is||| ||||||2222222 3334444. !

Now You Try 1.12. Write the decimal number and the word name for

the following Egyptian hieroglyphic numerals.

(1) ||||22333(2) |||2246777!

"

Now You Try 1.13. Write the following numbers using Egyptian hi-

eroglyphic numerals.

(1) Eighty-three.

(2) Two hundred nineteen.

(3) Six thousand, seven hundred nine.

(4) Three million, four hundred seventy-two thousand, six hundred thir-

teen.

"

According to Cajori, a multiplicative principle also came into use around

1600–1200 bc. He gives the following two examples.

one hundred and twenty thousand 4 223 120" 1000

two million, eight hundred thousand6

||||||||

!! 28" 100,000

In the first example, we see that a nonstandard (to the left) placement of the

glyph 4 for thousand implies multiplication by one thousand. In the second

1.3. EGYPTIAN AND ROMAN NUMBER SYSTEMS 43

one | two ||three ||| four ||||five ||| || six |||

|||

seven ||||||| eight ||||

||||

nine ||| |||||| ten 2

twelve ||2 twenty-one |22twenty-seven ||||

||| 22 sixty 222222

one hundred 3 one hundred one |3eight hundred fifty 222

2233333333 nine hundred ninety-nine ||| ||

||||222 222222

333 333333

And four thousand, three hundred seventy-nine is||| ||||||2222222 3334444. !

Now You Try 1.12. Write the decimal number and the word name for

the following Egyptian hieroglyphic numerals.

(1) ||||22333(2) |||2246777!

"

Now You Try 1.13. Write the following numbers using Egyptian hi-

eroglyphic numerals.

(1) Eighty-three.

(2) Two hundred nineteen.

(3) Six thousand, seven hundred nine.

(4) Three million, four hundred seventy-two thousand, six hundred thir-

teen.

"

According to Cajori, a multiplicative principle also came into use around

1600–1200 bc. He gives the following two examples.

one hundred and twenty thousand 4 223 120" 1000

two million, eight hundred thousand6

||||||||

!! 28" 100,000

In the first example, we see that a nonstandard (to the left) placement of the

glyph 4 for thousand implies multiplication by one thousand. In the second

1.3. EGYPTIAN AND ROMAN NUMBER SYSTEMS 43

one | two ||three ||| four ||||five ||| || six |||

|||

seven ||||||| eight ||||

||||

nine ||| |||||| ten 2

twelve ||2 twenty-one |22twenty-seven ||||

||| 22 sixty 222222

one hundred 3 one hundred one |3eight hundred fifty 222

2233333333 nine hundred ninety-nine ||| ||

||||222 222222

333 333333

And four thousand, three hundred seventy-nine is||| ||||||2222222 3334444. !

Now You Try 1.12. Write the decimal number and the word name for

the following Egyptian hieroglyphic numerals.

(1) ||||22333(2) |||2246777!

"

Now You Try 1.13. Write the following numbers using Egyptian hi-

eroglyphic numerals.

(1) Eighty-three.

(2) Two hundred nineteen.

(3) Six thousand, seven hundred nine.

(4) Three million, four hundred seventy-two thousand, six hundred thir-

teen.

"

According to Cajori, a multiplicative principle also came into use around

1600–1200 bc. He gives the following two examples.

one hundred and twenty thousand 4 223 120" 1000

two million, eight hundred thousand6

||||||||

!! 28" 100,000

In the first example, we see that a nonstandard (to the left) placement of the

glyph 4 for thousand implies multiplication by one thousand. In the second

1.3. EGYPTIAN AND ROMAN NUMBER SYSTEMS 43

one | two ||three ||| four ||||five ||| || six |||

|||

seven ||||||| eight ||||

||||

nine ||| |||||| ten 2

twelve ||2 twenty-one |22twenty-seven ||||

||| 22 sixty 222222

one hundred 3 one hundred one |3eight hundred fifty 222

2233333333 nine hundred ninety-nine ||| ||

||||222 222222

333 333333

And four thousand, three hundred seventy-nine is||| ||||||2222222 3334444. !

Now You Try 1.12. Write the decimal number and the word name for

the following Egyptian hieroglyphic numerals.

(1) ||||22333(2) |||2246777!

"

Now You Try 1.13. Write the following numbers using Egyptian hi-

eroglyphic numerals.

(1) Eighty-three.

(2) Two hundred nineteen.

(3) Six thousand, seven hundred nine.

