# Is there a particular reason why segment addition postulate and partition postulate are two different things?

I could be wrong but those two ideas sound the same, just that the partition postulate is more general. There is also the angle addition postulate.

The segment addition postulate states that if three points A, B, and C are collinear such that B lies between A and C, then the sum of the lengths of segment AB and segment BC is equal to the length of the entire segment AC.

Partition postulate states that the whole is equal to the sum of its parts

For context, I learned the partition postulate and not the segment addition postulate

• The reason why an axiom, instead of an alternative, belongs to an axiomatic system depends on the axiomatic system and not just the axiom in isolation. That makes an answer difficult. --However, usually in an axiomatic system, one seeks axioms that are simple (not compound). I don't think "The whole is equal to the sum of its parts" qualifies as simple. -- BTW, this seems a math. logic question, not a question about teaching. -- Commented Oct 3, 2023 at 23:31
• Is the partition postulate supposed to state something about "wholes" that are not necessarily segments, such as angles (interior angle) or even areas or solids? Commented Oct 5, 2023 at 15:15
• well partition postulate holds that measures are additive (this is coming from measure theory) Commented Oct 9, 2023 at 15:16

In the Elements, Euclid did not use lengths. But in contemporary high school geometry, we typically find lengths of segments being represented as real numbers. The "ruler postulate" was introduced in the nineteenth or twentieth century. It states that the points on a line can be put into correspondence with the real numbers. It is only in this "modern" context that the segment addition postulate makes any sense. Similarly, there is a "protractor postulate" to put the measures of angles into correspondence with real numbers.

I think the difference between the "segment addition postulate" and the "partition postulate" is that the partition postulate fits best when discussing how Euclid framed geometry in the Elements, whereas in Euclidean geometry with the ruler postulate in play, the partition postulate becomes updated in the form of segment addition postulate.

Chapter 3 of "Geometry: Euclid and Beyond" by Robin Hartshorne, it is good reference.

• OH that makes sense - I remember Birkhoff/SMSG axioms now Commented Oct 11, 2023 at 22:45

Without going down into the rabbit hole of axiomatic systems, I think the high-level answer you're looking for is that

1. yes, the segment addition postulate is a more specific case of the partition postulate, and

2. the reason why a geometry course might present both postulates is that the segment addition postulate clarifies what is meant in a case where the partition postulate is vague enough to lead to ambiguity.

To elaborate on item 2 -- it could be tempting to claim that the partition postulate supports the following statement, which is not true in general:

Given three points A, B, and C, the sum of the lengths of segment AB and segment BC is equal to the length of the entire segment AC.

The segment addition postulate specifies the additional conditions that are required for the statement to hold true in general:

Given three collinear points A, B, and C such that B lies between A and C, the sum of the lengths of segment AB and segment BC is equal to the length of the entire segment AC.

• Yes: the segment addition postulate is rigorous, whereas that the whole is the sum of its part can be understood to mean roughly whatever one wishes, true or not. Commented Oct 4, 2023 at 5:13