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does long division method or the other manual algorithms that are used to calculate square root of any number , gives the exact value of square root of any number OR they just approximate it as accurately as they can ? Please guide me.

Sameer Nilkhan
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  • What do you mean by "exact"? These are irrational numbers... The Babylonian method (aka Newton's method) $\textit {converges}$ to the exact value. Meaning that you can get as accurate an answer as you desire through sufficient iterations. Does that answer your question? – lulu Jan 07 '20 at 01:08
  • that is what i want to ask , is long division method as accurate as the babylonian method (aka newtons method) ? – Sameer Nilkhan Jan 07 '20 at 01:10
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    Then you should ask that question. What you wrote is quite different. Any of the numeric methods will get you as close as you want to a solution. Some take more iterations than others to converge to a given accuracy. – Ross Millikan Jan 07 '20 at 01:11
  • @Sameernilkhan That's a very different question than you actually asked. – lulu Jan 07 '20 at 01:12
  • so does that mean that the long division method does not gives us the exact value of square root of any number ? – Sameer Nilkhan Jan 07 '20 at 01:13
  • What? As @RossMillikan correctly observed, all these numerical methods converge to the exact answer, though possibly at different rates. Perhaps this survey article will help. – lulu Jan 07 '20 at 01:17
  • I don't think you have asked a clear question here. How could any method give us the exact value as it is an infinite nonrepeating decimal? – Ross Millikan Jan 07 '20 at 02:22

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Any numeric method will give an approximation to the square root. As the decimal expansion goes on forever, you can't write down an exact value. Generally you can go as long as you want to get a value that is as accurate as you want, but you can't get the exact value. There is no way to write the number that is $\sqrt 2$ exactly unless you just write $\sqrt 2$ or some equivalent expression. Any of the numeric methods will give you something like $1.414213$, which is not exact.

Ross Millikan
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