# Number Words and Number Symbols: A Cultural History of by Karl Menninger

By Karl Menninger

Vintage learn discusses quantity series and language and explores written numerals and computations in lots of cultures.

"The historian of arithmetic will locate a lot to curiosity him the following either within the contents and standpoint, whereas the informal reader may be intrigued via the author's enhanced narrative ability." — Library Journal. 282 illustrations.

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84 (1966), 537-554. _ : Algebraic classification of some knots of codimension .. two, Comm. Math. Helv. 45 (1970), 185-198. : Homology, Springer-Verlag, Berlin, 1963. : A duality theorem for Reidemeister torsion, Annals of Math. 76 (1962), 137-147. : Polynomial invariants and the integral homology of coverings of knots and links, Inv. Math. 15 (1972), 78-90. Trotter, H. : On the algebraic classification of Seifert matrices, Proceedings of the Georgia Topology Conference 1970, University of Georgia, 92-103.

The upper left corner of V(Pl,P2"") is then b+ 1 ] b+c+1 Now subtract the first row from the second, change the sign of the first row, and perform the corresponding column operations. The resulting matrix is integrally congruent to the original and has the 2 x 2 matrix a [ I b+ 1 a+ 1 in its upper left corner, and is otherwise unchanged. It is the same as V(P2' Pl'''') except that the entries a and a+1 are reversed in position. Jre unchanged [10]. The Alexander polynomial is det(tV - V'). Every non-zero term in the determinant of any tridiagonal matrix M must contain mi,i-t 1 if it contains mi+l,i' so the determinant is not affected if the two elements are exchanged.

S. , 75 (1969), 169-171. 7] 2-spheres in 4-s~ce, , The second homotopy group of Ann. of Math. 90 (1969), 199-204. 81 Lomonaco, S. , The second homotopy group of a spun knot, Topology, 8 (1969), 95-98. 91 Fox, R. , Free differential calculus I. Derivations in the free group ring, Ann. of Math. 57 (1953), 547-560. 10] Whitehead, J. H. , The secondary boundary operator, Proc. Nat. Acad. Sci. :J6 (1950), 55-60. _. ,A certain exact sequence, Ann. of Math. 52 (1950), 51-110. 12] Eilenberg, Samuel and Saunders Mac Lane, On the Groups H(7T, n), I, Ann.