SO SIMPLE, YET SO HARD: THE SQUARE ROOT DRAMA

Ken Abbott, who published his opinion on (Chowdhury, 2014) by the 8^th of January of 2014, did make us think about the square root drama:

“The fallacy occurs immediately at the second line "1= Sqrt(1)".
  The function Sqrt(1) has 2 values, one is 1 and the other is -1.
  So you simple CANNOT say "1= Sqrt(1)". It's just not true!”

We indeed learned that the square root of a number was the number that multiplied by itself would give us the radicand (A), just like we see at (Math.com, 2000-2005). 

We obviously would have to accept then that both -3 and 3 are square roots of 9.

However, when we learned the definition of square root, we were on our Year 1 at school, we reckon.


That means that we had not seen negative numbers yet.

We are sure that we learned that taking the square root was an operation, therefore something that should return only one result.

We did find a few websites, such as Dr. Math's (Drexel University, 1994-2014), that say the expected, that 3 is the square root of 9, but we also found a few websites, such as Wolfram's (Weisstein, 1999-2004), where they say that both 3 and -3 are square roots of 9.

This debate made us doubt both our learning and our thinking.

We went after more.

We found very simple arguments (the calculator gives us 3 with no hesitation!), such as the one used by Prof. Art DiVito (Harold Washington College: One more person to post their opinion at (Chowdhury, 2014), one more trying to explain that the result is only the non-negative one):

“Some folks and books will use y = arc sin(x) to mean the relation x = sin(y), and then call this a “multi-valued function.” Then they go on to say that y = Arc sin(x) is the usual arc sin function: x = sin(y) AND -π/2 ≤ x ≤ π/2. But this is really bad form. Firstly, “multi-valued function” is an oxymoron. Secondly, notations should be reserved for functions. That is why, for example, when you ask your calculator for sqrt(4), it returns 2. It doesn’t flash back and forth between 2 and -2.”

All non-specialized dictionaries that we have consulted returned (A) as a world reference for square root (see, for instance, (Merriam-Webster, 2014)).

(Hiob, 2006) finally speaks like the people from Brazil, for instance, and establishes that the square root of a positive number is the positive number that squared returns the positive number inside of the radical, that is, the radicand.

The convention we have is that x^2 = 36, for instance, implies that x=6 or x= -6, but sqrt(36)=6.

To be sincere with you, after reading all the above over and over, we saw ourselves experiencing a bit of conflict.

If we raise 36 to 1/2, however, and that should be the same as doing sqrt(36), it seems that we have no doubts: 6 is the answer.

This means that the way we present the information makes a huge difference: When we tell people that square root is the number that multiplied by itself gives the radicand, almost nobody can agree that the square root of 36 is only 6 after they have contact with the negative integers. When we tell them that the square root of a number is a fractional power, and that it is actually the power ½, however, they can understand immediately that the square root of 36 is only 6 because, first of all, they can physically cancel the power 2 of the number 6 with the 2 in the denominator of the power they initially see:

(36)^1/2 = (6²)^1/2

However, 2 x ½ =1 and, therefore, (36)^1/2 = (6²)^1/2 =  6¹ = 6.

We then believe that we should never define square root as the number that multiplied by itself would give us the radicand.

We now believe that the sequence created by the people from Modern Mathematics should be slightly changed therefore.

Square Root should always be presented as a function, as an exponential of power 1/2.


A more technical reason for the even root of a non-negative number not to be both negative and positive is that multiplication is a commutative operation and, after we write the root as a power, we could be thinking of swapping the exponents, what would then create inconsistency, since there is no even root of negative numbers. 

(36)^1/2 = ((-6)²)^1/2 = ((-6)^1/2)² 

Sqrt(-6) does not exist in the set of the real numbers, so that this is not a real number, and we are obliged to say it is unsolvable. 

(36)^1/2 = (6²)^1/2 = 6
We would then have two possible answers, conflicting answers, to the same operation. 
With this, even roots would not be considered mathematical operations, since we must have only one corresponding image for each element of the domain to have an operation. 
We have defined them as mathematical operations, however.


For matters of consistency, we are then obliged to never allow for the negative answer to exist in operations of the just-mentioned type, involving even roots of non-negative numbers.

References:

Chowdhury, S. R. (2014). Tell me if I am wrong. Retrieved January 10 2014 from http://www.linkedin.com/groupAnswers?viewQuestionAndAnswers=&discussionID=5825118495100121089&gid=158723&commentID=5827082260955828224&trk=view_disc&fromEmail=&ut=2uAsl9pU66km41

 

Math.com. (2000-2005). Square roots. Retrieved January 10 2014 from http://www.math.com/school/subject1/lessons/S1U1L9GL.html

Drexel University. (1994-2014). Square Roots Without a Calculator. Retrieved January 10 2014 from http://mathforum.org/dr.math/faq/faq.sqrt.by.hand.html

 

Weisstein, E. W. (1999-2004). Square Root. Retrieved January 10 2014 from http://mathworld.wolfram.com/SquareRoot.html

 

Merriam-Webster. (2014). Square root. Retrieved January 10 2014 from http://www.merriam-webster.com/dictionary/square%20root

Hiob, E. (2006). The square root function. Retrieved January 10 2014 from http://mathonweb.com/help_ebook/html/functions_5.htm

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