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## How do you divide this without a calculator?

2
√5

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

f(x) = √(x) gets very flat as x gets large.

√(5) = √(500)/10

22 < √(500) < 23

√(484) < √(500) < √(529)

√(500) ≈ 22 + (500-484)/(529-484) = 22+(16)/45

√(5) = √(500)/10 ≈ (22+16/45)/10

2/√(5) ≈ 20/(22+(16)/45) ≈ 10/(11+8/45) ≈ 450/(495+8) ≈ 450/503

Divide by 503:

450    | 0.8
402.4
–––
47.6   | 0.09
45.27
–––––
2.33   | 0.004
2.012
–––––
0.318  | 0.0006
0.3018
––––––
0.0162 etc.

2/√(5) ≈ 0.8946

check: calculator result ≈ 0.89442719099992

We could have gotten a more accurate approximation
by using √(5) = √(50000)/100.
Most problems in this format have you simplify by putting the radical in the numerator
2√5
5

You could try to use a number close to √5
For example, 22 is 4, 32 is 9
2.32 is 5.59
2.22 is 4.84
2.252 is 5.0625

2/√5 is close to 2/2.25 which you can divide by long division
2/2.25 rounds to 0.89
You would have to create your own value for the square root of 5.  You would have to guess and check values for multiplying decimals together to find approximate values.

2.3     2.2     2.23         2.24       2.235
*2.3   *2.2   *2.23        *2.24     *2.235
5.29   4.84   4.9729     5.0176  4.995225

Once you get as close as you want, you long divide 2 by your value.

2/2.235 ≈ 0.8949
You multiply it by 1... or in this case...√5 divided by √5. That will get rid of the exponent on the bottom and leave you with (2*√5) divided by 5. Can you do that on your own?