Thirty-six is a positive integer that follows 35 and precedes 37. It is even and commonly known as three dozen. In Roman numerals it is written XXXVI. Because it equals 6 × 6, 36 is the sixth perfect square and is also the eighth triangular number, making it one of the numbers that is simultaneously square and triangular.
Mathematical properties
In prime factor form 36 = 2^2 × 3^2. Its full set of positive divisors is 1, 2, 3, 4, 6, 9, 12, 18 and 36, so it has nine divisors in total. The sum of those divisors is 91, and the sum of proper divisors (55) exceeds 36, which classifies 36 as an abundant number. Because 36 has more divisors than any smaller positive integer at the time of its occurrence, it is also a highly composite number.
Geometry and patterns
36 appears in basic geometry and polygonal angles: the central angle of a regular decagon is 36 degrees (360/10), and related 36° angles occur in pentagonal and star-shaped constructions linked to the golden ratio. In positional notation, 36 is written 100100 in binary and 44 in octal.
History, culture, and uses
The count of thirty-six crops up in many cultural contexts. A group of three dozen is a common packaging unit. Traditions such as the Jewish concept of the lamedvav (thirty-six) righteous people and the Chinese "Thirty-Six Stratagems" show the number’s symbolic presence. For further context see general reference and cultural uses.
Examples and notable distinctions
- Square: 36 = 6^2.
- Triangular: 36 = 1+2+...+8 (the 8th triangular number).
- Square triangular number: one of the integers that are both perfect squares and triangular.
- Highly composite and abundant: many divisors and proper-divisor sum greater than itself.
For more mathematical details and tables, consult specialized resources: number theory, divisor functions, and polygon geometry.