# 50 (number)
**50 (fifty)** is the natural number following 49 and preceding 51. It is a composite number with prime factorization 2 × 5², the Roman numeral L, the atomic number of the element tin, and the anniversary marked by a Golden Jubilee.[1](https://www.numberempire.com/50) • [2](https://periodic-table.rsc.org/element/50/tin) • [3](https://commonslibrary.parliament.uk/research-briefings/cbp-11044/)
| Key fact | Detail |
|---|---|
| Factorization | 2 × 5²; divisors 1, 2, 5, 10, 25, 50 (six divisors, summing to 93)[1](https://www.numberempire.com/50) |
| Classification | Deficient number: the sum of its proper divisors (1 + 2 + 5 + 10 + 25 = 43) is less than 50; also a regular (Hamming) number[4](https://www.positiveintegers.org/50) • [1](https://www.numberempire.com/50) |
| Sum of two squares | Smallest positive integer expressible as a sum of two squares in two distinct ways: 5² + 5² and 7² + 1²[5](https://proofwiki.org/wiki/Sum_of_2_Squares_in_2_Distinct_Ways/Examples/50) |
| Other bases | Binary 110010, octal 62, duodecimal 42, hexadecimal 32, Roman numeral L[1](https://www.numberempire.com/50) |
| Chemistry | Atomic number of tin (Sn), which has relative atomic mass 118.710[2](https://periodic-table.rsc.org/element/50/tin) |
| Anniversary | A Golden Jubilee marks 50 years; the first British royal jubilee marked George III's 50th anniversary in 1809[3](https://commonslibrary.parliament.uk/research-briefings/cbp-11044/) |
| Neighboring primes | Lies between the primes 47 and 53; the 50th prime is 229[1](https://www.numberempire.com/50) |
## Arithmetic properties
The factorization 50 = 2 × 5² gives it six divisors: 1, 2, 5, 10, 25, and 50, whose sum is 93.[1](https://www.numberempire.com/50) Because the sum of its proper divisors, 43, falls short of 50, it is a deficient number.[4](https://www.positiveintegers.org/50) Its only prime factors are 2 and 5, which makes it a regular number, also called a Hamming number, meaning it has no prime factor other than 2, 3, or 5.[1](https://www.numberempire.com/50) [Euler's totient function](https://www.edgepedia.org/eulers-totient-function), which counts the integers from 1 to n that share no factor with n, gives φ(50) = 20.[1](https://www.numberempire.com/50)
In positional notation, 50 is written 110010 in binary, 1212 in ternary, 62 in octal, 42 in duodecimal, and 32 in hexadecimal.[1](https://www.numberempire.com/50) The binary form 110010 expands as 32 + 16 + 2, and its Hamming weight, the count of 1 bits, is 3.[1](https://www.numberempire.com/50)
## Sums of two squares
50 is the smallest positive integer that can be written as a sum of two square numbers in two distinct ways:[5](https://proofwiki.org/wiki/Sum_of_2_Squares_in_2_Distinct_Ways/Examples/50)
\[ 50 = 5^2 + 5^2 = 7^2 + 1^2 \]
The count of signed, ordered representations is given by the formula for \( r_2(n) \): since 50 = 2 · 5², with the prime 5 congruent to 1 mod 4 appearing to the second power, \( r_2(50) = 4(1+2) = 12 \). Four of these are the solutions with \( x = y = \pm 5 \), and sign changes and order swaps account for the rest, leaving the two essentially distinct representations above.[6](https://www.math.uwaterloo.ca/~f2alfais/notes/pm340-notes.pdf) ProofWiki derives the pair of representations from the Brahmagupta-[Fibonacci](https://www.edgepedia.org/fibonacci) identity applied to 50 = 5 × 10, since 5 = 1² + 2² and 10 = 1² + 3² are themselves sums of two squares.[5](https://proofwiki.org/wiki/Sum_of_2_Squares_in_2_Distinct_Ways/Examples/50)
## Roman and Babylonian notation
In [Roman numerals](https://www.edgepedia.org/roman-numerals), 50 is written with the single letter L. The Roman system is non-positional and assigns a dedicated symbol to 50, alongside I (1), V (5), X (10), C (100), D (500), and M (1000).[7](https://romandigits.com/50-in-roman-numerals) The symbol's ultimate origin is uncertain; Etruscan numeral signs, which preceded and resembled the Roman ones, included a sign for 50 that was later abbreviated into the letter form L.[8](https://en.wikisource.org/wiki/Page%3AA_History_Of_Mathematical_Notations_Vol_I_%281928%29.djvu/50)
Babylonian cuneiform had no single sign for 50. The system used a vertical wedge for 1 and a character for 10, so 50 was written as five repetitions of the ten symbol, and values up to 59 were built this way before the next power of 60 began.[9](https://mathshistory.st-andrews.ac.uk/HistTopics/Babylonian_numerals/) • [10](https://math.libretexts.org/Bookshelves/Applied_Mathematics/Contemporary_Mathematics_%28OpenStax%29/04%3A__Number_Representation_and_Calculation/4.01%3A__Early_Numeration_Systems) [The Babylonian](https://www.edgepedia.org/the-babylonian) system was positional in base 60, so a 50 could also appear as the digit of a higher place value, multiplied by 60, 3600, or a higher power.[9](https://mathshistory.st-andrews.ac.uk/HistTopics/Babylonian_numerals/)
