Number

40

40 — Forty days

40 is a composite number, even, a calendar year.

Forty runs through the Abrahamic scriptures as the standard measure of an ordeal. Forty days and nights of rain; forty years wandering the desert; Moses forty days on Sinai; Jesus forty days in the wilderness, remembered in Lent's forty. Muhammad received his first revelation at forty. Scholars generally read these as idiom rather than arithmetic — in Hebrew and Arabic usage, forty is less a count than a way of saying “a long while,” about a generation.

That scriptural weight left a fossil in modern public health. In 1377 the city-state of Ragusa — today's Dubrovnik — ordered arriving ships to lie at anchor thirty days before anyone could land, a trentino. Venice pushed the wait to forty: quaranta giorni. The phrase survives, worn smooth, as the English word quarantine — a medical term that still carries the shape of a biblical number.

A curiosity to close on: forty is the only number whose English name has its letters in alphabetical order — f, o, r, t, y.

Sources https://en.wikipedia.org/wiki/Quarantine · https://en.wikipedia.org/wiki/40_(number)

Abundant Number Curated Evil Number Harshad / Niven Octagonal Pentagonal Pyramidal Pernicious Number Practical Number Recamán's Sequence Refactorable Number Semiperfect Number

Interestingness

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7 notable facts · grade B

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Historical context — 40 AD

Calendar year

AD 40 (XL) was a leap year starting on Friday of the Julian calendar.

Excerpt from Wikipedia (en) ↗ · Licensed CC BY-SA 4.0 Read the full article on Wikipedia →

Historical context — 40 BC

Calendar year

Year 40 BC was either a common year starting on Thursday, Friday or Saturday or a leap year starting on Thursday or Friday of the Julian calendar and a common year starting on Friday of the Proleptic Julian calendar.

Excerpt from Wikipedia (en) ↗ · Licensed CC BY-SA 4.0 Read the full article on Wikipedia →

Year facts

Year type: Leap year
Divisible by 4 and not by 100; February has 29 days.
Days in year: 366
ISO weeks: 52
Started on: Sunday
January 1, 40
Ended on: Monday
December 31, 40
Friday the 13ths: 3
3 Friday the 13ths this year.
Decade: 40s
40–49
Century: 1st century
1–100
Millennium: 1st millennium
1–1000
Years ago: 1,986
1986 years before 2026.

In other calendars

Hebrew: 3800 / 3801 AM
Rosh Hashanah falls in September/October.
Chinese: Year of the zodiac:Metal zodiac:Rat
Sexagenary cycle position 37 of 60. Lunar new year falls in late January / mid-February.
Buddhist Era: 583 BE
Counted from the parinirvana of the Buddha (Theravada / Thai / Sri Lankan convention).
Ethiopian: 32 / 33 ET
Year boundary at Enkutatash (September 11/12).
Indian National (Saka): -38 / -39 Saka
Indian national calendar; year starts in March.

Cultural significance

Christian sacred

Recurring number of "a long time" in scripture.

40 days of the Flood; 40 years of wandering; 40 days of Lent; Jesus fasted 40 days in the wilderness.

Sourced from Wikipedia (Numerology, Chinese numerology, Gematria, and per-culture articles).

Properties

Parity: Even
Digit count: 2
Digit sum: 4
Digit product: 0
Digital root: 4
Palindrome: No
Bit width: 6 bits
Reversed: 4
Recamán's sequence: a(111) = 40
Square (n²): 1,600
Cube (n³): 64,000
Divisor count: 8
σ(n) — sum of divisors: 90
φ(n) — Euler's totient: 16
Sum of prime factors: 11

Primality

Prime factorization: 2 ³ × 5

Nearest primes: 37 (−3) · 41 (+1)

Divisors & multiples

All divisors (8)

1 · 2 · 4 · 5 · 8 · 10 · 20 (half) · 40

Aliquot sum (sum of proper divisors): 50

Factor pairs (a × b = 40)

1 × 40

2 × 20

4 × 10

5 × 8

First multiples

40 · 80 (double) · 120 · 160 · 200 · 240 · 280 · 320 · 360 · 400

Sums & aliquot sequence

As a sum of two squares: 2² + 6²

As consecutive integers: 6 + 7 + 8 + 9 + 10

Aliquot sequence: 40 → 50 → 43 → 1 → 0 — terminates at zero

Continued fraction of √n

√40 = [6; (3, 12)]

Period length 2 — the block in parentheses repeats forever.

