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← 999999 1000000 1000001 →
Cardinalone million
Ordinal1000000th
(one millionth)
Factorization26 × 56
Greek numeral
Roman numeralM, m
Binary111101000010010000002
Ternary12122102020013
Senary332333446
Octal36411008
Duodecimal40285412
HexadecimalF424016
Egyptian hieroglyph𓁨
Look up million in Wiktionary, the free dictionary.

1,000,000 (one million), or one thousand thousand, is the natural number following 999,999 and preceding 1,000,001. The word is derived from the early Italian millione (milione in modern Italian), from mille, "thousand", plus the augmentative suffix -one.[1]

It is commonly abbreviated:

In scientific notation, it is written as 1×106 or 106.[9] Physical quantities can also be expressed using the SI prefix mega (M), when dealing with SI units; for example, 1 megawatt (1 MW) equals 1,000,000 watts.

The meaning of the word "million" is common to the short scale and long scale numbering systems, unlike the larger numbers, which have different names in the two systems.

The million is sometimes used in the English language as a metaphor for a very large number, as in "Not in a million years" and "You're one in a million", or a hyperbole, as in "I've walked a million miles" and "You've asked a million-dollar question".

1,000,000 is also the square of 1000 and the cube of 100.

Visualisation of powers of ten from 1 to 1 million

Visualizing one million

[edit]

Even though it is often stressed that counting to precisely a million would be an exceedingly tedious task due to the time and concentration required, there are many ways to bring the number "down to size" in approximate quantities, ignoring irregularities or packing effects.

  • Information: Not counting spaces, the text printed on 136 pages of an Encyclopædia Britannica, or 600 pages of pulp paperback fiction contains approximately one million characters.
  • Length: There are one million millimetres in a kilometre, and roughly a million sixteenths of an inch in a mile (1 sixteenth = 0.0625). A typical car tire might rotate a million times in a 1,900-kilometre (1,200 mi) trip, while the engine would do several times that number of revolutions.
  • Fingers: If the width of a human finger is 22 mm (7⁄8 in), then a million fingers lined up would cover a distance of 22 km (14 mi). If a person walks at a speed of 4 km/h (2.5 mph), it would take them approximately five and a half hours to reach the end of the fingers.
  • Area: A square a thousand objects or units on a side contains a million such objects or square units, so a million holes might be found in less than three square yards of window screen, or similarly, in about one half square foot (400–500 cm2) of bed sheet cloth. A city lot 70 by 100 feet is about a million square inches.
  • Volume: The cube root of one million is one hundred, so a million objects or cubic units is contained in a cube a hundred objects or linear units on a side. A million grains of table salt or granulated sugar occupies about 64 mL (2.3 imp fl oz; 2.2 US fl oz), the volume of a cube one hundred grains on a side. One million cubic inches would be the volume of a small room 8+1⁄3 feet long by 8+1⁄3 feet wide by 8+1⁄3 feet high.
  • Mass: A million cubic millimetres (small droplets) of water would have a volume of one litre and a mass of one kilogram. A million millilitres or cubic centimetres (one cubic metre) of water has a mass of a million grams or one tonne.
  • Weight: A million 80-milligram (1.2 gr) honey bees would weigh the same as an 80 kg (180 lb) person.
  • Landscape: A pyramidal hill 600 feet (180 m) wide at the base and 100 feet (30 m) high would weigh about a million short tons.
  • Computer: A display resolution of 1,280 by 800 pixels contains 1,024,000 pixels.
  • Money: A U.S. dollar bill of any denomination weighs 1 gram (0.035 oz). There are 454 grams in a pound. One million dollar bills would weigh 1 megagram (1,000 kg; 2,200 lb) or 1 tonne (just over 1 short ton).
  • Time: A million seconds, 1 megasecond, is 11.57 days.

In Indian English and Pakistani English, it is also expressed as 10 lakh. Lakh is derived from lakṣa for 100,000 in Sanskrit.

