Sunday, November 8, 2009

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)

3 comments:

Tammy said...

Wow ~ I dont think I have ever read a nerdier blog ... and gotten such a big smile on my face because you are the coolest ;)

G-Unit said...

lol I love that you posted this :) Seriously though, that's a cool method. I've wondered the same thing myself.

Magoo said...

You can do your "squarish root" in your head? I need a calculator to get your 2 decimal place precision. Thus the reason I bought my TI-89. It can do it in fewer steps:)

I love the nerdiness! That's why I kept you around.