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The title of the book is a reference to a quotation attributed to mathematician Leopold Kronecker, who once wrote that "God made the integers; all else is the work of man." [ 2 ] Content
Possible tally marks made by carving notches in wood, bone, and stone appear in the archaeological record at least forty thousand years ago. [9] [10] These tally marks may have been used for counting time, such as numbers of days or lunar cycles, or for keeping records of quantities, such as numbers of animals or other valuable commodities.
In fact Kronecker might be the most famous of the Pre-Intuitionists for his singular and oft quoted phrase, "God made the natural numbers; all else is the work of man." Kronecker goes in almost the opposite direction from Poincaré, believing in the natural numbers but not the law of the excluded middle.
The synthetic elements are those with atomic numbers 95–118, as shown in purple on the accompanying periodic table: [1] these 24 elements were first created between 1944 and 2010. The mechanism for the creation of a synthetic element is to force additional protons into the nucleus of an element with an atomic number lower than 95.
For example, the integers are made by adding 0 and negative numbers. The rational numbers add fractions, and the real numbers add infinite decimals. Complex numbers add the square root of −1. This chain of extensions canonically embeds the natural numbers in the other number systems. [6] [7] Natural numbers are studied in different areas of math.
An even number is an integer that is "evenly divisible" by two, that is divisible by two without remainder; an odd number is an integer that is not even. (The old-fashioned term "evenly divisible" is now almost always shortened to "divisible".) Any odd number n may be constructed by the formula n = 2k + 1, for a suitable integer k.
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Leopold Kronecker was born on 7 December 1823 in Liegnitz, Prussia (now Legnica, Poland) in a wealthy Jewish family. His parents, Isidor and Johanna (née Prausnitzep), took care of their children's education and provided them with private tutoring at home—Leopold's younger brother Hugo Kronecker would also follow a scientific path, later becoming a notable physiologist.