# Between

Pronunciation: /bəˈtwin/ Explain

Point B is between two other points A and C if it is on line segment AC. In his book "The Foundations of Geometry", German mathematician David Hilbert organized the concept of between into five axioms called the Axioms of Order.

Axioms of Order
AxiomManipulativeDiscussion
II,1
 Click on the blue points and drag them the change the figure. Note that point B is between point A and C, and the B is between points C and A and B is distinct from A and C. Manipulative 1 - Hilbert Axiom ii1 Created with GeoGebra.
"If A, B, and C are points of a straight line and B lies between A and C, then B lies also between C and A."
This tells us that if point B is between A and C, then it is also between points C and A.
II,2
 Click on the blue points and drag them to change the figure. There always exists a point B on line AC such that C lies between A and B. Manipulative 2 - Hilbert Axiom ii2 Created with GeoGebra.
"If A and C are two points of a straight line, then there exists at least one point B lying between A and C and at least one point D so situated that C lies between A and D."
This axiom establishes that between any two points on a line, there exists another point. A consequence of this axiom is that an infinite number of points lie between any two points on a line.
II,3
 Click on the blue points and drag them to change the figure. Can you arrange the points on the line so that two of them are between other points? Manipulative 3 - Hilbert Axiom ii3 Created with GeoGebra.
"Of any three points situated on a straight line, there is always one and only one that lies between the other two."
This says that if point B lies between points A and C then point A can not lie between points B and C.
II,4
 Click on the blue points and drag them to change the figure This axiom establishes that it is possible to order points on a line. Manipulative 4 - Hilbert Axiom ii4 Created with GeoGebra.
"Any four points A, B, C, D of a straight line can always be arranged so that B shall lie between A and C and also between A and D, and, furthermore, that C shall lie between A and D and also between B and D."
This axiom establishes that points on a line can always be put in order.
II,5
 Click on the blue and red points and drag them to change the figure. Is it possible for the red line to cross one of the sides of the triangle without crossing either of the other sides? Manipulative 5 - Hilbert Axiom ii5 Created with GeoGebra.
"Let A, B, C be points not lying in the same straight line and let a be a straight line lying in the plane ABC and not passing through any of the points A, B, and C. Then, if a straight line passes through a point of segment AB, it will always pass through either a point of the segment BC or a point of the segment AC."
This axiom establishes that a straight line that intersects one side of a triangle, must also intersect a second side of the same triangle.

### References

1. McAdams, David E.. All Math Words Dictionary, between. 2nd Classroom edition 20150108-4799968. pg 25. Life is a Story Problem LLC. January 8, 2015. Buy the book

McAdams, David E. Between. 4/12/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/b/between.html.

### Revision History

4/12/2019: Corrected manipulative for axiom ii-2. (McAdams, David E.)
4/12/2019: Changed equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)