Representations

In words: forty
Ordinal: 40th
Roman numeral: XL
Binary: 101000
Octal: 50
Hexadecimal: 0x28
Base64: KA==
One's complement: 215 (8-bit)
Scientific notation: 4 × 10¹

In other bases

ternary (3) 1111

quaternary (4) 220

quinary (5) 130

senary (6) 104

septenary (7) 55

nonary (9) 44

undecimal (11) 37

duodecimal (12) 34

tridecimal (13) 31

tetradecimal (14) 2c

pentadecimal (15) 2a

Palindromic in base 3

As an angle

40°

40° ≈ 0.698 rad

Compass bearing: NE (northeast)

Why does a circle have 360 degrees? →

Historical numeral systems

Babylonian (base 60): 𒌋𒌋𒌋𒌋
Egyptian hieroglyphic: 𓎆𓎆𓎆𓎆
Greek (Milesian): μʹ
Mayan (base 20): 𝋢·𝋠
Chinese: 四十
Chinese (financial): 肆拾

In other modern scripts

Eastern Arabic ٤٠ Devanagari ४० Bengali ৪০ Tamil ௪௦ Thai ๔๐ Tibetan ༤༠ Khmer ៤០ Lao ໔໐ Burmese ၄၀

Digit at this position in famous constants

π — Pi (π): Digit 40 = 7
e — Euler's number (e): Digit 40 = 7
φ — Golden ratio (φ): Digit 40 = 0
√2 — Pythagoras's (√2): Digit 40 = 9
ln 2 — Natural log of 2: Digit 40 = 5
γ — Euler-Mascheroni (γ): Digit 40 = 2

Also seen as

Goldbach decomposition

Goldbach's conjecture says every even integer greater than 2 is the sum of two primes. For 40, here are decompositions:

3 + 37 = 40
11 + 29 = 40
17 + 23 = 40

ASCII character

As an ASCII codepoint, 40 is (. Printable ASCII character (.

Hex color

#000028

RGB(0, 0, 40)

IPv4 address

As an unsigned 32-bit integer, this is the IPv4 address 0.0.0.40.

Address: 0.0.0.40
Class: reserved
IPv4-mapped IPv6: ::ffff:0.0.0.40

Unspecified address (0.0.0.0/8) — "this network" placeholder.

US numbered highway

Matches numbered highway designations:

I-40 — Barstow, CA to Wilmington, NC.
US 40 — Atlantic City, NJ to Park City, UT — the historic National Road.

Interstate 40 on Wikipedia ↗ US 40 on Wikipedia ↗

Calculator-display word

Type 40 on a seven-segment calculator, flip it 180°, and the display reads:

A staple of calculator humor since pocket calculators put digits in front of bored students.

Musical pitch

Heard as a frequency, 40 Hz is closest to:

Concert pitch (A4 = 440 Hz): D♯1 (38.9 Hz, +49¢ — about midway to E1)
Scientific pitch (C4 = 256 Hz): E1 (40.3 Hz, -14¢)
Baroque pitch (A4 = 415 Hz): E1 (38.9 Hz, +50¢ — about midway to F1)

(very low — many laptop and phone speakers can't reproduce it)

As a MIDI note number, 40 is E2 (82.4 Hz at concert pitch). (very low — many laptop and phone speakers can't reproduce it)

Atomic number

On the periodic table, atomic number 40 is Zirconium (Zr) — period 5.

Zirconium on Wikipedia ↗