(4) Three million, four hundred seventy-two thousand, six hundred thir-

teen.

"

According to Cajori, a multiplicative principle also came into use around

1600–1200 bc. He gives the following two examples.

one hundred and twenty thousand 4 223 120" 1000

two million, eight hundred thousand6

||||||||

!! 28" 100,000

In the first example, we see that a nonstandard (to the left) placement of the

glyph 4 for thousand implies multiplication by one thousand. In the second

Three hundred twenty-four

Thirteen million, one hundred one thousand, twenty-three

1.3. EGYPTIAN AND ROMAN NUMBER SYSTEMS 41

A’hmosè Papyrus (or Ahmes Papyrus) after the Egyptian scribe who copied

it about 1650 bc. A’hmosè attributed the material to some document from

the Middle Kingdom about two hundred years prior. The papyrus begins

“Correct method of reckoning, for grasping the meaning of things and know-

ing everything that is, obscurities and all secrets,” which is often used as

its title. The Ahmes contains 87 mathematics problems, mostly of a practi-

cal nature, but some, for instance Problems 26 and 27, belong to the small

group of “aha” problems (“aha” in Egyptian refers to the unknown quantity)

that are purely mathematical. (See examples 4.5 and 4.6.) Only part of the

Ahmes has been found. The part bought by Rhind has been at the British

Museum since 1865, while another section was discovered in the Egyptian

collection of the New York Historical Society in 1922 and is currently at the

Brooklyn Museum of Art [40].

Another of the more prominent extant Egyptian texts is the Moscow

Mathematical Papyrus which contains 25 problems. It can be found at the

Moscow Museum of Fine Arts. We will see examples from the Ahmes and

the Moscow papyri later on.

1.3.1.3. Number system. The Egyptian number system is a base 10 sys-

tem, but Cajori [30, p. 11] tells us that “traces of other systems, based on the

scales of 5, 12, 20, and 60, are believed to have been discovered.” Here are

the Egyptian hieroglyphs used for the powers of 10 from one to ten million.

| 2 3 4(sta! or stroke) (heel bone or loaf) (coiled rope or snake) (lotus flower)

one ten hundred thousand

5 6(bent or pointed finger) (burbot or tadpole)

ten thousand hundred thousand

7 !(astonished man) (rising sun)

million ten million

\ .

2

4

\ 8

\ 16

34 × 25

\ . 34

2 68

4 136

\ 8 272

\ 16 544

34 × 25

\ . 34

2 68

4 136

\ 8 272

\ 16 544

34 × 25

34 × 25

= 34 + 272 + 544

= 850

You Try

135 × 12

. 135

2 270

\ 4 540

\ 8 1080

You Try

135 × 12

= 540 + 1080

= 1620

Chabert et al., A History of Algorithms, p. 16.

Chabert et al., A History of Algorithms, p. 17.

Chabert et al., A History of Algorithms, p. 17.

⁹⁄₁₀⁹⁄₁₀

1⁴⁄₅

3³⁄₅

7¹⁄₅

Exercise

Show that every counting number can be expressed as a sum of powers of 2.

• Kushyar ibn Labban (fl. ca. 1000).

• Indian mathematician.

• In Principles of Hindu Recokning shows one method of multiplication.

• The method was most likely carried out on a “dust board” or some other easily erasable surface.

Ibn Labban

• Positional.

• Base 10.

• Uses ten symbols of numeration, corresponding to the base (ten) of the system.

• Place-holder symbol that is also a number (zero).

• Separatrix (decimal point) that separates the integer of a numeral from the fraction part.

The evolution of the Indo-Arabic numerals from India to Europe. Image from Karl Menninger, Zahlwort und Ziffer, Vandenhoeck & Reprect, Gottingen (1958), Vol. II p. 233. <http://books.google.com/books?id=W01gQYIrG24C&pg=PA47>

Pierre Simon de Laplace (1749–1827)

It is from the Indians that there has come to us the ingenious method of expressing all numbers, in ten characters, by giving them, at the same time, an absolute and a place value; an idea fine and important, which appears indeed so simple, that for this very reason we do not sufficiently recognize its merit. But this very simplicity, and the extreme facility which this method imparts to all calculation, place our system of arithmetic in the first rank of the useful inventions. How difficult it was to invent such a method one can infer from the fact that it escaped the genius of Archimedes and of Apollonius of Perga, two of the greatest men of antiquity.

