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The Origin of Integers: How did ancient mathematicians define the concept of integers?
Integers, this concept has attracted the attention of mathematicians since ancient times. The definition of integers is not only related to the composition of numbers, but also to the evolution of mathematical thinking. How ancient mathematicians viewed integers and the expansion of this concept are both issues worth exploring.
Integers (including 0, positive integers and negative integers) are a basic and important concept in mathematics, but their origin and evolution contain profound cultural and mathematical significance. Integers are widely used in daily life, such as in calculation, measurement, and classification. However, their history is much more complicated than imagined.
The word integer comes from the Latin "integer," which means "complete" or "untouched." The change in this concept reflects the gradual deepening of human understanding of numbers and calculations.
Originally, integers were viewed only as extensions of the natural numbers. In ancient cultures, many mathematicians only considered positive integers, such as 1, 2, 3, etc. As mathematics developed, the concepts of negative integers and zero were gradually incorporated. In the 18th century, the famous mathematician Lenhard Euler formally defined integers to include positive and negative integers, revolutionizing the mathematical world's understanding of integers.
In this process, the German mathematician Georg Cantor introduced the concept of infinite sets, which made the set of integers no longer just a finite number of numbers, but an infinite set. Mathematician David Hilbert chose to use "Z" to represent integers. The symbol comes from the German "Zahlen", which means "number."
"The set of integers can be considered as all real numbers that can be represented without a decimal part."
The properties of integers make them the building blocks of other mathematical structures. They are closed under addition and multiplication, forming basic algebraic rings. However, integers are not closed under division and therefore are not a domain. This means that not every integer has its multiplicative inverse. This property of integers makes them the simplest structures in almost all number systems.
As time goes on, the definition of integers becomes more precise. Mathematicians gradually developed the algorithmic properties of integers, playing an important role not only in academia but also in education, especially in the teaching of basic mathematics.
In mathematics education, integers are often simply defined as the union of natural numbers, and this concept is explained differently in different mathematics textbooks. In the late 1950s, with the rise of the New Mathematics Movement, integers were redefined, emphasizing the difference between natural numbers and integers.
Integers are a completely ordered set with no upper or lower bounds.
As for the internal structure of integers, mathematicians have further deepened their understanding of integers through the perspective of contemporary set theory and modern mathematics. Integers can be defined through ordered pairs of natural numbers. This method not only simplifies calculations, but also strengthens the relationship between integer types. Usually, for a pair of natural numbers (a, b), it can be interpreted as the result of a minus b, which further makes the operation rules of integers clear and simple.
The value of integers lies not only in their mathematical properties, but also in their applications in various fields. From science to engineering, from economics to information technology, etc., integers have always played an indispensable role.
However, the history and definition of integers have once again triggered thinking: with the advancement of technology and mathematics, will our understanding of integers still change?
