High-Temperature Modification of Saponite-Containing Material

Drozdyuk, Tatiana; Ayzenshtadt, A. M.; Korolev, Evgeniy

doi:10.31659/0585-430x-2021-797-11-30-35

Cited by 2 publications

(9 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For this example, Fibonacci multiplies 7243 with 10 4 and finds the integer part of the square root of 7234 using the same technique as for the other examples. We find the spelling Liber Abbaci and Liber Abaci in the literature, though Abaci is used more often today [1] (p. 37).…”

Section: Introductionmentioning

confidence: 88%

“…The correction is N−a 2 2a ≈ 1 4 . The approximation of √ 72,340,000 is then 8505 1 4 and the approximation of √ 7234 is 85 + 1 20 + 1 400 .…”

Section: Computing the Fractional Partmentioning

confidence: 99%

“…For a = 3 1 5 + 1 = 2 then a + (16/5 − a 2 )/(2a) = 1 4 5 and 3 1 5 is a little less so √ 10 will be a little less than 1 4 5 and 4 + √ 10 is a little less than 5 4 5 . Another explanation is given by in Høyrup [31] (p. 28) which uses √ 10 a little less than 3 1 6 and for a = 2 we have for 3 1 6 that a − a 2 −3…”

mentioning

confidence: 97%

“…Fibonacci first gives a geometric proof and then verifies the computation with approximate numbers [13] (p. 500). Fibonacci states that √ 448 is a little less than 21 1 6 , and that 21 1 6 + 23 = 44 1 6 is a little less than 6 = 14 1 4 . The sum the three terms 16, 3 1 6 , and 14 1 4 , is a little more than 33 and for a =…”

mentioning

confidence: 99%

“…Fibonacci states that √ 448 is a little less than 21 1 6 , and that 21 1 6 + 23 = 44 1 6 is a little less than 6 = 14 1 4 . The sum the three terms 16, 3 1 6 , and 14 1 4 , is a little more than 33 and for a =…”

mentioning

confidence: 99%

See 4 more Smart Citations

On the Square Root Computation in Liber Abaci and De Practica Geometrie by Fibonacci

Steihaug

2024

Mathematics

View full text Add to dashboard Cite

We study the square root computation by Leonardo Fibonacci (or Leonardo of Pisa) in his MSS Liber Abaci from c1202 and c1228 and De Practica Geometrie from c1220. In this MSS, Fibonacci systematically describes finding the integer part of the square root of an integer in numerous examples with three to seven decimal digits. The results of these examples are summarized in a table in the paper. Liber Abaci also describes in detail finding an approximation to the fractional part of the square root. However, in other examples in Liber Abaci and De Practica Geometrie, only the approximate values of the fractional part of the square roots are stated. This paper further explores these approximate values using techniques like reverse engineering. Contrary to many claims that Fibonacci also used other methods or approximations, we show that all examples can be explained using one digit-by-digit method to compute the integer part of the square root and one approximation scheme for the fractional part. Further, it is shown that the approximation scheme is tied to the method to compute the integer part of the square root.

show abstract

Section: Introductionmentioning

confidence: 88%

“…The correction is N−a 2 2a ≈ 1 4 . The approximation of √ 72,340,000 is then 8505 1 4 and the approximation of √ 7234 is 85 + 1 20 + 1 400 .…”

Section: Computing the Fractional Partmentioning

confidence: 99%

mentioning

confidence: 97%

mentioning

confidence: 99%

mentioning

confidence: 99%