By Irving Adler

More than a hundred workouts with solutions and two hundred diagrams light up the textual content. lecturers, scholars (particularly these majoring in arithmetic education), and mathematically minded readers will delight in this extraordinary exploration of the position of geometry within the improvement of Western clinical thought.

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Locate on the vertical line the point whose scale number on this line is b. We use this point to represent the complex number a + bi. In the diagram above we show the points that represent the numbers 3, 2i, 2 + 2i, and 3 − 2i Finite Fields The rational number system, the real number system, and the complex number system are all fields. Each of these three fields contains infinitely many members. It is an interesting fact, first discovered by Evariste Galois (1811–1832) that there are also fields that contain only finitely many members.

The corresponding statements that express the information given in the second and the third lines are related to each other in the same way, that is, each is the dual of the other. The corresponding statement that expresses the information given in the first line is its own dual. What we have observed here is a foretaste of a more general symmetry known as duality which we shall discuss in Chapter 10. The duality revealed in the table is not accidental. It corresponds to a significant geometric relationship: The centers of the faces of a regular polyhedron are the vertices of another regular polyhedron inscribed in it.

5. x(yz) = (xy)z. (Associative Law of Multiplication) 6. xy = yx. (Commutative Law of Multiplication) 7. There exists a member 1 such that, for every x in the system, x · 1 = 1 · x = x. 8. If x ≠ 0, there exists a member denoted by that has the property 9. x(y + z) = xy + xz. (Distributive Law) Any system of elements for which two operations, addition and multiplication, are defined that satisfy conditions 1 to 9 is known as a field. The Real Number System The construction shown below locates a point P to the right of O whose distance from O is the length of the diagonal of a square whose side is 1 unit.