How can I calculate the square root of 26?

The square root of 26 is a number which when multiplied by itself equals 26.

The first question to ask is “Does the square root of 26 exist?”

And the answer is that for all real numbers greater than 0 there exists a number which is a root (2 , 3, 4, 5 …) for that number.

So for positive numbers, yes.

Also for every positive root there is a negative root of the same value.

so we set up our first equation as

x * x = 26

This seems too simple , but it is what we are talking about.

Try

x = 4

4 * 4 = 16

; too low

5 * 5 = 25

; too low

6 * 6 = 36

; too high , so

5 < \sqrt{26} < 6

5 < x < 6

still simple but now we have something to work with.

Let us split the difference

This is called "Hero's Method".

S

is our number.

x

is our guess for square root of

S

e

is our error.

e ≪ x

S = (x + e)^{2}

S = (x + e)^{2} = x^{2} + 2 e x + e^{2}

S - x^{2} = 2 e x + e^{2}

S - x^{2} = e (2 x + e)

e = \frac{S - x^{2}}{2 x + e}

e

is considered insignificant compared to

x

, (even less to

2 x

) so

\frac{S - x^{2}}{2 x + e} \approx \frac{S - x^{2}}{2 x}

x + e \approx x + \frac{S - x^{2}}{2 x}

multiply

x

\frac{2 x}{2 x}

by to combine terms —

x = x * \frac{2 x}{2 x} = \frac{2 x^{2}}{2 x}

\approx \frac{2 x^{2}}{2 x} + \frac{S - x^{2}}{2 x}

\approx \frac{S + 2 x^{2} - x^{2}}{2 x}

\approx \frac{S + x^{2}}{2 x}

\approx \frac{\frac{S}{x} + x}{2}

\frac{\frac{S}{x} + x}{2}

is not the simplest form, but it is the easiest to use for the calculations
we need to make.

Take an educated guess:

5 < x < 6

start approximating:

S = 26, x = 5

1) x \approx \frac{\frac{26}{5} + 5}{2} = 5.100000

2) x \approx \frac{\frac{26}{5.1} + 5.1}{2} = 5.0990196

3) x \approx \frac{\frac{26}{5.0990196} + 5.0990196}{2} = 5.0990195

4) x \approx \frac{\frac{26}{5.0990195} + 5.0990195}{2} = 5.0990195

I got the same answer twice, so I am satisfied with 6 decimal places.
I have 7 places, but that last 5 might be rounded up.

This means my error figure

e < 0.000001

after 4 iterations. Pretty good.

\sqrt{26} = ⟨ 5.0990195, - 5.0990195 ⟩

(both positive an negative values)

Calculator check:

\sqrt{26} = 5.09901951359

NAIRAFAME graduated from Osun State University (Mathematics Department).