## Tin, element 50
The chemical element tin has atomic number 50, meaning every atom of tin contains 50 protons.[11](https://pubchem.ncbi.nlm.nih.gov/element/Tin) Tin (symbol Sn) sits in group 14 and period 5 of the periodic table, with a relative atomic mass of 118.710 and a melting point of 231.928 °C.[2](https://periodic-table.rsc.org/element/50/tin) It is a soft, pliable metal used to coat steel in tin cans, in alloys such as solder, pewter, and bronze, and in a niobium-tin alloy for superconducting magnets.[2](https://periodic-table.rsc.org/element/50/tin)
## Jubilees and the fiftieth anniversary
A Golden Jubilee marks 50 years, and the word itself carries the number in its history. It comes from the Hebrew *yobel*, the ram's horn that announced a jubilee year of emancipation and restoration held every fifty years, described in the book of Leviticus.[12](https://www.worldwidewords.org/tw-jub1.html) Through Greek and Latin the word reached English with the sense of a fiftieth anniversary or its celebration.[12](https://www.worldwidewords.org/tw-jub1.html)
In Britain, the first royal jubilee marked King George III's 50th anniversary in 1809, combining public and private events that became a feature of later jubilees.[3](https://commonslibrary.parliament.uk/research-briefings/cbp-11044/) [George III](https://www.edgepedia.org/george-iii) celebrated on 25 October 1809, the start of the 50th year of his reign, with processions, feasts, and a 50-gun salute fired from the [Tower of London](https://www.edgepedia.org/tower-of-london); since [Queen Victoria](https://www.edgepedia.org/queen-victoria)'s reign, Golden Jubilees have been celebrated after the completion of 50 years of service.[13](https://www.hrp.org.uk/tower-of-london/history-and-stories/a-history-of-royal-jubilees/) Victoria's Golden Jubilee began on 20 June 1887, exactly 50 years after she succeeded to the throne, and her banquet at Buckingham Palace was attended by 50 foreign kings and princes.[3](https://commonslibrary.parliament.uk/research-briefings/cbp-11044/) The 50-year milestone has also been adopted outside royalty: India marked the Golden Jubilee of its independence on 15 August 1997, completing 50 years since 15 August 1947.[14](https://www.indiaculture.gov.in/sites/default/files/annual-reports/MoC%20Annual%20Report%201997-1998.pdf)
## References
1. [Properties of the number 50, Number Empire](https://www.numberempire.com/50)
2. [Tin, element information, properties and uses, Royal Society of Chemistry](https://periodic-table.rsc.org/element/50/tin)
3. [How are royal jubilees celebrated in the UK?, House of Commons Library](https://commonslibrary.parliament.uk/research-briefings/cbp-11044/)
4. [The Positive Integer 50](https://www.positiveintegers.org/50)
5. [Sum of 2 Squares in 2 Distinct Ways/Examples/50, ProofWiki](https://proofwiki.org/wiki/Sum_of_2_Squares_in_2_Distinct_Ways/Examples/50)
6. [Lecture 27: Sums of Squares and Pythagorean Triples, PMATH 340 notes, University of Waterloo](https://www.math.uwaterloo.ca/~f2alfais/notes/pm340-notes.pdf)
7. [50 in Roman Numerals, Roman Digits](https://romandigits.com/50-in-roman-numerals)
8. [A History of Mathematical Notations, Vol. I (1928), page 50, Wikisource](https://en.wikisource.org/wiki/Page%3AA_History_Of_Mathematical_Notations_Vol_I_%281928%29.djvu/50)
9. [Babylonian numerals, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/HistTopics/Babylonian_numerals/)
10. [Early Numeration Systems, Mathematics LibreTexts (OpenStax)](https://math.libretexts.org/Bookshelves/Applied_Mathematics/Contemporary_Mathematics_%28OpenStax%29/04%3A__Number_Representation_and_Calculation/4.01%3A__Early_Numeration_Systems)
11. [Tin (Sn), PubChem Elements, NIH](https://pubchem.ncbi.nlm.nih.gov/element/Tin)
12. [Jubilee: The History of Fifty Year Celebrations, World Wide Words](https://www.worldwidewords.org/tw-jub1.html)
13. [A history of Royal Jubilees, Historic Royal Palaces](https://www.hrp.org.uk/tower-of-london/history-and-stories/a-history-of-royal-jubilees/)
14. [Celebrations of the Golden Jubilee of India's Independence, Ministry of Culture Annual Report 1997-1998](https://www.indiaculture.gov.in/sites/default/files/annual-reports/MoC%20Annual%20Report%201997-1998.pdf)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Arithmetic and number systems › Integer sequences and partitions › Special and named integers › Named individual integers*
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