Are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the country's third largest city"). In common mathematical terminology, "cardinal numbers" are words colloquially used for counting, and "ordinal numbers" are words used for ordering. Natural numbers can sometimes appear as a convenient set of codes (labels or "names"); that is, as linguists call nominal numbers, forgetting in a mathematical sense many or all of the properties of being a number. The symbol is often denoted by a set of natural numbers.
The use of numerals to represent numbers was the first great advance in abstraction. This allowed systems to be developed for large numbers to be recorded. A powerful number system with distinct hieroglyphs for 1, 10, and all powers from 10 to over 1 million was developed by the ancient Egyptians. A stone carving from Karnak, dating back to about 1500 BCE and now at the Louvre in Paris, depicts 276 as two hundreds, seven tens, six tens, and 4,622. Using base sixty, the Babylonians had a place-value system basically based on the numerals for 1 and 10, so that the symbol for sixty was the same as the symbol for one, its value being determined by context.
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There was a mathematical and philosophical debate in 19th century Europe about the exact nature of natural numbers. A school[Which one? Naturalism has stated that natural numbers are a direct consequence of the human psyche. Like Leopold Kronecker, who summarized his belief as "God made the integers, everything else is the work of man," Henri Poincaré was one of his advocates.
The constructivists saw a need to enhance the logical rigor in the foundations of mathematics in opposition to the Naturalists.[h] In the 1860s, Hermann Grassmann proposed a recursive definition for natural numbers, thus stating that they were not really natural, but a consequence of definitions. Subsequently, two classes of such formal definitions were built; later on, they were shown to be equivalent.
Four-digit numbers are often named in American usage using multiples of "hundred" and combined with tens and ones: "eleven hundred three," "twelve hundred twenty-five," "four thousand forty-two," or "ninety-nine hundred ninety-nine." This style is common in British usage for multiples of 100 between 1,000 and 2,000, but not for higher numbers ( e.g. 1,500 as "fifteen hundred.").
Without saying "hundred" and inserting "oh" for zero tens, Americans can pronounce four-digit numbers with non-zero tens and ones as pairs of two-digit numbers: "twenty-six fifty-nine" or "forty-one oh five". This use probably evolved from the distinctive use for years; "nineteen-eighty-one" or from four-digit numbers originally used in the American telephone numbering system.