How do you find the square root of 193?

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How do you find the square root of 193?

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Answer 1 · 2016-08-30T19:38:32

$\sqrt{193} \approx 13.8924439894498$ $sqrt(193) ~~ 13.8924439894498$ is an irrational number.

We can find approximations to it using a Newton Raphson method.

Explanation:

$193$ $193$ is a prime number, so its square root does not have any simpler form. It is an irrational number a little less than $14$ $14$ (since $14^{2} = 196$ $14^2 = 196$ ). That is, it is not expressible in the form $\frac{p}{q}$ $p/q$ for any integers $p, q$ $p, q$ .

We can find approximations to it using a kind of Newton Raphson method.

Given a number $n$ $n$ and an initial approximation $a_{0}$ $a_0$ to $\sqrt{n}$ $sqrt(n)$ , derive progressively more accurate approximations by using the formula:

$a_{i + 1} = \frac{a_{i}^{2} + n}{2 a_{i}}$ $a_(i+1) = (a_i^2 + n)/(2a_i)$

I like to reformulate this slightly using integers $p_{i}$ $p_i$ and $q_{i}$ $q_i$ where $a_{i} = \frac{p_{i}}{q_{i}}$ $a_i = p_i/q_i$ . Then use these formulae to iterate:

$p_{i + 1} = p_{i}^{2} + n q_{i}^{2}$ $p_(i+1) = p_i^2+n q_i^2$

$q_{i + 1} = 2 p_{i} q_{i}$ $q_(i+1) = 2p_i q_i$

If the resulting $p_{i + 1}$ $p_(i+1)$ and $q_{i + 1}$ $q_(i+1)$ have a common factor, then divide both by that factor before the next iteration.

Let $n = 193$ $n=193$ , $p_{0} = 14$ $p_0 = 14$ and $q_{0} = 1$ $q_0 = 1$

Then:

$p_{1} = p_{0}^{2} + n q_{0}^{2} = 14^{2} + 193 \cdot 1^{2} = 196 + 193 = 389$ $p_1 = p_0^2+n q_0^2 = 14^2+193*1^2 = 196+193 = 389$

$q_{1} = 2 p_{0} q_{0} = 2 \cdot 14 + 1 = 28$ $q_1 = 2p_0 q_0 = 2*14+1 = 28$

If we stopped here then we would have:

$\sqrt{193} \approx \frac{389}{28} = 13.89 ¯ ¯¯¯¯¯¯¯¯¯¯ ¯ 285714$ $sqrt(193) ~~ 389/28 = 13.89bar(285714)$

Next iteration:

$p_{2} = p_{1}^{2} = n q_{1}^{2} = 389^{2} + 193 \cdot 28^{2} = 151321 + 151312 = 302633$ $p_2 = p_1^2 = n q_1^2 = 389^2 + 193*28^2 = 151321+151312 = 302633$

$q_{2} = 2 p_{1} q_{1} = 2 \cdot 389 \cdot 28 = 21784$ $q_2 = 2p_1 q_1 = 2*389*28 = 21784$

So:

$\sqrt{193} \approx \frac{302633}{21784} \approx 13.892444$ $sqrt(193) ~~ 302633/21784 ~~ 13.892444$

Actually:

$\sqrt{193} \approx 13.8924439894498$ $sqrt(193) ~~ 13.8924439894498$

but as you can see this method converges quite rapidly.

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How do you find the square root of 193?

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