It’s the number you get when you take the square of another number (hence the name “square” root). For example, if we want to find the square root of 25, we can write it as:

(25) = 5

But what happens when we have a decimal instead? Let’s say we want to find the square root of 16.9. Well, you might be tempted to use your calculator and get an answer like this:

(16.9) = 4.12358…

But hold on a sec what if we don’t have a calculator handy? How can we find the square root of a decimal without one? That’s where decimal arithmetic comes in!

Here’s how it works: you take your decimal number and write out its digits as far right as possible. For example, 16.9 would look like this:

16.900…

Then, you find the largest perfect square that is less than or equal to your starting number (in this case, we’re looking for a number with a square root of 16). That number is 144. Let’s call it x:

x = 12 * 12 + 10 * 5 + 9

Now, let’s find the difference between our starting decimal and this perfect square (in other words, we want to know how far away from 16.9 we are). That number is called the “error”:

Error = Starting Decimal Perfect Square

In our case:

Error = 16.900… 144

The error will always be a decimal, since we’re subtracting two numbers that have different units (one is in squares and the other is not).

To do this, let’s divide both sides of the equation by the error:

Starting Decimal / Error = Perfect Square / Error

This gives us a new number that represents how many times our perfect square is “closer” to our starting decimal than it actually is. Let’s call it y:

y = (12 * 12 + 10 * 5 + 9) / (16.9 144)

Now, let’s find the square root of this new number! We can do that by repeating the same process we did before: finding a perfect square that is less than or equal to y and calculating an error between them. Let’s call this new error e:

e = y (10 * 2 + 9)

And let’s find our new number z by dividing both sides of the equation by e:

z = (10 * 2 + 9) / e

We can keep repeating this process over and over again, each time getting closer to our original decimal. We have found the square root using only basic arithmetic operations no calculators required!

Of course, this method is not perfect (hence the name “approximate” square roots). But it’s a great way to get an idea of what the answer might be without having to rely on technology. And who knows? Maybe one day you’ll become so good at decimal arithmetic that you won’t need a calculator anymore!