Bombelli’s method of computing square roots

Bombelli’s method of computing square roots

In the following, all numbers are positive real numbers.

The equations

100

⁢

a

=

10

⁢

a

and

a

100

=

a

10

imply, that if the

decimal point

of the

radicand

of a

square root

is moved two steps to the right or to the left, then the value of the square root changes only such that its decimal point moves one step in the same direction.

If the

integer part

of a number has one or two

http://

planetmath

.org/node/3313

digits, the integer part of its square root has evidently one digit.  Accordingly, one may infer the following rule:

If the integer part of the radicand is , starting from the decimal point, into pieces of two digits (when the leftmost piece may consist of only one digit), then the number of the pieces expresses the number of digits in the integer part of the square root.

We now illustrate the computing of square root by using

2238.9

as an example, and denote its first digits by

x

,

y

,

z

,

…

The integer part of

22.389

has one digit, which is

x

.  This is the

http://planetmath.org/node/6118

greatest one-digit

integer

whose square is at most 22.  Hence,

x

=

4

.

By the above rule, the integer part of

2238.9

has two digits and thus equals to

10

⁢

x

+

y

=

40

+

y

.  The number

y

is the greatest of the one-digit integers such that the square

(

10

⁢

x

+

y

)

2

=

100

⁢

x

2

+

2

⋅

10

⁢

x

⁢

y

+

y

2

=

100

⁢

x

2

+

y

⁢

(

10

⋅

2

⁢

x

+

y

)

does not exceed 2238, i.e. such that the product

y

⁢

(

10

⋅

2

⁢

x

+

y

)

=

y

⁢

(

80

+

y

)

is at most the

remainder

2238

-

100

⁢

x

2

=

638

.

We see that

y

=

7

.

We can continue similarly and determine next the digit

z

.  The calculations may be organised right from the start as follows:

22

38

.

90

=

4

7

.

3

1

7

…

16

4

6

38

8

7

6

09

7

29

90

9

4

3

28

29

3

1

61

00

9

4

6

1

94

61

1

66

39

00

9

4

6

2

7

66

23

89

7

15

11

…

…

The algorithm:

1

∘

.  Starting from the decimal point, the radicand is into pieces of two digits in either directions.

2

∘

.  The first digit (4) of the square root is the greatest one-digit integer whose square does not exceed the first piece.  The

difference

of the first piece and that square is concatenated with the second piece, giving the

first remainder

(638); the sum of the factors (4 and 4) of the square is the

first sum

(8).

3

∘

.  The second digit (7) of the square root is the greatest one-digit integer such that the product of it and with it concatenated first sum (87) does not exceed the first remainder.  The difference of the first remainder and that product is concatenated with the third piece, giving the

second remainder

(2990); the sum of the factors (87 and 7) of the product is the

second sum

(94).

The procedure continues similarly.  The decimal point of the square root is put when the last piece of the integer part of the radicand has been used.

Note that this algorithm produces the right digits of the square root one by one.

References

1

Raffaele Bombelli:

L’

Algebra

.

Bologna (1572—1929). (See

http://it.

wikipedia

.org/wiki/Metodi_per_il_calcolo_della_radice_quadrata

algoritmo di Bombelli).

2

K. Väisälä:

Algebran oppi- ja esimerkkikirja I

.  Fifth edition. Werner Söderström osakeyhtiö, Porvoo & Helsinki (1952).

Title

Bombelli’s method of computing square roots

Canonical

name

BombellisMethodOfComputingSquareRoots

Date of creation

2013-03-22 19:05:59

Last modified on

2013-03-22 19:05:59

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

15

Author

pahio (2872)

Entry type

Algorithm

Classification

msc 01A40

Related topic

BabylonianMethodOfComputingSquareRoots

Related topic

SquareRootOfPolynomial

Related topic

KalleVaisala

Related topic

PrimitiveRecursiveNumber

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