34 × 25

3 4

2 5

34 × 25

3 4

2 5

2 × 3 = 66 + 0 = 6

34 × 25

6 3 4

2 5

2 × 3 = 66 + 0 = 6

34 × 25

6 3 4

2 5

5 × 3 = 15

34 × 25

6 4

2 5

5 × 3 = 1515 + 60 = 75

34 × 25

7 5 4

2 5

5 × 3 = 1515 + 60 = 75

34 × 25

7 5 4

2 5

34 × 25

7 5 4

2 5

2 × 4 = 88 + 75 = 83

34 × 25

8 3 4

2 5

2 × 4 = 88 + 75 = 83

34 × 25

8 3 4

2 5

5 × 4 = 20

34 × 25

8 3

2 5

5 × 4 = 2020 + 830 = 850

34 × 25

8 5 0

2 5

5 × 4 = 2020 + 830 = 850

You Try

135 × 12

1 3 5

1 2

You Try

135 × 12

1 1 3 5

1 2

You Try

135 × 12

1 2 3 5

1 2

You Try

135 × 12

1 2 3 5

1 2

You Try

135 × 12

1 5 3 5

1 2

You Try

135 × 12

1 5 6 5

1 2

You Try

135 × 12

1 5 6 5

1 2

You Try

135 × 12

1 6 1 5

1 2

You Try

135 × 12

1 6 2 0

1 2

Division

\ . 34

2 68

4 136

\ 8 272

\ 16 544

Egyptian

34 × 25

= 34 + 272 + 544

= 850

135

270

540 \

1080 \

1620 ÷ 135

. 135

2 270

4 540 \

8 1080 \

1620 ÷ 135

. 135

2 270

4 540 \

8 1080 \

1620 ÷ 135

1620 ÷ 135

= 4 + 8

= 12

• Better known today as Fibonacci (only since 19th cent.).

• Well known in history of banking, e.g., credited with introducing “present value.”

• Liber Abaci (Book of Calculation; 1202, 1228) on mathematics of trade, valuation, and commercial arbitrage, and one of the first to introduce the Indo-Arabic number system to Europeans and to demonstrate the number system’s practical and commercial use with many examples.

• In math perhaps best known for the “Fibonacci sequence”: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,.…

