How many fixed points does a Kelvin scale have? I have a book that says:

In the absolute Kelvin scale, the triple point of water is assigned the value of 273.16 K. The absolute zero is taken as the other fixed point.

But, then another section in the same book says:

On the Kelvin scale, the lower fixed point is taken as 273.15 K and the upper fixed point as 373.15 K.

What does this all mean? Can anyone clarify?
All I want to know is: How many fixed points does a Kelvin scale have?
 A: A linear scale for measuring something - temperature, pressure, whatever - has to have two fixed points, because two points are necessary and sufficient to define a line.
If a scale is defined with more than two fixed points, then calibration using any two of them is expected to produce the same result (up to experimental error), and which two are used is a matter of convenience.  The SI definition of Kelvin only has two fixed points (0K and 273.16K = the triple point of water), but the ITS-90 thermometer calibration standard adds 13 more, because most thermometers don't include both 0K and 273.16K in their range, and it's useful to have fixed points in the middle as well as near the ends of a thermometer's range, to verify linearity.
Your book seems to be making a distinction between "Kelvin" and "absolute Kelvin". There is no such distinction; they are the same thing.  I think it is (sloppily) trying to emphasize the historical development of the temperature scale, which started out defined as "Celsius shifted so that zero lies at absolute zero" and thus had Celsius's fixed points embedded in its definition, plus absolute zero.
Note also that "the boiling point of water" is deliberately not used in either the SI definition or ITS-90 specification, because it depends on the ambient pressure (whereas "the triple point of water" fixes the ambient pressure).
A: The Kelvin scale is based on the Celsius scale in which 0 °C and 100 °C were defined to be the freezing and boiling points of water.
When the absolute minimum temperature was discovered to be -273 °C, scientists began simplifying matters by simply adding 273 and using the Kelvin scale in which 0 °C becomes 273 K. By defining the Kelvin scale in terms of absolute zero and a 273 K melting point, the 373 K boiling point is fixed incidentally.
A: In a sense, there is only one fixed point. Look to the definition (http://www.bipm.org/en/CGPM/db/13/4):

The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.

There's a lot hiding in that simple definition. One key concept is that of thermodynamic temperature. A temperature scale that does not have zero at absolute zero is not a thermodynamic temperature scale. So in this sense, the Kelvin scale has two fixed points, absolute zero and the triple point of water.
Another issue hiding in that definition is that it uses the triple point of water. Purified water from an East African lake, from the oceans, or Antarctic ice have slightly different triple points, slightly different freezing points, and slightly different boiling points. The isotopic composition of water varies with latitude. To get around this issue, the water used in determining the triple point of water must have a very specific isotopic composition specified by the Vienna Standard Mean Ocean Water (VSMOW) standard.


But , then another section in the same book says :
On the Kelvin Scale, the lower fixed point is taken as 273.15 K and the upper fixed point as 373.15 K.

This portion of your text is extremely outdated. The triple point of water has been used as the sole fixed point in the SI temperature scale since 1954. Using the freezing and boiling point of water as defining temperature is a discarded concept. Textbooks are oftentimes a decade or so out-of-date; that happens all the time. On the other hand, a textbook that is over six decades out-of-date is unacceptable.
Even using the triple point of water as a fixed point will most likely soon be discarded, at least in theory. If all goes according to plan, the only fixed point will be absolute zero. Temperature in the International System of Units will soon be defined by making the Boltzmann constant a defined value.
In practice, measuring temperature is rather difficult. The International Temperature Scale, which attempts to be a practical realization of the SI concept of the kelvin, has 14 different fixed points (excluding absolute zero). The melting and boiling points of water are not amongst those fixed points. The triple point of water (VSMOW) is.
A: The definition for a Kelvin scale I was given in Thermodynamics was:


*

*0 on the scale is absolute zero.

*The scale is the same as Celsius system.
Thus this definition defines this linear scale through the point slope method and not the two fixed points method. So there is only 1 fixed point.
