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The term numbers or numerals or digits often implies only these symbols, however this can only be inferred from context. Europeans first learned of Arabic numerals c. the 10th century , though their spread was a gradual process.
It follows from the definition that each natural number is equal to the set of all natural numbers less than it. This definition, can be extended to the von Neumann definition of ordinals for defining all ordinal numbers, including the infinite ones: "each ordinal is the well-ordered set of all smaller ordinals."
This led to expressions involving the square roots of negative numbers, and eventually to the definition of a new number: a square root of −1, denoted by i, a symbol assigned by Leonhard Euler, and called the imaginary unit. The complex numbers consist of all numbers of the form + where a and b are real numbers.
Happy number. Tree showing all happy numbers up to 100. In number theory, a happy number is a number which eventually reaches 1 when replaced by the sum of the square of each digit. For instance, 13 is a happy number because , and . On the other hand, 4 is not a happy number because the sequence starting with and eventually reaches , the number ...
The integers arranged on a number line. An integer is the number zero (0), a positive natural number (1, 2, 3, . . .), or the negation of a positive natural number (−1, −2, −3, . . .). [ 1 ] The negations or additive inverses of the positive natural numbers are referred to as negative integers. 2 The set of all integers is often denoted ...
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a ...
Definition. An abundant number is a natural number n for which the sum of divisors σ(n) satisfies σ(n) > 2n, or, equivalently, the sum of proper divisors (or aliquot sum) s(n) satisfies s(n) > n. The abundance of a natural number is the integer σ(n) − 2n (equivalently, s(n) − n).
Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The n th triangular number is the number of dots in the triangular arrangement with n dots on each side, and is equal to the sum of the n natural numbers from 1 to n. The sequence of triangular numbers, starting with the 0th triangular number ...