Leonardo of Pisaca. 1170–1240

Wikipedia: en.wikipedia.org/wiki/Fibonacci

Liber Abaci(1) Here Begins the First Chapter

(2) On the Multiplication of Whole Numbers

(3) On the Addition of Whole Numbers

(4) On the Subtraction of Lesser Numbers from Greater Numbers

(5) On the Divisions of Integral Numbers

(6) On the Multiplication of Integral Numbers with Fractions

(7) On the Addition and Subtraction and Division Of Numbers with

Fractions and the Reduction of Several Parts to a Single Part

(8) On Finding the Value of Merchandise by the Principal Method

(9) On the Barter of Merchandise and Similar Things

(10) On Companies and Their Members

(11) On the Alloying of Monies

(12) Here Begins Chapter Twelve

(13) On the Method of Elchataym and How with It Nearly All Problems

of Mathematics Are Solved

(14) On Finding Square and Cubic Roots, and on the Multiplication,

Division, and Subtraction of Them, and On the Treatment of Binomials

and Apotomes and their Roots

(15) On Pertinent Geometric Rules And on Problems of Algebra

Wikipedia: en.wikipedia.org/wiki/Liber_Abaci

Liber Abaci(1) Here Begins the First Chapter

(2) On the Multiplication of Whole Numbers

(3) On the Addition of Whole Numbers

(4) On the Subtraction of Lesser Numbers from Greater Numbers

(5) On the Divisions of Integral Numbers

(6) On the Multiplication of Integral Numbers with Fractions

(7) On the Addition and Subtraction and Division Of Numbers with

Fractions and the Reduction of Several Parts to a Single Part

(8) On Finding the Value of Merchandise by the Principal Method

(9) On the Barter of Merchandise and Similar Things

(10) On Companies and Their Members

(11) On the Alloying of Monies

(12) Here Begins Chapter Twelve

(13) On the Method of Elchataym and How with It Nearly All Problems

of Mathematics Are Solved

(14) On Finding Square and Cubic Roots, and on the Multiplication,

Division, and Subtraction of Them, and On the Treatment of Binomials

and Apotomes and their Roots

(15) On Pertinent Geometric Rules And on Problems of Algebra

Wikipedia: en.wikipedia.org/wiki/Liber_Abaci

365 ÷ 2And if one will wish to divide 365 by 2, then ...2 he puts beneath the 5, and he begins by dividing the 3 by the 2, namely the last figure, saying ½ of 3 is 1, and 1 remains; he writes the 1 beneath the 3, and the 1 which remains he writes above, as is displayed in the first illustration; and the remaining 1 couples with the 6 that is next to the last given figure, making 16; he takes ½ of the 16 which is 8; 2 he therefore puts the 8 beneath the 6 put before, the 1 the 3, as is displayed in the second illustration; and as there is no remainder in the division of the 16, one divides the 5 by the 2; the quotient is 2 and the remainder 1; he writes the 2 under the 5, and the 1 which remains he writes over the whole; and before the ½ he writes the quotient coming 2 from the division, namely 182, as one shows in the last illustration.

365 ÷ 2And if one will wish to divide 365 by 2, then ...2 he puts beneath the 5, and he begins by dividing the 3 by the 2, namely the last figure, saying ½ of 3 is 1, and 1 remains; he writes the 1 beneath the 3, and the 1 which remains he writes above, as is displayed in the first illustration; and the remaining 1 couples with the 6 that is next to the last given figure, making 16; he takes ½ of the 16 which is 8; 2 he therefore puts the 8 beneath the 6 put before, the 1 the 3, as is displayed in the second illustration; and as there is no remainder in the division of the 16, one divides the 5 by the 2; the quotient is 2 and the remainder 1; he writes the 2 under the 5, and the 1 which remains he writes over the whole; and before the ½ he writes the quotient coming 2 from the division, namely 182, as one shows in the last illustration.

3 6 52

365 ÷ 2And if one will wish to divide 365 by 2, then ...2 he puts beneath the 5, and he begins by dividing the 3 by the 2, namely the last figure, saying ½ of 3 is 1, and 1 remains; he writes the 1 beneath the 3, and the 1 which remains he writes above, as is displayed in the first illustration; and the remaining 1 couples with the 6 that is next to the last given figure, making 16; he takes ½ of the 16 which is 8; 2 he therefore puts the 8 beneath the 6 put before, the 1 the 3, as is displayed in the second illustration; and as there is no remainder in the division of the 16, one divides the 5 by the 2; the quotient is 2 and the remainder 1; he writes the 2 under the 5, and the 1 which remains he writes over the whole; and before the ½ he writes the quotient coming 2 from the division, namely 182, as one shows in the last illustration.

3 6 52

365 ÷ 2And if one will wish to divide 365 by 2, then ...2 he puts beneath the 5, and he begins by dividing the 3 by the 2, namely the last figure, saying ½ of 3 is 1, and 1 remains; he writes the 1 beneath the 3, and the 1 which remains he writes above, as is displayed in the first illustration; and the remaining 1 couples with the 6 that is next to the last given figure, making 16; he takes ½ of the 16 which is 8; 2 he therefore puts the 8 beneath the 6 put before, the 1 the 3, as is displayed in the second illustration; and as there is no remainder in the division of the 16, one divides the 5 by the 2; the quotient is 2 and the remainder 1; he writes the 2 under the 5, and the 1 which remains he writes over the whole; and before the ½ he writes the quotient coming 2 from the division, namely 182, as one shows in the last illustration.

13 6 5

21

365 ÷ 2And if one will wish to divide 365 by 2, then ...2 he puts beneath the 5, and he begins by dividing the 3 by the 2, namely the last figure, saying ½ of 3 is 1, and 1 remains; he writes the 1 beneath the 3, and the 1 which remains he writes above, as is displayed in the first illustration; and the remaining 1 couples with the 6 that is next to the last given figure, making 16; he takes ½ of the 16 which is 8; 2 he therefore puts the 8 beneath the 6 put before, the 1 the 3, as is displayed in the second illustration; and as there is no remainder in the division of the 16, one divides the 5 by the 2; the quotient is 2 and the remainder 1; he writes the 2 under the 5, and the 1 which remains he writes over the whole; and before the ½ he writes the quotient coming 2 from the division, namely 182, as one shows in the last illustration.

