Whats the square root of X?

Vader

You know how it is, your in the pub and everyone becomes a philosopher or some sort of fúcking genius and stupid questions are flying all over the place. Well its the next morning and Im wondering, what IS the square root of X?

I was thinking: Where Y and Z are factors of X, and Y=Z, then Y is the square root of X. Will that do?

ecksor

Originally posted by Vader
Well its the next morning and Im wondering, what IS the square root of X?

The word 'root' always brings to mind an equation to be solved to me, so I'd say that the square root of X is whatever value of Y solves Y^2 = X

I was thinking: Where Y and Z are factors of X, and Y=Z, then Y is the square root of X. Will that do?

Factors suggests whole numbers to me, so perhaps if Z*Y = X and Y=Z is closer to what you want, unless you do mean to restrict to the whole numbers.

DMT

The root of X is that number, which when multiplied by itself, equals X.

D-Generate

Would it not be +/- Y or +/- Z considering both are equal to each other?

DMT

The root of a number is always positive.

Syth

The root of X is that number, which when multiplied by itself, equals X.

That's a pretty good definition.

Would it not be +/- Y or +/- Z considering both are equal to each other?

Getting the square root of a number is applying the square root function to the number, which will always return the positive square root. The reason is something to do with the strict mathematical definition of a function which says that a function can only return one number. So you can't get two number back out of the function.

It can be confusing because when you have something like x^2 = 4 and x = sqrt(4), initially one thinks that they are the same, but they actucally aren't, because in one case x = 2 or -2 and in the other x = 2. So it's wrong to go from x^2 = 4 to x = sqrt(4).

If you want messed up question just think about irrational number like the square root of 2. You can't write it as a fraction or a recurring decimal, so you can never actually write down the digits of the sqaure root of 2, so does it really exist? Two corners of a square with length 1, are root 2 apart, so you can have the sqaure root of two, but you can't write it down, but it there ... (cue cycle to insanity)

Capt'n Midnight

Ignore for a moment the magnitude of a number when thinking of it's root.

Imagine a circle 360 degrees.
0,0 at the centre - one axis is real numbers the other axis is imaginary numbers.

If it was a unit circle - radius one
at 0 degrees (3 O'Clock) you have 1
at 90 degrees (6 O'clock) you have -i
at 180 (9) you have -1
at 270 degrees (12) you have +i

To get the square roots of a number divide the circle into 2 - the roots will be 180 degrees apart - to get the fourth root the numbers will be 90 degrees apart

eg: the fourth roots of 1 are 1,-1,+i , -i


The point being that there are N different Nth roots of a number.

============================
So THE square root of a number is 0

For every other number there are TWO square roots eg: 2 is A square root of 4 (it is not THE only one.)

Syth

For every other number there are TWO square roots eg: 2 is A square root of 4 (it is not THE only one.)

Course it's just semantics..,

ecksor

Originally posted by Capt'n Midnight
Imagine a circle 360 degrees.
0,0 at the centre - one axis is real numbers the other axis is imaginary numbers.

I believe this is a method for cutting a cake using the fundamental theorem of algebra?

I'm starting a complex analysis course at the moment and this stuff is pretty fukkin dedly.

There's a really nice introduction to complex numbers that includes the geometric interpretation of multiplication, division, roots etc. here: http://www.clarku.edu/~djoyce/complex/

Delphi91

Originally posted by DMT
The root of a number is always positive.

Untrue.

The graph of x vs root(x) is u-shaped curve, minimum at (0,0) and symmetric about the y-axis.

If you compare what you said above with your previous post, then youo'll see that the root of a number can be positive or negative.

E.g. Root(9) = +/- 3 because

1. (+3) x (+3) = 9
2. (-3) x (-3) = 9

I think what you mean to say is that the SQUARE of a number is always positive. You do run into problems, however, when you try to get the root of a negative number.

Mike

Delphi91

Originally posted by ecksor
....There's a really nice introduction to complex numbers that includes the geometric interpretation of multiplication, division, roots etc. here: http://www.clarku.edu/~djoyce/complex/

Read "Fermat's Last Theorem" by Simon Singh - its got one of the best introductions to numbers I've come across in ages, especially the number line, the coordinate system and imaginary numbers.

Mike

ecksor

Originally posted by Delphi91
Untrue.

I interpreted what DMT said as the definition of the square root function of a positive real as returning positive.

i.e, contrary to the habit that some have of writing sqrt(4) = +/- 2, the function actually returns just 2.

seamus

Originally posted by ecksor
I interpreted what DMT said as the definition of the square root function of a positive real as returning positive.

i.e, contrary to the habit that some have of writing sqrt(4) = +/- 2, the function actually returns just 2.

Is that not a dodgy area though? For all intents and purposes, it's comes out at 2, but for calculus, the ± part comes in to determine both arms of the line, should they exist.
I always remember that leaving out the ± lost a lot of marks in the Leaving because you were disregarding an entire half of an answer as it were.

ecksor

[ I don't know what DMT meant, I'm just saying what my interpretation was ]

f(x) = x^2 will draw a U graph as delphi91 describes but that's really a graph of the square of x, not of the square root of the y axis. +2 and -2 will both give 4 if you square them but that's thinking in the wrong direction too.

However, sqrt(4) = 2. Leaving cert mandates that you write down sqrt(4) = ±2 but if you think about it that's nonsense because the result of a function must be unique for a given input. I.e, it must be one-to-one or many-to-one. Given the history of the function, "what's the length of the side of a square witih area x" the positive result makes most sense.

I'm also not advocating that people not write down the ± on their leaving cert paper, they probably have their own good reasons for making you do that

Actually, I've just noticed that Syth posted similar.

ecksor

Of course, strictly speaking I've contradicted my first post.

I'm going to weasel out of that one by claiming that it depends upon who you're talking to (I probably wouldn't get a leaving cert student bogged down in this now if I was talking to them about it) and more importantly whether or not you restrict to integers, reals, naturals, positive reals, complex numbers but preferably something that makes that equation I posted not have more than one solution

Capt'n Midnight

Vader
You know how it is, your in the pub and everyone becomes a philosopher or some sort of fúcking genius and stupid questions are flying all over the place.

Originally posted by Syth
Course it's just semantics..,

Just a fancy word for pub talk

The really weird stuff happens with MrHawkings and calculations around the big bang, IIRC he ended up using complex numbers etc. - imaginary time and all that.

Another way of looking at square roots is removing a dimension, you take an area and convert it to a line (or convert a hypercube to an area - depending on how many dimensions you can juggle)

For a random walk - where you take a step of fixed size in a random direction after N steps you will be on average(?) (positive) root(N) away from where you started. Or in pub terms - if your central nervous system has been stewed by closing then it's going to take a long time to stagger home.