When we deal with speeds, they are ratios of counting numbers, which means they are all rational numbers. This would be fractions of the kind 1/2, 2/5, 3/4, 7/6 etc...
When we plot this on the REAL number line, we get a number of gaps, left by the irrational numbers. Technically, there is no way we can get a speed of pi.
However, when we deal with the motion's representation, we say that they are shown as (x,y,z,t) or (s, tx, ty, tz) for the two alternative systems (spatial and temporal reference systems). These numbers are taken to be real, or even complex, both of which are continuous, and not discretized. Neither the ratio of two quantities, nor the quantities themselves, are really real!
So what about the irrationals?
Gopi
Rational numbers and real numbers
Rational numbers and real numbers
if there is a quantum of motion (a smallest possible time and/or motion) then this problem goes away. I know Ed Fre(i?)dkin addressed this, and Wolfram, too, but not recalling
whether Mr. Larson addressed it or not.
Mathematically, A ratio of reals is real, in general. No problem there.
In "reality", I don't think we have the time to measure to "an infinite number of digits", so there are no numbers that are 'real' in the mathematical sense.
Interesting stuff.
Gopi (email removed) wrote:
Quote:
whether Mr. Larson addressed it or not.
Mathematically, A ratio of reals is real, in general. No problem there.
In "reality", I don't think we have the time to measure to "an infinite number of digits", so there are no numbers that are 'real' in the mathematical sense.
Interesting stuff.
Gopi (email removed) wrote:
Quote:
When we deal with speeds, they are ratios of counting numbers, which means they are all rational numbers. This would be fractions of the kind 1/2, 2/5, 3/4, 7/6 etc...
When we plot this on the REAL number line, we get a number of gaps, left by the irrational numbers. Technically, there is no way we can get a speed of pi.
However, when we deal with the motion's representation, we say that they are shown as (x,y,z,t) or (s, tx, ty, tz) for the two alternative systems (spatial and temporal reference systems). These numbers are taken to be real, or even complex, both of which are continuous, and not discretized. Neither the ratio of two quantities, nor the quantities themselves, are really real!
So what about the irrationals?
Gopi
BE the change that you want to see in the world.
Re: Rational numbers and real numbers
Gopi wrote:
For example, take a rectangular sign, 3 feet tall by 4 feet wide. Stick it out in the sunlight, so it casts a shadow at some oblique angle. Now go measure the shadow--you'll find it is not 3x4 feet, probably an irrational, real measurement, and the 90-degree angles of the rectangle do not measure out to be 90 degrees any more. That's what projection does--distorts, creating an illusion (shadow) of reality (sign).
If you want to measure an integer quantity as PI, then all you have to do is tilt the observer at the correct angle to the line of measure; somewhere in there you'll see "3" sitting at "3.14159..."
This is why I consider it of prime importance to account for both the observer principle and projective geometry in the study of the RS. When you know you're looking at a shadow, you have a much better chance of understanding the motion casting it.
Gopi wrote:
Once you are in a coordinate reference frame (rectangular, polar, or other), you are looking at and measuring shadows, not the actual speeds. The distortions introduced produce the irrational numbers.
Once you introduce the concept of a "number line", then you have moved outside of the realm of scalar speeds and into the realm of projective geometry, since the speed is now projected upon the line, with its various assumptions (zero, infinity, dimension, direction, etc).When we plot this on the REAL number line, we get a number of gaps, left by the irrational numbers. Technically, there is no way we can get a speed of pi.
For example, take a rectangular sign, 3 feet tall by 4 feet wide. Stick it out in the sunlight, so it casts a shadow at some oblique angle. Now go measure the shadow--you'll find it is not 3x4 feet, probably an irrational, real measurement, and the 90-degree angles of the rectangle do not measure out to be 90 degrees any more. That's what projection does--distorts, creating an illusion (shadow) of reality (sign).
If you want to measure an integer quantity as PI, then all you have to do is tilt the observer at the correct angle to the line of measure; somewhere in there you'll see "3" sitting at "3.14159..."
This is why I consider it of prime importance to account for both the observer principle and projective geometry in the study of the RS. When you know you're looking at a shadow, you have a much better chance of understanding the motion casting it.
Gopi wrote:
To quote Kosh, "then listen to the music; not the song."However, when we deal with the motion's representation, we say that they are shown as (x,y,z,t) or (s, tx, ty, tz) for the two alternative systems (spatial and temporal reference systems). These numbers are taken to be real, or even complex, both of which are continuous, and not discretized. Neither the ratio of two quantities, nor the quantities themselves, are really real!
Once you are in a coordinate reference frame (rectangular, polar, or other), you are looking at and measuring shadows, not the actual speeds. The distortions introduced produce the irrational numbers.
Every dogma has its day...