Square Root Polarity

By Mike Turnbull

Table of Contents

1. Introduction
2. Square Roots of a Real Numerical Value
3. Square Roots of Positive Real Variables of Unknown Absolute Numerical Value
4. Square Root of the Square of Real Variables of Unknown Polarity and Absolute Numerical Value
5. he Quadratic Formula as a Practical Example

Introduction

We are taught at school that, for instance, the square root of (say) $4$ , is $2$ - because $2·2=4$ - obviously. However, it is also true that $-2·-2=4$ . So, in reality, the square root of $4$ can be either $+2$ or $-2$ . That is, the square root of $4$ is equal to $\pm 2$ .

However, there may be particular situations where the square root of an algabraic term, in the context of the situation, can only be positive, or can only be negative.

In this paper we will discuss what the polarity of the square roots of various numerical values or algabraic terms can or cannot be.

Square Roots of a Real Positive Numerical Values

As briefly stated in the introduction, either $+2$ or $-2$ can be square roots of $4$ . That is, a, rather than the square root of $4$ is equal to $\pm 2$ - unless the context of the overall mathematical statement that includes that numerical value suggests otherwise.

Therefore both $2$ and $-2$ are square roots of $4$ , because $2^2=-2^2=4$ ; and this situation can be extended to include all other known positive real numerical values. The Square Root of the Square of Real Values of Negative Polarity

Square Roots of Positive Real Variables of Unknown Absolute Numerical Value

In mathematics a square root of a positive real variable of unknown absolute value (that is, unspecified, or currently indeterminate in the context of the mathematical statement) number $x$ is any number $y$ such that $y^2=x$ ; in other words, any number $y$ whose square (that is, the result of multiplying the number by itself $y·y$ ), is equal to the positive real number of unknown or unspecified absolute value $x$ , and this includes the positive and the negative values of $y$ . therefore when $x$ is a positive real variable of unknown, unspecified, or indeterminate absolute value:

Square Root of the Square of Real Variables of Unknown Polarity and Absolute Numerical Value

The wording in what follows is somewhat convoluted, so it needs to be read quite carefully; or, otherwise, the meaning may be lost.

Consider the situation in the processing of algebraic development where we start with a real variable of indeterminate sign and absolute numerical value. For the purooses of this discussion we will label this variable $x$ .

Throughout the algebraic development this variable may take on any real value and either polarity; however, even though, at any particular instance the sign and value are, in the global sense, indeterminate, in the intrinsic instantaneous context this variable must, by its very nature, actually have a definite sign, and a definite numerical value.

Consider the situation where, as part of the algebraic development, this variable is first squared and then, some time following, becomes part of a square root - either on its own or in combination with other real numbers or variables (either known or indeterminate).

Considering the variable in question on its own.

Following the squaring and following square rooting of this variable it would be illogical for the end result to be anytning but the original variable - both in sign and absolute value! This may be expressed algebraically as follows.

This precludes the possibility that the the resulting variable will have any polarity other than that of the original variable. If the original variable was positive, then the resulting variable will also be positive. If the original variable was negative, then the resulting variable will also be negative.

The Quadratic Formula as a Practical Example

A very good example of where the principles described above come into play is the algebraic development of the Quadratic Formula, and the reader is encouraged to study that example.

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