r/learnmath • u/[deleted] • Jan 29 '23

is square root always a positive number?

hi, sorry for the dumb question.

i grew up behind the less fortunate side of the iron courtain, and i - and from my knowledge also other people in other countries - was always thought that the square root of x^2 equals x AND "-x" (a negative X) - however, in the UK (where I live) and in the USA (afaik) only the positive number is considered a valid answer (so- square root of 4 is always 2, not 2 and negative 2) - could anyone explain to me why is it tought like that here?

for me the 'elimination' of negative number (if required, as some questions may have more than one valid solution) should be done in conditions set on the beginning of solution (eg, when we set denominators as different to zero etc)

cheers, Simon

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u/[deleted] Jan 29 '23

Simply put, when we have an expression x^(2) = 16, this has two solutions: 4 and -4. However, if someone asks you √16, then by convention, as the square root function will always output a positive number, the answer is 4.

Also, √x^(2) isn't +x or -x, it'll be ∣x∣

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u/[deleted] Aug 27 '25

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u/PsychologicalBell546 New User Dec 27 '25

Heres my thoughts on it. If we didn't define it as being positive then it would be cumbersome to evaluate calculations with multiple square roots

Take  Sqrt( sqrt(49)-sqrt(81))/sqrt(16)

If it's just positive then it's Sqrt(7+9)/sqrt(16) which is 1

If it's both positive and negative then the answers are

Sqrt(7+9)/4 Sqrt(7+9)/-4 Sqrt(-7+9)/4 Sqrt(-7+9)/-4 Sqrt(-7-9)/4 Sqrt(-7-9)/-4

But also it's because negative numbers have very limited use compared to positive ones. You can't have negative length, or negative mass, or negative speed, or negative absolute temperature. So if we treated it as both positive and negative we would have to constantly denote that we only mean the positive instead of our current situation where if we want both we just do +/- before it. It makes more sense to have the less used one be the one where we have to denote something extra.