Why is equating these two symbols an error?

In summary, The conversation discusses the concept of square roots and the convention of using the symbol \pm when encountering a square root in an equation. The book "Calculus (7th ed)" explains that a number is called a square root of A if its square is A, and every positive real number has two square roots - one positive and one negative. The symbol \sqrt{a} always signifies a non-negative number, while -\sqrt{a} is always a non-positive number. The book also clarifies that the symbol \sqrt{a} is shorthand for "the principal square root of a" and equating it to \pm a is an error.
  • #1
EngWiPy
1,368
61
Hello,

My calculus book says that readers who are writting [tex]\sqrt{9}[/tex] as [tex]\pm3[/tex] must stop doing that, because it is incorrect. The question is: why is it incorrect?

Regards
 
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  • #2
Because the square root of a number A is DEFINED to be the unique, non-negative number whose square equals A.
 
  • #3
arildno said:
Because the square root of a number A is DEFINED to be the unique, non-negative number whose square equals A.

I didn't understand. For any real number a there are two square roots: a positive square root, and a negative square root. How is the square root is unique?
 
  • #4
When you encounter a square root in an equation, use the +/- thing. If somebody asks you, "What is the square root of four", say "two". Its just what mathematicians have decided we will mean, when "square root" is used in each of those contexts.
 
  • #5
S_David said:
I didn't understand. For any real number a there are two square roots: a positive square root, and a negative square root. How is the square root is unique?

Incorrect.

For any non-negative number "a", the equation:
[tex]x^{2}=a[/tex]
has two SOLUTIONS:
[tex]x_{1}=\sqrt{a},x_{2}=-\sqrt{a}[/tex]

The [itex]\sqrt{a}[/itex] is a non-negative number.
 
  • #6
To expand on arildno's point: S David, would you say that the solution to [itex]x^2= 5[/itex] is [itex]\sqrt{5}[/itex] or [itex]\pm \sqrt{5}[/itex]? I suspect you will say the latter and the point is that the whole reason we need the "[itex]\pm[/itex]" is because [itex]\sqrt{5}[/itex] itself only gives one of them: the positive root.
 
  • #7
arildno said:
Incorrect.

For any non-negative number "a", the equation:
[tex]x^{2}=a[/tex]
has two SOLUTIONS:
[tex]x_{1}=\sqrt{a},x_{2}=-\sqrt{a}[/tex]

The [itex]\sqrt{a}[/itex] is a non-negative number.

Referreing to the book whose name is: Calculus (7th ed), for Anton, Bivens, and Davis, Appendix B at the bottom of the page, it says the following:

Recall from algebra that a number is called a square root of A if its square is A. Recall also that every positive real number has two square roots, one positive and one negative; the positive square root is [tex]\sqrt{A}[/tex] and the negative square root is [tex]-\sqrt{A}[/tex]. For example, the positive square root of 9 is [tex]\sqrt{9}=3[/tex] and the negative square root of 9 is [tex]-\sqrt{9}=-3[/tex]

After this review, it says the thing I started with. Is this differ from what I said in post #3 in this thread?
 
  • #8
The SYMBOL [itex]\sqrt{a}[/itex] always signifies a non-negative number.

Therefore, [itex]-\sqrt{a}[/itex] is always a non-positive number.

Colloquially, we call this "the negative square root of a", whereas if we want to be über-precise, we ought to call it "the (additive) negative OF the square root of a"

(alternatively, "minus square root of a", in complete agreement of calling -2 for "minus two")
 
  • #9
The symbol [itex]\sqrt{9}[/itex] is shorthand for "the principal square root of 9" (not simply "a square root of 9") where the principal square root is a function. A function has only a single output for each input, therefore equating it to the symbols [itex]\pm 9[/itex] which is shorthand for the set {9, -9} is an error.
 

What is a square root?

A square root is the number that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5, because 5 x 5 = 25.

How do you find the square root of a number?

The most common way to find the square root of a number is to use a calculator. However, you can also estimate the square root by "guess and check" or use long division to find it.

What is the difference between a perfect square and a non-perfect square?

A perfect square is a number whose square root is a whole number. For example, 25 is a perfect square because its square root is 5. A non-perfect square is a number whose square root is not a whole number, such as 12 or 37.

What are some real-world applications of square roots?

Square roots are used in many fields, including engineering, physics, and finance. They can be used to calculate the length of the sides of a square or to find the distance between two points on a coordinate plane. In finance, square roots are used in the calculation of the standard deviation, a measure of risk in investments.

What is the difference between a positive and negative square root?

When finding the square root of a positive number, there are two possible answers: a positive and a negative square root. For example, the square root of 25 can be either 5 or -5. However, when finding the square root of a negative number, the answer is always imaginary, and it is denoted by the letter "i." For example, the square root of -25 is 5i.

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