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 = 9Squares can be represented with squares. Go figure. Here's a 3x3:

000We'll define

000

000

AThen let's define:_{1}= x^{2}= 9 (area of square)

x = 3 (side of square)

AThe new square has four sections:_{2}= (x + Δx)^{2}= 10 (area of new square)

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

1113And 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:

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

A_{section 0}= x^{2}= 9

A_{section 1}= x(Δx)

A_{section 2}= x(Δx)

A_{section 3}= (Δx)^{2}

111If we set the total area of that shape to 10, then we can easily find Δx:

0002

0002

0002

A_{2}= x^{2}+ 2x(Δx)

... (math) ...

Δx = (A_{2}- x^{2}) / 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 = 8,3872. Take the difference of N and x

100^{2}= 10,000 (too big)

50^{2}= 2,500 (too small)

90^{2}= 8,100 (perfect)

x = 90

^{2}(and call it d)

d = N - x3. This equation:^{2}

= 8387 - 8100

= 287

(x + a)tells us the difference effected in a square by incrementing the root by a. For example:^{2}= x^{2}+ 2xa + a^{2}

2So choose a to get as close as you can to d without going over (and call that value b):^{2}= 4

(2 + 1)^{2}= 2^{2}+ 2(2)(1) + 1^{2}

= 2^{2}+ 5 (so adding 1 to the root increases the square by 5)

= 9 = 3^{2}

(2 + 5)^{2}= 2^{2}+ 2(2)(5) + 5^{2}

= 2^{2}+ 45 (so adding 5 to the root increases the square by 45)

= 49

2xa + aAdd your chosen a to make a new x:^{2}

2(90)(1) + 1^{2}= 181

2(90)(2) + 2^{2}= 364 (too big)

b = 181

a = 1

x = 914. If you want more precision (x is already within 1 of the answer), subtract b from d:

d = 287 - 181then divide that number by 2x:

= 106

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

106 / 2x = 106 / 2(91)

= 106 / 182

= 53 / 91

≈ 5/9

≈ .55

√8387 ≈ 91.55 (estimate)

= 91.58 (real answer)

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

On that one, if I knew that 11

√127 = ?

10^{2}= 100

127 - 100 = 27

27 / (2*10) = 1 + 7/20

7/20 = .35

√127 ≈ 11.35 (estimate; not bad)

= 11.26 (actual answer)

^{2}= 121, then

11^{2}= 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:

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 ;)

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

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.

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