# How do you find the square root of 193?

Aug 30, 2016

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

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

#### Explanation:

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

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

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

${a}_{i + 1} = \frac{{a}_{i}^{2} + n}{2 {a}_{i}}$

I like to reformulate this slightly using integers ${p}_{i}$ and ${q}_{i}$ where ${a}_{i} = {p}_{i} / {q}_{i}$. Then use these formulae to iterate:

${p}_{i + 1} = {p}_{i}^{2} + n {q}_{i}^{2}$

${q}_{i + 1} = 2 {p}_{i} {q}_{i}$

If the resulting ${p}_{i + 1}$ and ${q}_{i + 1}$ have a common factor, then divide both by that factor before the next iteration.

Let $n = 193$, ${p}_{0} = 14$ and ${q}_{0} = 1$

Then:

${p}_{1} = {p}_{0}^{2} + n {q}_{0}^{2} = {14}^{2} + 193 \cdot {1}^{2} = 196 + 193 = 389$

${q}_{1} = 2 {p}_{0} {q}_{0} = 2 \cdot 14 + 1 = 28$

If we stopped here then we would have:

$\sqrt{193} \approx \frac{389}{28} = 13.89 \overline{285714}$

Next iteration:

${p}_{2} = {p}_{1}^{2} = n {q}_{1}^{2} = {389}^{2} + 193 \cdot {28}^{2} = 151321 + 151312 = 302633$

${q}_{2} = 2 {p}_{1} {q}_{1} = 2 \cdot 389 \cdot 28 = 21784$

So:

$\sqrt{193} \approx \frac{302633}{21784} \approx 13.892444$

Actually:

$\sqrt{193} \approx 13.8924439894498$

but as you can see this method converges quite rapidly.