Squarish Roots

I love math and I love solving clever math problems. This guy is pretty cool. Well, he's a nerd, but he's a pretty cool nerd. Also, Donald encounters square roots here.

Anyway, for the longest time I've wondered how to quickly find the ballpark square root of a number. The square root of 9 is easy, but what about 10, or 11, or 8,387? Chace once said that his dad knew of a way that was pretty quick, but we couldn't figure it out. Or maybe Chace figured it out and never told me :)

Well, I finally found a way! So.... here we go:

How I figured it out:
What's the square root of 10? It's probably just bigger than 3 because

3 * 3 = 9

Squares can be represented with squares. Go figure. Here's a 3x3:

000
000
000

We'll define

A1 = x2 = 9 (area of square)
x = 3 (side of square)

Then let's define:

A2 = (x + Δx)2 = 10 (area of new square)
x + Δx = ? (side of new square - this is the answer to the original question)

The new square has four sections:

1113
0002
0002
0002

section 0: area encompassing the original 3x3 square
section 1: additional area on top
section 2: additional area on the side
section 3: additional area in the top right corner

Asection 0 = x2 = 9
Asection 1 = x(Δx)
Asection 2 = x(Δx)
Asection 3 = (Δx)2

And here's the cool part! (You didn't think there'd be a cool part, did you?) Δx is less than 1, because otherwise x + Δx = 4 (or more) which squares to 16. And since I just want the ballpark answer, I can ignore the area of section 3. That leads to:

111
0002
0002
0002

If we set the total area of that shape to 10, then we can easily find Δx:

A2 = x2 + 2x(Δx)
... (math) ...
Δx = (A2 - x2) / 2x
  = (10 - 9) / 2(3)
  = 1 / 6
  = 0.16

√10 ≈ x + Δx
   ≈ 3.16

Which is close to the real answer:

3.16227766

Steps for any number:
Given any number (let's call it N):
1. Choose a number, x, whose square is just less than N

N = 8,387
1002 = 10,000 (too big)
502 = 2,500 (too small)
902 = 8,100 (perfect)
x = 90

2. Take the difference of N and x² (and call it d)

d = N - x2
 = 8387 - 8100
 = 287

3. This equation:

(x + a)2 = x2 + 2xa + a2

tells us the difference effected in a square by incrementing the root by a. For example:

22 = 4

(2 + 1)2 = 22 + 2(2)(1) + 12
        = 22 + 5 (so adding 1 to the root increases the square by 5)
        = 9 = 32

(2 + 5)2 = 22 + 2(2)(5) + 52
        = 22 + 45 (so adding 5 to the root increases the square by 45)
        = 49

So choose a to get as close as you can to d without going over (and call that value b):

2xa + a2
2(90)(1) + 12 = 181
2(90)(2) + 22 = 364 (too big)

b = 181
a = 1

Add your chosen a to make a new x:

x = 91

4. If you want more precision (x is already within 1 of the answer), subtract b from d:

d = 287 - 181
  = 106

then divide that number by 2x:

106 / 2x = 106 / 2(91)
         = 106 / 182
         = 53 / 91
         ≈ 5/9
         ≈ .55

Add that result to x and you've got a pretty precise answer:

√8387 ≈ 91.55 (estimate)
      = 91.58 (real answer)

So here's one done really quickly with even less precision:

√127 = ?
102 = 100
127 - 100 = 27
27 / (2*10) = 1 + 7/20
7/20 = .35
√127 ≈ 11.35 (estimate; not bad)
     = 11.26 (actual answer)

On that one, if I knew that 11² = 121, then

112 = 121
127 - 121 = 6
6 / (2*11) = 6 / 22
           = 3 / 11
           = 3 * (1 / 11)
           ≈ 3 * .09
           ≈ .27
√127 ≈ 11.27 (even better)
     = 11.26 (actual answer)