I was once told that for all real x,
Square root of (x^2) = modulus (x)
Thus if e.g. x = -2, x^2 = 4,
then square root of 4 would be modulus (-2) = 2
I was wondering if the above is the correct definition of a square root. If so, then the definition would imply that square roots always yield a positive number (i.e. although (-2)^2 = 4, square root of 4 = 2).
Please correct me if I’m wrong!
That’s certainly what we mean colloquially when we say “the square root.” But every number of course has two distinct square roots (except for zero, which brings up the minor nitpick that the principle square root is never negative rather than always positive). For some useful information, see here.
Hmm…I always thought MOD() was shorthand for a function defined for taking the modulus of a number. Specifically, I used to use MOD() to tell me the remainder from a division problem. For example, 5 MOD(5) = 0 because there would be no remainder. Likewise, 10 MOD(5) = 0, as well. However, 12 MOD(5) = 2, for example.
It is useful in some BASIC programs I used to write to predict when a very specific event of interest to me (in astronomy) would occur…as a sub-calculation to a larger program of astronomical events. Also, IIRC: Truncation, known as TRUNC(), was a mathematical function serving as the counterpart to MOD() yielding not the remainder, but the whole number of the answer to a division problem…truncating off the decimal portion.
Neither MOD() not TRUNC() yielded solutions to square roots…
Isn’t this correct? - Jinx
Modulus is an alternate term for absolute value.
It’s a bit of a convenient fiction that a number has only one square root, or cube root, or nth root. For the purposes of real variable calculus, a function must be single-valued, so we take the positive value of the square root.
Now, once you get into complex variable calculus, you can’t ignore the other values, but that gets really ugly really fast, so let’s leave that as is.
But ultrafilter, you can’t ignore the other roots even when doing simple high school algebra. They pound into your head that if you have a square root, you need to keep in mind that there can be up to two (real) roots.
Ultrafilter, just wondering… What do you know about my other meaning for Modulus, in math? How do mathematicians distinguish between these functions? - Jinx
There are quite a few places in mathematics where terms have two (or more) conflicting meanings. Generally, one has to pick the appropriate meaning from the context. A good author will ensure the context is unambiguous, or supply an explicit definition.
In the real world, it’s often convenient to take the positive square (or nth) root and discard the negative. For example, a square room is 36 square feet - how long is each wall? 6 feet of course. The “-6” answer doesn’t have a real world interpretation - how long is “-6 feet” as a distance? (as a displacement, perhaps, but that’s another story). For a real world interpretation of “-6 feet”, without cheating and interpreting the minus sign as a direction, you’ll need a science fiction writer 
“modulus” referring to absolute value is pretty rare, IME, except when you get to complex numbers. Your mod function is one of the more important things out there.
There’s usually very little room for confusion, as there are symbols–|z| and a mod b–for each operation, and the context makes them clear.
Yes, but when you’re doing calculus, sqrt(x) means the positive root. That’s what I had in mind.