13 6 5

21

365 ÷ 2And if one will wish to divide 365 by 2, then ...2 he puts beneath the 5, and he begins by dividing the 3 by the 2, namely the last figure, saying ½ of 3 is 1, and 1 remains; he writes the 1 beneath the 3, and the 1 which remains he writes above, as is displayed in the first illustration; and the remaining 1 couples with the 6 that is next to the last given figure, making 16; he takes ½ of the 16 which is 8; 2 he therefore puts the 8 beneath the 6 put before, the 1 the 3, as is displayed in the second illustration; and as there is no remainder in the division of the 16, one divides the 5 by the 2; the quotient is 2 and the remainder 1; he writes the 2 under the 5, and the 1 which remains he writes over the whole; and before the ½ he writes the quotient coming 2 from the division, namely 182, as one shows in the last illustration.

1 03 6 5

21 8

365 ÷ 2And if one will wish to divide 365 by 2, then ...2 he puts beneath the 5, and he begins by dividing the 3 by the 2, namely the last figure, saying ½ of 3 is 1, and 1 remains; he writes the 1 beneath the 3, and the 1 which remains he writes above, as is displayed in the first illustration; and the remaining 1 couples with the 6 that is next to the last given figure, making 16; he takes ½ of the 16 which is 8; 2 he therefore puts the 8 beneath the 6 put before, the 1 the 3, as is displayed in the second illustration; and as there is no remainder in the division of the 16, one divides the 5 by the 2; the quotient is 2 and the remainder 1; he writes the 2 under the 5, and the 1 which remains he writes over the whole; and before the ½ he writes the quotient coming 2 from the division, namely 182, as one shows in the last illustration.

1 03 6 5

21 8

365 ÷ 2And if one will wish to divide 365 by 2, then ...2 he puts beneath the 5, and he begins by dividing the 3 by the 2, namely the last figure, saying ½ of 3 is 1, and 1 remains; he writes the 1 beneath the 3, and the 1 which remains he writes above, as is displayed in the first illustration; and the remaining 1 couples with the 6 that is next to the last given figure, making 16; he takes ½ of the 16 which is 8; 2 he therefore puts the 8 beneath the 6 put before, the 1 the 3, as is displayed in the second illustration; and as there is no remainder in the division of the 16, one divides the 5 by the 2; the quotient is 2 and the remainder 1; he writes the 2 under the 5, and the 1 which remains he writes over the whole; and before the ½ he writes the quotient coming 2 from the division, namely 182, as one shows in the last illustration.

1 0 13 6 5

21 8 2

365 ÷ 2And if one will wish to divide 365 by 2, then ...2 he puts beneath the 5, and he begins by dividing the 3 by the 2, namely the last figure, saying ½ of 3 is 1, and 1 remains; he writes the 1 beneath the 3, and the 1 which remains he writes above, as is displayed in the first illustration; and the remaining 1 couples with the 6 that is next to the last given figure, making 16; he takes ½ of the 16 which is 8; 2 he therefore puts the 8 beneath the 6 put before, the 1 the 3, as is displayed in the second illustration; and as there is no remainder in the division of the 16, one divides the 5 by the 2; the quotient is 2 and the remainder 1; he writes the 2 under the 5, and the 1 which remains he writes over the whole; and before the ½ he writes the quotient coming 2 from the division, namely 182, as one shows in the last illustration.

½182

1 0 13 6 5

21 8 2

You Try

12532 ÷ 11

You Try

12532 ÷ 11

1

1 4 0 3

1 2 5 3 2

1 1

1 1 3 9 ³⁄₁₁1139

• Division by a composite number with two digits.

• If the divisor has a factor that is also a factor of the dividend.

• Division by prime numbers with three digits.

• Division by numbers with four or more digits.

Galley

Piano Regolatore Sociale Comune di Genova: www.pianoregolatoresociale.comune.genova.it/portal/page/categoryItem?contentId=498336

• Also called scratch method in England.

• Originated in India.

• Popularized in Europe by Luca Pacioli in Summa (1494). Of the four methods of division Pacioli presented, he considered this method “the swiftest … just as the galley is the swiftest ship.”

• Used in Europe until as late as the 17th century.

