Country Boy

For a given positive number, a, both x=√a and -x= -√a satisfy the equation [math]x^2= a[/math] and are called "square roots of a". The square root function, √x, however, in order to be a function, must have a single value and that is defined to be the positive value. "The" square root of x^2 is |x|.

No! Especially since you used the word "technically". Technically, [math]\sqrt{x^2}= |x|[/math]. That is x if [math]x\ge 0[/math] and -x if [math]x< 0[/math],

a= -2. a2= 4. What is the square root of 4? (Remember that the square root is a FUNCTION so can have only one value.)

Physics is the study of mass, motion, and the relations between them. Chemistry is the study of elements, compounds, and the relations between them. Economics is the study of goods, prices and the relations between them. ... Mathematics is the study of relations in the abstract.

All motion is relative to some fixed point. You van is traveling at 60 mph relative to the road. The fly is traveling at some much lower speed relative to the van. It speed relative to the road is very close to 60 mph (slightly more when it is flying forward, slightly less when it is flying back) but since it is flying in the enclosed air in the vehicle the fly doesn't notice it. (If the car were to accelerate the fly might wind up plastered to the back window- acceleration is not relative.)

You said that the train's "rest length" is 100 m. Then in the train's frame of reference the two lightening strikes are 100 m apart. Of course, in the track's frame of reference, the two lightening strikes are 1 m apart. Yes, that is the same question as the "tunnel" question. In the train's frame of reference the tunnel is 100 m long. In the track's frame of reference it is 1 m long. (Who would build a tunnel one meter long?!) I don't know what you mean by "abstract distance". Any distance, pretty much by definition, can be measured, as well by a meter stick as any other way.

No, there shouldn't. While, for any positive number, a, there exist two values of x, one positive and one negative, such that $x^2= a$, [tex]\sqrt{a}[/tex] is, by definition, the positive one only. That is why, when writing the solutions to

"Relatively stationary"? Relative to what? "Receding light source"? Receding from what? I assume you are referring to an observer who is, of course, stationary relative to himself, observing a light source receding relative to him. The measurement of the speed of light, with light from whatever source is probably the most repeated experiment in physics.

I think that ConciousEnergy, saying "in space" is confusing mathematics and physics. In mathematics a point is defined by its position. It has NO "area" or "volume". Talking about a "point in space" you are talking about physics and when you apply mathematics to physics it only matches approximately,

If 0/0= a then 0 X a= 0 is also "well defined". a/0, for non-zero a, is "undefined" because if we set a/0= b then a= bX0= 0 which is not true. 0/0 is not defined because if we set 0/0= b then 0= bX0 for any b. Many people say 0/0 is "undetermined" rather than "undefined". But there is no good reason to set it equal to 0.

Swansont: Yes, but like all such things it is not useful until everyone has agreed on a specific interpretation. That was done with "up" and "down" on earth long ago! Sirflappington: I would argue that in space there is NO "up" and "down"

Few people will be willing to down load multiple files with no idea what is in them. Is it really too much trouble to type your problem? Then I guess it is not very important.

It's NOT a law, it is a definition. A geometric point is not a physical object. It is an a mental concept that identifies a position and has no other properties.

B says that, at some time in the future, I will have made 3 cakes and that I willstill have all three. A say that, at some time in the future, I will have made 3 cakes but that I might have disposed of (ate, gave away, threw away) one or more of them,

11, our standar "decimal system" is base 10. We have 10 digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and create all numerals using those. For example immediately after "9" is 10= 1*10+ 0, then 11= 1*10+ 1, then 12= 1*10+ 2, etc. In "binary" we have only two digits, 0 and 1 and create all numerals using the. For example immediately after 1 is 10 which can be read as "2 and 0", then 11= 2+ 1= 3, then 100= 4+0*2+ 0*1, then 101= 4+ 0*2+ 1*1= 5, then 110= 4+ 1*2+ 0*1, etc. Notice that, just as 10,100, 1000, 10000 are "powers of 10" so in binary, 10= 2, 100= 4, 1000= 8 are "powers of 2".