There is a mathematical way to find the square root of a number without a calculator.
First of all write 2/√5 as 2√5/5, or (2/5)√5 which is 0.4√5
Now find the value of √5
Do the following:
√5.00 00 00 00 00 (double 0's for as many digits as you want to find)
What is the largest integer squared that is less than 5 ? The answer is 2.
2. 2 3 6
______________
√5. 00 00 00 00 00
2 4
__
1 00 (bring down the first 00 and double 2 to get 4)
84
_____ (divide 4x into 100, replacing x with a digit and this same digit goes above the first 00)
42 16 00 ( bring down the next 00 and double 22 to get 44)
1329
443 ________ (divide 44x into 1600, replacing x with a digit and this same digit goes above the 00)
271 00 (bring down the next 00 and double 223 to get 446)
4466 26796
___________ (divide 446x into 27100, replacing x with a digit and this digit goes above the 00)
304 00 ( bring down the next 00 and double 2236 to get 4472
4472 continue this procedure for as many digits as you want for √5
now we can get the answer to the problem; remember we wrote the number as 0.4√5
now multiply 0.4*2.236 to get 0.8944 (which is of course an approximation)
2/√5≈0.8944
Another solution is by trial and error:
2^2=4
3^2=9
the answer is between 2 and 3 and closer to 2
2.1^2=4.41
2.2^2=4.84(2.3^2=5.29-too big !)
2.21^2=4.8841
2.22^2=4.9284
2.23^2=4.9729
2.231^2, 2.232^2, 2.233^2, 2.234^2, 2.235^2, and finally 2.236^2=4.999696
the next digit will be 0, 2.2360
the next digit will be 6, 2.23606^2=4.999964324
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