Universität Bayreuth Lehrstul für Mathematik und ihre Didaktik:did.mat.uni-bayreuth.de/mmlu/duerer/lu/bio_pacioli.htm

12532 ÷ 11

1

1 4 0 3

11 1 2 5 3 2 1 1 3 9

1 1 1 3 9

1 3 9

12532 ÷ 11

11 1 2 5 3 2

12532 ÷ 11

11 1 2 5 3 2 1

1 1

12532 ÷ 11

1

11 1 2 5 3 2 1

1 1

12532 ÷ 11

1

11 1 2 5 3 2 1 1

1 1 1

1

12532 ÷ 11

1 4

11 1 2 5 3 2 1 1

1 1 1

1

12532 ÷ 11

1 4

11 1 2 5 3 2 1 1 3

1 1 1 3

1 3

12532 ÷ 11

1

1 4 0

11 1 2 5 3 2 1 1 3

1 1 1 3

1 3

12532 ÷ 11

1

1 4 0

11 1 2 5 3 2 1 1 3 9

1 1 1 3 9

1 3 9

12532 ÷ 11

1

1 4 0 3

11 1 2 5 3 2 1 1 3 9

1 1 1 3 9

1 3 9

1139³⁄₁₁

You Try

67892 ÷ 176

You Try

67892 ÷ 176

1

1 0 3

1 5 0 1 2

176 6 7 8 9 2 385

5 2 8 8 0

1 4 0 8

8

385¹³²⁄₁₇₆

Square Roots

• Babylonian (ca. 2000 BC)

• Jiuzhang Suan Shu (Chinese, ca. 150 BC)

• Theon of Alexandria (fl. AD 375)

• Heron of Alexandria (3rd century AD)

• Nicolas Chuquet (d. 1487)

• Bakhshali Manuscript (uncertain, 3rd–12th century)

• Babylonian (ca. 2000 BC)

• Jiuzhang Suan Shu (Chinese, ca. 150 BC)

• Theon of Alexandria (fl. AD 375)

• Heron of Alexandria (3rd century AD)

• Nicolas Chuquet (d. 1487)

• Bakhshali Manuscript (uncertain, 3rd–12th century)

Babylonian√a2 + b ≈ a+ b

2a

Babylonian√a2 + b ≈ a+ b

2a

a

a a2

b/2a

b/2a√N =

√a2 + b =⇒ b = N − a2

a

a a2

b/a

b

Heron

It is in Metrica, in the course of finding the area of a triangle using what we now call Heron’s formula,

that Heron gave a method for approximating the square root of a nonsquare number as well as we please.

area =�

s(s− a)(s− b)(s− c),

In Metrica Heron writes,

“Since 720 has not its side rational, we can obtain its side within a very small difference as follows. Since the next succeeding square number is 729, which has 27 for its side, divide 720 by 27. This gives 26⅔. Add 27 to this, making 53⅔, and take half of this or 26½¹⁄₃. The side of 720 will therefore be very nearly 26½¹⁄₃. In fact, if we multiply 26½¹⁄₃ by itself, the product is 720¹⁄₃₆, so that the difference (in the square) is ¹⁄₃₆.

“If we desire to make the difference still smaller than ¹⁄₃₆, we shall take 720¹⁄₃₆ instead of 729 [or rather we should take 26½¹⁄₃ instead of 27], and by proceeding in the same way we shall find that the resulting difference is much less than ¹⁄₃₆.”

In Metrica Heron writes,

“Since 720 has not its side rational, we can obtain its side within a very small difference as follows. Since the next succeeding square number is 729, which has 27 for its side, divide 720 by 27. This gives 26⅔. Add 27 to this, making 53⅔, and take half of this or 26½¹⁄₃. The side of 720 will therefore be very nearly 26½¹⁄₃. In fact, if we multiply 26½¹⁄₃ by itself, the product is 720¹⁄₃₆, so that the difference (in the square) is ¹⁄₃₆.

“If we desire to make the difference still smaller than ¹⁄₃₆, we shall take 720¹⁄₃₆ instead of 729 [or rather we should take 26½¹⁄₃ instead of 27], and by proceeding in the same way we shall find that the resulting difference is much less than ¹⁄₃₆.”

Heron’s method amounts to the iteration formula

where is the next square number succeeding α0 A.

√A ≈ αn =

1

2

�αn−1 +

A

αn−1

�,

xn = xn−1 −f(xn−1)

f �(xn−1)

Newton’s method

xn = xn−1 −f(xn−1)

f �(xn−1)

xn = xn−1 −x2n−1 −A

2xn−1=

1

2

�xn−1 +

A

xn−1

�

f �(x) = 2xf(x) = x2 −A,

Newton’s method

√AFor let

xn = xn−1 −f(xn−1)

f �(xn−1)

xn = xn−1 −x2n−1 −A

2xn−1=

1

2

�xn−1 +

A

xn−1

�

√A ≈ αn =

1

2

�αn−1 +

A

αn−1

�

f �(x) = 2xf(x) = x2 −A,

Newton’s method

√AFor let

Heron’s method

Nicolas Chuquet• French physician (d. 1487) by profession.

