Question

Describe how to estimate a non-perfect square root to the hundredths place without using a calculator.

Answer 1

....evil, comes to mind. And anyways, you don't usually do more than say it is between two integers. There is no simple way to find it to the hundreths place that I know of.

Answer 2

you do a bunch of guess and check

Answer 3

theres a long hand divisiony process

Answer 4

I would like to see that amistre, I don't actually know it.

Answer 5

my way is just take a nice guess and compare, usually find it in 5 tries

Answer 6

I mean unless you mean doing a prime factorization and multiplying by a common irrational root that you happen to know the digits of

Answer 7

sqrt(4351.234)

now the idea is that the biggest perfect square of a single digit is 9*9 = 81, 2 positions

from the decimal start partitioning it into 2 digit parts


sqrt(43 51. 23 40)

now we are going to work off of (x+y)^2 = x^2 + 2xy + y^2

we want to find x^2 that divides 43 and subtract it .... like long division


6
------------
sqrt(43 51. 23 40)
36
---
7 51 <-- drop down


now comes the 2xy part, we have x already given as 6

2*6 = 12 + y some single digit soo


6 y
------------
sqrt(43 51. 23 40)
36
---
12+y| 7 51


(12+y)*y < 751

Answer 8

Btw this is about estimating it not doing it on paper. i cant do that in my head.

Answer 9

the problem says nothing about getting to use paper, just no calculator?

Answer 10

also, very very interesting amistre! When did you learn that?

Answer 11

The lesson is on estimating

Answer 12

you can use paper to estimate

Answer 13

ie. find square root of 167, well it's between 12 and 13, closer to thirteen, I'd bet like 12.9 ish but I can't do two places without testing it by writing it on paper

Answer 14

estimating is not always the same as mental math

Answer 15

y = 5 is just right

125 * 5 = 625


6 5
------------
sqrt(43 51. 23 40)
36
---
125| 7 51
625
-----
126 23



2(65) = 130

130y < 12623 ; 9 works

1309*9 = 11781



6 5 . 9
------------
sqrt(43 51. 23 40)
36
---
7 51
625
-----
126 23
11781
-------
842 40

2(659) = 1318

1318y*y < 84240 ; 7?6? , 6 is good

13186*6 =

6 5 . 9 6
------------
sqrt(43 51. 23 40)
36
---
7 51
625
-----
126 23
11781
-------
842 40
79116
-------
5124


dbl chk

(65.96)^2 = 4350.7216

Answer 16

when? i cant remember that far back lol

Answer 17

lol, I would love to know why that works. I don't think I've ever seen it

Answer 18

its based off of

(x+y)^2 = x^2 + 2xy + y^2

finding xs and ys along the way the are "closest" and the rest is long divisiony

Answer 19

But, how do you find s, I missed that part. (I don't really follow any of the logic though to be honest)

Answer 20

when we divide long hand, whats the first thing we do? try to find a multiple that is closest to the digits we are assessing right?

Answer 21

yea, but I don't get why you split it up the way you did, I mean I know it has to be a two digit square, but I guess I just don't understand how that works.

Answer 22

like could you prove that this method works for every number?

Answer 23

9 is the largest digit, it makes the largest square, 9*9 = 81 so we need at least 2 digits of freedom to play with

Answer 24

after partitioning, it would make sense that since your reasoning is the biggest 1 digit number squared is 2 digits, then why do we then assume a long division protocol for it. I don't get why it works, I want to know why

Answer 25

ok

start with x^2 we are finding x^2 that is closest to the required values right? then subtracting the difference

Answer 26

(x+y)^2 = 1 42

x^2 + 2xy + y^2 = 1 42

x^2 + (2x + y)y = 1 42

(2x + y)y = 1 42 - x^2
but x^2 is a positional placement,

Answer 27

But that doesn't explain why that works. You are performing a function not on the actual numbers as you are splitting it up, it becomes \(4300+51+.23+.004\) then finding the closest square to 4300. But I mean, how does that translate to (x+y)^2 then finding x and y

Answer 28

x^2*100 is our positional value

Answer 29

"sqrt(4351.234)

now the idea is that the biggest perfect square of a single digit is 9*9 = 81, 2 positions

from the decimal start partitioning it into 2 digit parts


sqrt(43 51. 23 40)

now we are going to work off of (x+y)^2 = x^2 + 2xy + y^2
"

Why do we do this? Why does it work?

Answer 30

10(2x) + 1y is our other positional setup, for the subsequent findings

Answer 31

I don't understand what you are saying. I don't understand your terminology

Answer 32

ugh ..... we dont have just "numbers" they are working on positions as well.

as you noted: 42 isnt 42, its actually 4200

x^2 just gets us to 81 tops, x^2(100) is positionally correct

Answer 33

I've never heard of positionally correct

Answer 34

when we long hand divide:

------
3 | 234


we find 7*3 = 21

but its not 7*3 we are finding, its 70*3