# Calculate Square Root with a Slide Rule

## Continued Fraction Method

Some slide rule has a “A” scale for x², which can be used to look up square root values quickly. However there are plenty of slide rules that doesn’t come with “A” scale.

How can you calculate square root, simply using division? Either by hand, or in this article, by a slide rule.

First, to derive the recursive math formula for √n

Now replace √n in the denominator on the left side of the equation again and again, we get a continued fraction formula for square root of `n`

:

Let’s put some number to the test. For example, to calculate √5, we get

`a`

= 2 and `b`

= 1.

Once you put them into the continued fraction formula, you can calculate the square root on a slide rule.

Here are the steps:

First calculate `b/2a`

= 1/4. You put 1 on D scale, 4 on C scale. Now read D scale to be 0.25. (2.5 => 0.25 of course)

Secondly, add (2 a = 4) to 0.25. You now put 4.25 on C scale, and 1 still on D. **This moves the slide towards left**. Now D scale reads 0.235.

Again, adjust C scale to be 4.235. **This push the slide a little towards right**. This will read a slightly larger number on D scale 0.236, which in turn pushes the slide towards left in the next step.

With this perfect negative feedback loop, the slide rule zooms in on the “final” value of 0.236 quickly. So the square root is √5 = 2 + 0.236 = 2.236

Just to emphasis that `b`

doesn’t have to be 1, here is another example √47

a = 6 and b = 11

11/12 → 0.915

11/12.915 → 0.855

11/12.855 → 0.855

So the square root √47 = 6.855

Note: J language generalized continued fraction operator: `(+`% /) a b 2a b 2a b 2a b 2a …`