# What is the square root of 145?

Oct 18, 2015

$145 = 5 \cdot 29$ is the product of two primes and has no square factors, so $\sqrt{145}$ is not simplifiable.

$\sqrt{145} \approx 12.0416$ is an irrational number whose square is $145$

#### Explanation:

You can find approximations for $\sqrt{145}$ in a number of ways.

My current favourite is using something called continued fractions.

$145 = 144 + 1 = {12}^{2} + 1$ is of the form ${n}^{2} + 1$

sqrt(n^2 + 1) = [n;bar(2n)] = n + 1/(2n+1/(2n+1/(2n+1/(2n+...))))

So

sqrt(145) = [12;bar(24)] = 12 + 1/(24+1/(24+1/(24+...)))

We can get an approximation by just truncating the repeating continued fraction.

For example:

sqrt(145) ~~ [12;24] = 12 + 1/24 = 12.041dot(6)