POC: A common core method of square roots proved wrong.

[REDACTED]
9/29/15
Proof of concept: Incorrect square root formula

My name is [REDACTED], and as I was having a discussion with [REDACTED] regarding square roots (I am currently doing them in geometry), I found that the formula for getting the square root of an imperfect square varied.

After having [REDACTED] explain the formula, I tested it more in depth, and found that it gives you a very close answer to the square root, but not the real one, proving that the formula, is false when talking about the actual definite square root.

Below I will work the same problem with both formulas, then work them backwards to check the work.

Method explained by [REDACTED]:
Find the two closest square roots
Square root the one less than the target square root
Subtract the lower square from the target square and use the square of the lower as your whole number, and the leftover of the subtraction as your numerator
Subtract the lower square from the higher square, the difference being the denominator

Finding the square root of 27
√27
The two closest perfect roots to 27, would be 25 and 36.
√25 = 5
√27 - √25 = 2
5 being our whole number, 2 being our numerator
5 2/?
√36 - √25 = 11

5 2/11 is our final answer according to this formula

Let’s check our work.

2/11 is .18 repeating, so it is safe to assume that we can call this 5.18

By our math 5 2/11 AKA 5.18, is the square root of 27

5.18 * 5.18 or 5.18^2 = 26.8324

We can see that it is NOT the square root of 27, but rather very close.

Let us try with another number, let’s say 48.
√48
The two closest perfect squares are 36 and 49
√36 = 6, √48 - √36 being 12
6 12/?
√49 - √36 = 13
6 12/13

Now again, let’s work backwards.
12/13 = 0.9230769230769231
6.9230769230769231
6.923076923076923^2 or 6.923076923076923 * 6.923076923076923 = 47.92899408284024

Again, not the actual square root of the number, close, but still wrong.


Now I will demonstrate  & explain the method that I have learned through my years of schooling, that works 100% correctly.

[REDACTED]’s (My) method:
Factor the square root under the radical into a perfect square times a number EG: √18 being √9*2
Square root the perfect square, and leave the imperfect square under the radical
That is your answer, and is correct.

Let’s find the square root of 27 using this method.
√27
√9*3
3√3

3√3 is our answer, let’s check that.

√3 = 1.732050807568877
3*1.732050807568877 = 5.196152422706632
5.196152422706632 * 5.196152422706632 or 5.196152422706632^2 = 27 exactly

Again, let’s try it with 48.
√48
√16*3
4√3

Let’s check our answer.

√3 = 1.732050807568877
4*1.732050807568877 = 6.928203230275508
6.928203230275508^2 or 6.928203230275508*6.928203230275508 = 47.99999999999998, which would round up, but again, is far closer, and makes sense.

In conclusion, I have found the formula explained by [REDACTED] to be inconsistent, faulty, and rather hard to explain and hard to understand, as well as it takes far more steps to complete, and in geometry, real world math, wouldn’t apply.

The method creates confusion and inconsistency in mathematical formulas, the formula I have demonstrated is, from what I understand explained by my own teacher, is how Pythagoras himself explained it, but again do not quote me on that, but I would view it as being logical to believe that statement, as it bares a closer resemblance to the Pythagorean Theorem.