Forty (number)

Mathematical characteristics

Forty has the prime factorization 2^3·5. Its positive divisors are 1, 2, 4, 5, 8, 10, 20 and 40. Notable arithmetic properties include:

• Number of divisors: 8. Forty is divisible by this count, so it is a refactorable (or tau) number.
• Sum of divisors (σ): 90; sum of proper divisors: 50, which makes 40 an abundant number (proper divisors sum to more than the number).
• Euler's totient: φ(40) = 16, so sixteen positive integers less than 40 are relatively prime to it.
• 40 = 6^2 + 2^2, so it can be expressed as the sum of two squares.
• It is a Harshad (Niven) number in base ten because 40 is divisible by the sum of its digits (4 + 0 = 4).
• Semiperfect: 40 equals a sum of some of its proper divisors (for example, 20 + 10 + 8 + 2 = 40).

Notation and etymology

The English word "forty" derives from Old English feowertig, built from feower (four) plus a tens suffix and historically meant "four tens." Many languages use a compound equivalent to "four tens," but modern English uses the unique form "forty." The common misspelling "fourty" is not standard.

History, culture and idioms

The number 40 appears frequently in cultural and religious contexts. In several Abrahamic traditions, periods of forty days or forty years are symbolic in narratives of trial, testing or transition (for example, fasts and periods of wandering). The word "quarantine" derives from the Italian quaranta, meaning forty, referring to a forty-day isolation used in medieval port cities during plague outbreaks. In everyday English idioms, "forty winks" denotes a short nap, and phrases like "life begins at forty" treat the age as a cultural milestone. The 40-yard dash is a standard sprint test in American football scouting.

Science, measurement and practical notes

In science, 40 is the atomic number of zirconium, a transition metal used in alloys and nuclear applications. In human biology, a full-term pregnancy is commonly described as about forty weeks. Because forty is a multiple of ten, it serves as a convenient round figure in packaging, time intervals and percentages (for example, 40-minute periods or 40 percent thresholds).

Notable curiosities

• Alphabetical ordering: "forty" is often cited as the only English number whose letters are in alphabetical order.
• Roman numeral XL is frequently used as a concise symbol for forty in clocks, inscriptions and product names.
• Forty appears in many traditions as a symbolic period, reinforcing its presence across religion, folklore and common speech.