The first calculation was: 1/2(10 + 2/10) = 5.1

The second calculation was: 1/2(5.1 + 2/5.1) = 2.746078431

The third calculation was: 1/2(2.746078431 + 2/2.746078431) = 1.737194874

The fourth calculation was: 1/2 (1.737194874 + 2/1.737194874) = 1.444238095

The fifth calculation was: 1/2 (1.444238095 + 2/1.444238095) = 1.414525655

The sixth calculation was: 1/2 (1.414525655 + 2/ 1.414525655 ) = 1.414213597

5. Repeat the whole procedure again, but this time start with x = 1.

The first calculation was: 1/2 (1 + 2/1) = 1.5

The second calculation was: 1/2 (1.5 + 2/1.5) = 1.416666667

The third calculation was: 1/2 (1.416666667 + 2/1.416666667) = 1.414215686

The fourth calculation was: 1/2 (1.414215686 + 2/1.414215686) = 1.414213562

The fifth calculation was: 1/2 (1.414213562 + 2/1.414213562) = 1.414213562

The sixth calculation was: 1/2 (1.414213562 + 2/1.414213562) = 1.414213562

6. Finally repeat the whole procedure again, but this time start with x = 2.

The first calculation was: 1/2 (2 + 2/2) = 1.5

The second calculation was: 1/2 (1.5 + 2/1.5) = 1.4166666667

The third calculation was: 1/2 (1.416666667 + 2/1.416666667) = 1.414215686

The fourth calculation was: 1/2 (1.414215686 + 2/1.414215686) = 1.414213562

The fifth calculation was: 1/2 (1.414213562 + 2/1.414213562) = 1.414213562

The sixth calculation was: 1/2 (1.414213562 + 2/1.414213562) = 1.414213562

7. What can be deduced from the results of 4, 5 and 6?

For number 5 and 6, the answers are the same, however, for number 4, it isn't. Moreover, every time the equation goes through the "machine", the final number gets smaller, or sometimes, it just stays the same.

8. By experimenting changes to 1/2(x + 2/x), find out how to calculate decimal approximations to other radicals such as √3, √5, √11.

For √3:

The first calculation was: 1/2 (√3 + 2/√3) = 1.443375673

The second calculation was: 1/2 (1.443375673 + 2/1.443375673) = 1.41450816

The third calculation was: 1/2 (1.41450816 + 2/1.41450816 ) = 1.414213593

The fourth calculation was: 1/2 (1.414213593 + 2/1.414213593) = 1.414213562

The fifth calculation was: 1/2 (1.414213562 + 2/1.414213562) = 1.414213562

The sixth calculation was: 1/2 (1.414213562 + 2/1.414213562) = 1.414213562

For √5:

The first calculation was: 1/2 (√5 + 2/√5) = 1.565247584

The second calculation was: 1/2 (1.565247584 + 2/1.565247584) = 1.421500357

The third calculation was: 1/2 (1.421500357 + 2/1.421500357) = 1.414232239

The fourth calculation was: 1/2 (1.414232239 + 2/1.414232239) = 1.414213562

The fifth calculation was: 1/2 (1.414213562 + 2/1.414213562) = 1.414213562

The sixth was: 1/2 (1.414213562 + 2/1.414213562) = 1.414213562

9. Prove that the method described above will always calculate decimals approximations to radicals. Your proof should involve finding √k, say.

This method which was describe above will always calculate decimal approximations to radicals because the prime number (?) roots such as √5, √3, etc. are always in decimal form.

10. Challenge: Design a number crunching machine which can calculate decimal approximation to numbers like 3√2 and 3√23. To do this you must be successful at 9.

2*2*2*23*23*23=97336

Homemade Crunching Machine: 97336 (x+2/x) = Answer