Question

Ques = 1 + 24/60 + 51/60² + 10/60³


I know ans. is
=1.41421296
2ed = 1.41421356

only ans for step by step.​

Answer 1

Answer:

In modern characters, this would be written approximately as 1 ; 24, 51, 10, where a semicolon is used to separate the integer and fractional parts, and the comma is used as a separatrix for the sexagesimal positions. This Babylonian calculation of √2 is performed as 1 + 24/60 + 51/60² + 10/60³, which is approximately 1.414212963.

It is defined by C.B. Boyer in his “History of Mathematics” [2] as:6

“Let x = √a be the root desired, and let a₁ be a first approximation to this root; let a second approximation b₁ be found from the equation b₁=a/a₁. If a₁ is too small, then b₁ is too large , and vice versa. Hence the arithmetic mean a₂=½ (a₁ + b₁) is a plausible next approximation. Inasmuch as a₂ always is too large, the next approximation b₂=a/a₂ will be too small, and one takes the arithmetic mean a₃=½ (a₂ + b₂) to obtain a still better result; the procedure can be continued indefinitely.”

The technique is equivalent to Heron’s method for extracting a square root, found in Metrica I 8 [4]. Here is an example for calculating √2, with N=2, and r₀=1 as the initial guess (blue digits represent those accurately calculated).

r₁ = (r₀ + N/r₀)/2 = (1 + 2/1)/2 = 1.5

r₂ = (r₁ + N/r₁)/2 = (1.5 + 2/1.5)/2 = 1.416666

r₃ = (r₂ + N/r₂)/2 = (1.416666 + 2/1.416666)/2 = 1.4142156

r₄ = (r₃ + N/r₃)/2 = (1.4142156 + 2/1.4142156)/2 = 1.4142135

Answer 2

Step-by-step explanation:

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