# What is the square root of 26?

Sep 6, 2015

You can only have an approximation: 5.09901951...

#### Explanation:

The square root of a number $x$ is a number $y$ such that ${y}^{2} = x$. So, we're looking for a number $y$ such that ${y}^{2} = 26$. Since ${5}^{2} = 25$ and ${6}^{2} = 36$, the square root of $26$ is between $5$ and $6$. Since there is no algorithm to compute it exactly, you can only have an approximation. A possible way is the following: we know that $\sqrt{26}$ is between $5$ and $6$. So, since ${5}^{2} = 25$ and ${5.1}^{2} = 26.01$, $\sqrt{26}$ must be between $5$ and $5.1$.

Iterating this process gives you all the decimal digits you need.

Sep 6, 2015

$\sqrt{26}$ does not simplify, but you can calculate an approximation efficiently using Newton Raphson method as:

$\sqrt{26} \approx \frac{54100801}{10610040} \approx 5.099019513592786$

#### Explanation:

$26 = 2 \cdot 13$ has no square factors, so $\sqrt{26}$ cannot be simplified.

If you want to calculate an approximation by hand, then I would recommend a form of Newton Raphson method, starting with first approximation ${a}_{0} = 5$.

To iterate you can use the formula:

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

where $n = 26$ is the number you are approximating the square root of.

Personally, I like to deal with these approximations as rational approximations in the form ${p}_{i} / {q}_{i} = {a}_{i}$ where ${p}_{i}$ and ${q}_{i}$ are integers as follows:

$n = 26$
${p}_{0} = 5$
${q}_{0} = 1$

Iterate using:

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

So:

${p}_{1} = {5}^{2} + 26 \cdot {1}^{2} = 25 + 26 = 51$
${q}_{1} = 2 \cdot 5 \cdot 1 = 10$

${p}_{2} = {51}^{2} + 26 \cdot {10}^{2} = 2601 + 2600 = 5201$
${q}_{2} = 2 \cdot 51 \cdot 10 = 1020$

${p}_{3} = {5201}^{2} + 26 \cdot {1020}^{2} = 27050401 + 27050400 = 54100801$
${q}_{3} = 2 \cdot 5201 \cdot 1020 = 10610040$

Stop when you think you have enough significant digits (typically about the number of significant digits of ${p}_{i}$ + the number of significant digits of ${q}_{i}$).

$\sqrt{26} \approx \frac{54100801}{10610040} \approx 5.099019513592786$

Actually $\sqrt{26} \approx 5.099019513592785$