Mathematics
Eight is the natural number that follows seven and precedes nine. It is an even composite number, the third positive power of two, and the first nonzero cube of an even integer. The numeral 8 also has major roles outside elementary arithmetic: it is the atomic number of oxygen, the basis of octal notation, and a recurring symbol in religious, scientific, and cultural systems.
Eight is an even composite integer with four positive divisors: 1, 2, 4, and 8. Its prime factorization is 23, so it is both a perfect cube and a power of two. The sum of its proper divisors is 7, making 8 a deficient number rather than a perfect or abundant number. It is also the eighth term when the Fibonacci sequence is conventionally written as 0, 1, 1, 2, 3, 5, 8, 13; indexing conventions can therefore give it a different position.1 In geometry, eight is the number of vertices and faces of a cube and an octahedron, respectively, while a regular octagon has eight equal sides.
Eight is the base of the octal numeral system, in which each digit represents three binary bits. Thus decimal 8 is written 10 in octal and 1000 in binary; octal notation once had a practical role in computer programming because it compactly represented groups of three bits.2 In positional notation, 8 is the smallest integer that requires two digits in base 8, but it remains a single digit in every base greater than 8. The number is also central to digital quantities such as an octet, conventionally a group of eight bits. In combinatorics, the eight subsets of a three-element set illustrate the general rule that a set with n elements has 2n subsets.
Eight is the atomic number of oxygen, meaning that a neutral oxygen atom contains eight protons in its nucleus.3 The element is a major component of Earth’s atmosphere and is essential to aerobic respiration, although oxygen also participates in combustion and corrosion. In particle physics, the Eightfold Way was a classification scheme for hadrons that organized particles into mathematical patterns related to the symmetry group SU(3); its success helped lead to the quark model.4 Chemistry also uses the “octet rule” as a simplified description of the tendency of many main-group atoms to achieve eight valence electrons. The rule is useful but has recognized exceptions, including electron-deficient and expanded-valence compounds.
Eight has acquired distinctive meanings in several traditions. Buddhism describes a Noble Eightfold Path of ethical conduct, mental discipline, and wisdom, while the Eightfold Way in physics borrowed its name from this religious phrase.5 In East Asian cultures, the pronunciation and visual associations of eight have often made it a favorable number, especially in commercial and ceremonial contexts. Less symbolically, eight is the smallest number that is neither prime nor a square, and it is the only positive cube that is also a nontrivial power of two. In music, an octave spans eight notes when both endpoint notes are counted in a diatonic scale; the interval itself corresponds to a frequency ratio of 2:1 in equal-tempered and other standard tuning systems.
Mathematical indexing and the counting of scale degrees can vary by convention; the article uses standard elementary-number and Western diatonic conventions.
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