Between

Pronunciation: /bɪˈtwin/ ?

Point B is between two other points A and C if it is on line segment AC[1]. In his book "The Foundations of Geometry"[2], German mathematician David Hilbert organized the concept of between into five axioms called the Axioms of Order.

Axioms of Order
AxiomManipulativeDiscussion
II,1
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Manipulative 1: Axiom of order II,1. Click on the blue points in the manipulative to change the figure.
"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."[2]
This tells us that if point B is between A and C, then it is also between points C and A.
II,2
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Manipulative 2: Axiom of order II,2. 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."[2]
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
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Manipulative 3: Axiom of order II,3. 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."[2]
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
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Manipulative 4: Axiom of order II,4. 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."[2]
This axiom establishes that point on a line can always be put in order.
II,5
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Manipulative 5: Axiom of order II,5. 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, 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."[2]
This axiom establishes that a straight line that intersects one side of a triangle, must also intersect a second side of the same triangle.

More Information

  • McAdams, David E.. Segment Addition Postulate. allmathwords.org. Life is a Story Problem LLC. 2010-03-16. http://www.allmathwords.org/article.aspx?lang=en&id=Segment Addition Postulate.

Cite this article as:


Between. 2010-01-08. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/b/between.html.

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Revision History


2010-01-08: Rewrote article from point of view of "Foundations of Modern Geometry". (McAdams, David.)
2008-11-20: Initial version (McAdams, David.)

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