• Wrote but never published Triparty en la science des nombres (Science of Numbers in Three Parts), credited as the earliest work on algebra of the Rennaisance.

• In Triparty he coined the terms billion for million million (1012) and trillion for million million million (1018), used in England and Germany, but not in France or the U.S.

• Generalized al-Khwarizmi’s methods for solving quadratic equations to solving equations of any degree that are of quadratic type.

• Noted in Triparty that “[to] find a number between two fractions, add numerator to numerator and denominator to denominator.” This is the key to his method for finding square roots.

Square root of 6

2 <√6 < 3

Square root of 6

2 <√6 < 3

2 < 2 12 < 3 and

�2 12

�2= 6 1

4 > 6

Square root of 6

2 <√6 < 3

2 < 2 12 < 3 and

�2 12

�2= 6 1

4 > 6

2 <√6 < 2 1

2

Square root of 6

2 <√6 < 3

2 < 2 12 < 3 and

�2 12

�2= 6 1

4 > 6

2 <√6 < 2 1

2

2 < 2 13 < 2 1

2 and�2 13

�2= 5 4

9 < 6

Square root of 6

2 <√6 < 3

2 < 2 12 < 3 and

�2 12

�2= 6 1

4 > 6

2 <√6 < 2 1

2

2 < 2 13 < 2 1

2 and�2 13

�2= 5 4

9 < 6

2 13 <

√6 < 2 1

2

2 13 <

√6 < 2 1

2

0 <a

b<

c

d=⇒ a

b<

a+ c

b+ d<

c

d

2 13 <

√6 < 2 1

2

2 13 < 2 2

5 < 2 12 and

�2 25

�2= 5 19

25 < 6

2 25 <

√6 < 2 1

2

0 <a

b<

c

d=⇒ a

b<

a+ c

b+ d<

c

d

2 13 <

√6 < 2 1

2

2 13 < 2 2

5 < 2 12 and

�2 25

�2= 5 19

25 < 6

2 25 <

√6 < 2 1

2

2 37 <

√6 < 2 1

2 2 49 <

√6 < 2 1

2

2 49 <

√6 < 2 5

11 2 49 <

√6 < 2 9

20

0 <a

b<

c

d=⇒ a

b<

a+ c

b+ d<

c

d

2 13 <

√6 < 2 1

2

2 13 < 2 2

5 < 2 12 and

�2 25

�2= 5 19

25 < 6

2 25 <

√6 < 2 1

2

2 37 <

√6 < 2 1

2 2 49 <

√6 < 2 1

2

2 49 <

√6 < 2 5

11 2 49 <

√6 < 2 9

20

2 920 − 2 4

9 = 0.0055 . . .

0 <a

b<

c

d=⇒ a

b<

a+ c

b+ d<

c

d

2 13 <

√6 < 2 1

2

2 13 < 2 2

5 < 2 12 and

�2 25

�2= 5 19

25 < 6

2 25 <

√6 < 2 1

2

2 37 <

√6 < 2 1

2 2 49 <

√6 < 2 1

2

2 49 <

√6 < 2 5

11 2 49 <

√6 < 2 9

20

2 920 = 2.45 and

√6 = 2.449 . . .

0 <a

b<

c

d=⇒ a

b<

a+ c

b+ d<

c

d

Thank you.John Thoo jthoo@yccd.edu

• Jean-Luc Chabert (editor) et al. A History of Algorithms: From the Pebble to the Microchip. Springer-Verlag, Berlin, 1999. Translator of the English Edition: Chris Weeks.

• Sir Thomas Heath. A History of Greek Mathematics Volume II: From Aristarchus to Diophantus. Dover Publications, Inc., New York, 1981.

• Annette Imhausen. Egyptian mathematics. In Victor J. Katz, editor, The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press, Princeton, 2007.

• Victor J. Katz. A History of Mathematics: An Introduction. Pearson Education, Inc., Boston, 3rd edition, 2009.

• Leonardo of Pisa (Fibonacci). Fibonacci’s Liber Abaci: A Translation into Modern English of Leonardo Pisano’s Book of Calculation. Springer, New York, 2002. Translated L. E. Sigler.