What's The Square Root Of Five?

It's about 2.24.

That's the quick answer. I just typed 5^0.5 into Google, and I got the result I wanted. I could have just as easily typed the problem into my calculator, or my phone, and found the same result.

According to Wikipedia, the first scientific calculator came out in 1968 and cost $32,000 in 2011 US Dollars. Without tables or a slide rule handy, in some cases, how did people calculate square roots?

Some were easy. The square root of 1 is 1, 4 is 2, 9 is 3 and so on. These numbers (1, 4, 9, ...) are called perfect squares because their square roots (1, 2, 3, ...) are integers.

Unfortunately, not every number is a perfect square. The first obvious example is 2. Another is 3. In fact, it seems obvious that there would be more non-perfect squares than there are perfect squares.

So how can we calculate the square root of a non-perfect square? The short answer is long division.

Yes, the same techniques learned way back in third grade (or so) to find quotients and remainders can be applied to finding square roots of numbers (albeit with slight modification).

The numbers 2 and 3 are between 1 and 4, so this means their square roots are between 1 and 2. Since we don't want to make any statements that aren't true, we'll use 5 so we don't run into any issues with the number 1 (since 1 is its own square and square root).

Where do we begin?

At the beginning, of course.

Now, we want to find an answer to two decimal places. Remember from long division that however many decimal places you want to find, you have to calculate one more. If you want one decimal place, find two. And so on. We want two decimal places, so we need to calculate our answer to three places.

For every digit after the decimal, we need to add two zeroes after the five's decimal point. Since we want two decimals, we need to calculate three. Since we need to calculate three, we'll need six zeroes.

 

Now we can start calculating. What's the biggest perfect square that goes into five? One goes into five, four goes into five, but nine does not go into five. So our biggest perfect square is 4.

What's the square root of four? Since the square root of 4 is 2, we write 2 above the five (like we'd do in long division).

Like long division, we have to subtract what we've already taken account of. In this case, we took care of 4 using that 2, so we're going to subtract away 4 from 5, giving us a difference of 1.

In long division, you would bring down a zero. In finding square roots, we're going to bring down two zeroes. Now he's where it starts to get tricky. 

Take the number on top (2) and double it (4). Bring the number (4) down beside the difference (1 with two zeroes, 100). Write a blank next to the new number (4) and put both that number (4) and the blank in parentheses. Your work should look like this:

 

This is the hardest part to think about. You need to put a number in that blank, so that if you multiply the number (4_) by the number you choose, it is less than the other number (100). Confused? Here's how it works. If you choose 1, you're going to multiply 41x1. If you choose 4, you're going to multiply 44x4. If you choose 9, you'll multiply 49x9. Make sense now? You're going to choose a number between 0 and 9, and you want the biggest product less than that 100. If we choose 1, we get 41x1 = 41. If we choose 2, we get 42x2 = 84. If we choose 3, we get 43x3=129. So the biggest one that works is 2. Now we write 2 in that blank from earlier.

We also write a 2 on top after the decimal place over our first two zeroes. As we said, 42x2 = 84, so we subtract that from 100, giving us a difference of 16.

Like earlier, we carry down the next two zeroes. We also take the top number (22), double it (44), and write it next to a blank space in parentheses like we did earlier.

We run through blanks again. If we choose 1, 441x1 = 441. If we choose 2, 442x2 = 884. If we choose 3, 443x3 = 1329. If we choose 4, 444x4 = 1776, which is too big. So we choose 3. As before, the 3 goes in the blank and above the next two zeroes.

As we said, 443x3 is 1329. So again we subtract the product and carry down the next two zeroes. We then take number on top (223), double it (446) and write it in parentheses with a blank (446_).

 

We try numbers again. If we choose 1, 4461x1 = 4461. If we choose 2, 4462x2 = 8924. If we choose 4, 4464x4 = 17856. If we choose 8, 4468x8 = 35744. If we choose 9, 4469x9 = 40221, which is too big.  So our rounding digit is 8.

 

In reality, we don't care about this digit. It only matters for rounding (since we want two places). All we need to check is the case of 5. In that case, 4465x5 = 22325. Since the number is less than 37100, we know to round up (since the digit is 5 or greater). If 4465x5 had been more than 37100, we would know to round down (since our digit would have to be 4 or less). Since the number we calculated is 2.238, we know the square root of 5, to two decimal places, is 2.24, which is what I calculated at the beginning of this entry.

Since we can easily calculate this with tools, this method's use is somewhat outdated, but it's a cool thing to know. Some other numbers to practice on include 2, 3, 6, 7 and 8. Once you've mastered those, try your hand at 10 and the double digits.

Anyway, I hope you got something out of this. If not, well, there's always next time.

If you have any questions, or if anything isn't clear, just let me know in the comments.

I learned this from Suzanne Ingram, who was my teacher for three years of math team, one year of Algebra II/Trig and one year of Precalculus in high school.

The images used here were produced by me using Noteshelf for iPad.