What is the actual purpose of a square root? I understand what a square root is and how it is used, but my question is why does it exist at all when it is basically a question that has to be answered. Why not simply write the answer?

There's no way to write the answer to a square root for advanced functions and complex numbers.

I am not sure I understand your question. Are you asking why instead of root(9) we don't simply write 3? We can of course do that, but in more abstract contexts square roots and similar expressions appear as functions of variables, not specific numbers. If you have something like root(x) you replace this expression by a number unless you specify what x is.

Ah shit yeah you're right I shouldn't have included "complex numbers" in there.

Feel free to write down the square root of 2 in decimals. Except you can't, not precisely, since it's an irrational number. Suddenly the square root is quite handy.

Otoh when the square roots are simple rational numbers (at least when they are integers) people tend to forgo the square root and just write the answer.

You want a natural motivation. There are probably many. Here is my favorite.

Pythagoras: a^2+b^2=c^2

So if you know two special sides of special triangle (a,b) and you want to compute c you have to compute the square root. And if you want to give a general formula, applicable to all possible values of variables a and b, you must write a symbol for the square root, which I can't write here, so I will write text:

c = sqrt(a^2+b^2)

Voila, you need to write sqrt.

I think what ThomasPierson is asking (and even if I'm wrong, I'd like to know) is why is it important to know a square root? Like, how is it used in a productive way? I get all the geometry & such, but how is it used to make our lives better?

You could just as well ask how does addition make our lives better. Or multiplication. Hell, even calculus. That's almost impossible to answer, because the answer is ridiculously broad. Mathematics is used all the time in all fields of activity known to mankind. A square root is just one of its basic tools.

Look around the room where you're currently sitting. For 99% of things you see there, a square root was involved in their creation in some way or another, be it in calculating their shapes, modelling their functions, setting up prices, what have you.

Yep, D^2= X^2 + Y^2 + Z^2 or really the difference between two points x,y,,z. But the forum software doesn't really manage that very well.

I hate people who ask that question. Seriously. So congratulations, I hate you now :-P

And I always thought square roots were for rabbits that didn't like round carrots.
Oh well, no wonder I never passed math exam. :(

Are we doing somebody's homework again?

and sqrt are very important in all aspects in trigonometry calculations and all disciplines derived from it.

The answer is clearly that square roots stack more efficiently than round roots. But that round roots are more efficient in terms of perimeter if you have limited materials.

Technically, complex numbers provide a simpler notation for square root of -1, so you can simplify it. The real answer are algebra (working with variables instead of actual number) and irrational numbers (infinite digits after the .).

Square root of 2 is the poster child for this.

If you resolve square roots at the earliest moment, you'll lose accuracy if it results in an irrational number (because you'll have to truncate some digits).

However, if you postpone resolving the square root until the last possible moment, you can reduce error and sometimes even maintain perfect accuracy (if you end up applying exponent 2 to it later on for example).

Here's a simplistic computer science example for it:

var X;

<Read X from a file, where in a particular case, X happens to be inputed as sqr(2)>

Y = X^2;

A naive way of implementing this would be to resolve the square root when you read the input and lose accuracy.

However, if you postpone resolving the square root until the last possible moment, then Y will resolve to the following:

Y = X^2.

Given X = sqr(2), Y = sqr(2)^2 = 2

I rest my case.

That is a good point. Beside maintaining best possible accuracy, sometimes, postponing calculation of a square root allows you to avoid calculating it at all, resulting in greater efficiency.


EDIT:

I'd also like to add that giving a square root rather than a truncation is giving more complete information.

Suppose for example that you are working in a team and the result of a calculation that everyone will use is sqr(2).

You could round it to the second digit, third digit, fourth digit or even 10th digit after the dot and give it to your partners.

However, each might make a different usage of the result and might need to round it at different digits to get the accuracy that they need.

By rounding it at an arbitrary digit and giving them the result, you make that decision for them.

It's a bad habit that is often encouraged in high school that you should always simplify everything to a decimal number.

There is nothing wrong with giving sqr(2) as an answer to your teacher.

In fact, it's preferable over rounding it yourself.