Why is the Square Root of a Real Number Positive?

In summary, the conversation discusses why the square root of a real number is always positive, even though every number has two square roots. The reason is that the principal square root function is defined to give the positive value that when squared gives the original number. This is done to differentiate between the two roots, with the notation for both being \pm \sqrt{x}. The link provided in the conversation contradicts this convention.
  • #1
brt
1
0
Hi,

Apologies for the trivialness of the question, but I'm not so great at this. I was wondering why the square root of a real number is positive. Why is sqrt(9) = 3, and not -3 as well, since (-3)² would give 9. Is it just a condition you set, that the function values must be positive? At least I thought it was, googling for it produces sites that tell the opposite, such as this one: http://thesaurus.maths.org/mmkb/entry.html?action=entryByConcept&id=1015 . Are they wrong?
 
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  • #2
The principal square root function of x is defined to give the positive value that when squared gives x. It is only a convention.
 
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  • #3
every number has two square roots, however to differentiate between them, [tex]\sqrt{x}[/tex] is defined to be positive. The notation for both roots is [tex]\pm \sqrt{X}[/tex]
 

Related to Why is the Square Root of a Real Number Positive?

1. Why is the square root of a positive real number always positive?

The square root of a number is defined as the number that, when multiplied by itself, gives the original number. Since a positive number multiplied by a positive number will always result in a positive number, the square root of a positive real number will also be positive.

2. Can the square root of a real number be negative?

Yes, the square root of a negative real number is a complex number. However, when dealing with only real numbers, the square root of a real number will always be positive.

3. Is the square root of 0 positive or negative?

The square root of 0 is neither positive nor negative. It is considered to be a special case and is defined as 0. This is because 0 multiplied by itself results in 0, so there is no positive or negative value that can give 0 as the answer.

4. Why is the square root of a negative number undefined?

The square root of a negative number is undefined because there is no real number that, when multiplied by itself, will result in a negative number. This is why we use complex numbers to represent the square root of a negative number.

5. Can the square root of a real number be 0?

Yes, the square root of 0 is defined as 0. This is because 0 multiplied by itself results in 0, so the square root of 0 is the number that, when multiplied by itself, gives 0 as the answer.

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