Twelve definitions of a square

How does mathematics define a math concept?

Definitions of concepts in mathematics are different from definitions of concepts in other discipline or subject area. A definition of a concept in mathematics give a list of properties of that concept. A mathematics object will only be an example of that concept if it fits ALL those requirements, not just most of them. Further, a definition is also stated in a way that the concept being defined belongs to an already ‘well-defined’ concept. On top of this, economy of words and symbols and properties are highly observed.

Does a math concept only have one definition? Of course, not. A concept can be defined in different ways, depending on your knowledge about other math objects. In a study by Zaskin and Leikin, they suggested that the definitions students give about a concept mirrors their knowledge of mathematics. Below are examples of definitions of squares from that research. Do you think they are all legitimate definitions?

What is a square?

A square is

1. a regular polygon with four sides
2. a quadrilateral with all the angles and all the sides are equal
3. a quadrilateral with all the sides equal and an angle of 90 degrees
4. a rectangle with equal sides
5. a rectangle with perpendicular diagonals
6. a rhombus with equal angles
7. a rhombus with equal diagonals
8. a parallelogram with equal adjacent angles and equal adjacent sides
9. a parallelogram with equal and perpendicular diagonals
10. a quadrilateral having 4 symmetry axes
11. a quadrilateral symmetric under rotation by 90 degrees
12. the locus of all the points in a plane for which the sum of the distances from two given perpendicular lines is constant. Click this link to visualize #12.

Making (not stating) definitions is a worthwhile assessment task.

Here’s three great references for definitions of mathematical concepts. The first is from no other than Dr. Math (The Math Forum Drexel University). The middle one’s for mom and kids – G is for Google and the third’s a book of definitions for scientists and engineers.

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