Casting Out Nines: An Explanation and Extensions

1990 ◽  
Vol 83 (8) ◽  
pp. 661-665
Author(s):  
Murray Lauber
Keyword(s):  
Middle Ages ◽  
Mathematics Curriculum ◽  
Western Culture ◽  
Computer Code ◽  
Modular Arithmetic ◽  
Mathematical Basis ◽  
Decimal System ◽  
Electronic Calculator ◽  
Mathematical Operations ◽  
The Middle Ages

The method of casting out 9s has been used for centuries, perhaps even as long as a millenium, for checking computations with integers involving the four mathematical operations. According to Eves (1980, 160–68) it was used by Hindu-Arabic scholars in the Middle Ages. It appears to have been imbibed by Western culture as a part of the decimal system of representation of numbers. With the invention of the electronic calculator its practical value has diminished. However, it is still an intriguing application of modular arithmetic that can be generalized to arithmetic in other bases. This article explains how casting out 9s is done, examines some reasons for including it as a topic for exploration in the mathematics curriculum, and uses modular arithmetic to explore its mathematical basis and its generalizability to computations in bases other than ten. A method of detecting errors in the transmission of computer code with some affinities to its analogues is also explored.

Divine Omnipotence and Laws of Nature

2018 ◽  
pp. 117-201
Author(s):  
Amos Funkenstein
Keyword(s):  
Middle Ages ◽  
Laws Of Nature ◽  
Mathematical Truth ◽  
Medieval Theology ◽  
Eternal Truths ◽  
God's Will ◽  
Divine Omnipotence ◽  
The World ◽  
Mathematical Operations ◽  
The Middle Ages

This chapter explores how eternal truths are created in a radical sense of the word; even mathematical theorems are contingent upon God’s will. What the most radical defenders of divine omnipotence in the Middle Ages hardly ever asserted, Descartes did without hesitation: that God could invalidate the most basic mathematical operations. Meanwhile, Spinoza argued that divine omnipotence and necessity of nature are one and the same, since all that is really possible in the world is also as necessary as any mathematical truth. The chapter shows how medieval theology introduced the distinction between the two aspects of God’s power so as to enlarge as far as possible the horizon of that which is possible to God without violating reason.

The Divine Order: Western Culture in the Middle Ages and the Renaissance

1969 ◽  
Vol 75 (2) ◽  
pp. 465
Author(s):  
F. Edward Cranz ◽  
Henry Bamford Parkes
Keyword(s):  
Middle Ages ◽  
Western Culture ◽  
Divine Order ◽  
The Middle Ages

Calfurnio's Identification of Pseudepigrapha of Ognibene, Fenestella, and Trebizond, and His Attack on Renaissance Commentaries

1988 ◽  
Vol 41 (1) ◽  
pp. 32-43 ◽  
Author(s):  
John Monfasani
Keyword(s):  
Middle Ages ◽  
Western Culture ◽  
Literary Works ◽  
Dionysius The Areopagite ◽  
Large Influence ◽  
Corpus Hermeticum ◽  
Pseudo Dionysius ◽  
The Middle Ages

Literary forgeries and pseudepigrapha have played an important role in Western culture since antiquity. One thinks of the large influence exercised in the Middle Ages and Renaissance by the writings of Pseudo-Dionysius the Areopagite, the Corpus Hermeticum, the Zohar, the Pseudo-Aristotelian Liber de causis, the Pseudo- Ciceronian Rhetorica ad Herennium, the correspondence between St. Paul and Seneca, and the vast sea of pseudonymous hagiographical literature. However, in the Renaissance the situation changed somewhat because printing did more than merely provide a new medium for the diffusion of pseudonymous literary works; it increased greatly the possibility of financial profit for the publishers, printers, and, eventually, authors of such works.

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