One million black dots (pixels) – each tile with white or grey background contains 1000 dots (full image)

Selected 7-digit numbers (1,000,001–9,999,999)

[edit]

1,000,001 to 1,999,999

[edit]
  • 1,000,003 = Smallest 7-digit prime number
  • 1,000,405 = Smallest triangular number with 7 digits and the 1,414th triangular number
  • 1,002,001 = 10012, palindromic square
  • 1,006,301 = First number of the first pair of prime quadruplets occurring thirty apart ({1006301, 1006303, 1006307, 1006309} and {1006331, 1006333, 1006337, 1006339})[10]
  • 1,024,000 = Sometimes, the number of bytes in a megabyte[11]
  • 1,030,301 = 1013, palindromic cube
  • 1,037,718 = Large Schröder number
  • 1,048,576 = 10242 = 324 = 165 = 410 = 220, the number of bytes in a mebibyte (previously called a megabyte)
  • 1,048,976 = smallest 7 digit Leyland number
  • 1,058,576 = Leyland number
  • 1,058,841 = 76 x 32
  • 1,077,871 = the amount of prime numbers between 0 and 16777216(2^24)
  • 1,081,080 = 39th highly composite number[12]
  • 1,084,051 = fifth Keith prime[13]
  • 1,089,270 = harmonic divisor number[14]
  • 1,111,111 = repunit
  • 1,112,083 = logarithmic number[15]
  • 1,129,30832 + 1 is prime[16]
  • 1,136,689 = Pell number,[17] Markov number[18]
  • 1,174,281 = Fine number[19]
  • 1,185,921 = 10892 = 334
  • 1,200,304 = 17 + 27 + 37 + 47 + 57 + 67 + 77 [20]
  • 1,203,623 = smallest unprimeable number ending in 3[21][22]
  • 1,234,321 = 11112, palindromic square
  • 1,246,863 = Number of 27-bead necklaces (turning over is allowed) where complements are equivalent[23]
  • 1,256,070 = number of reduced trees with 29 nodes[24]
  • 1,262,180 = number of triangle-free graphs on 12 vertices[25]
  • 1,278,818 = Markov number[18]
  • 1,290,872 = number of 26-bead binary necklaces with beads of 2 colors where the colors may be swapped but turning over is not allowed[26]
  • 1,296,000 = number of primitive polynomials of degree 25 over GF(2)[27]
  • 1,299,709 = 100,000th prime number
  • 1,336,336 = 11562 = 344
  • 1,346,269 = Fibonacci number,[28] Markov number[18]
  • 1,367,631 = 1113, palindromic cube
  • 1,388,705 = number of prime knots with 16 crossings
  • 1,413,721 = square triangular number[29]
  • 1,419,857 = 175
  • 1,421,280 = harmonic divisor number[14]
  • 1,441,440 = 11th colossally abundant number,[30] 11th superior highly composite number,[31] 40th highly composite number[12]
  • 1,441,889 = Markov number[18]
  • 1,500,625 = 12252 = 354
  • 1,539,720 = harmonic divisor number[14]
  • 1,563,372 = Wedderburn-Etherington number[32]
  • 1,594,323 = 313
  • 1,596,520 = Leyland number
  • 1,606,137 = number of ways to partition {1,2,3,4,5,6,7,8,9} and then partition each cell (block) into subcells.[33]
  • 1,607,521/1,136,689 ≈ √2
  • 1,647,086 = Leyland number
  • 1,671,800 = Initial number of first century xx00 to xx99 consisting entirely of composite numbers[34]
  • 1,679,616 = 12962 = 364 = 68
  • 1,686,049 = Markov prime
  • 1,687,989 = number of square (0,1)-matrices without zero rows and with exactly 7 entries equal to 1[35]
  • 1,719,900 = number of primitive polynomials of degree 26 over GF(2)[27]
  • 1,730,787 = Riordan number
  • 1,741,725 = equal to the sum of the seventh power of its digits
  • 1,771,561 = 13312 = 1213 = 116, also, Commander Spock's estimate for the tribble population in the Star Trek episode "The Trouble with Tribbles"
  • 1,864,637 = k such that the sum of the squares of the first k primes is divisible by k.[36]
  • 1,874,161 = 13692 = 374
  • 1,889,568 = 185
  • 1,928,934 = 2 x 39 x 72
  • 1,941,760 = Leyland number
  • 1,953,125 = 1253 = 59
  • 1,978,405 = 16 + 26 + 36 + 46 + 56 + 66 + 76 + 86 + 96 + 106 [37]

2,000,000 to 2,999,999

[edit]
  • 2,000,002 = number of surface-points of a tetrahedron with edge-length 1000[38]
  • 2,000,376 = 1263
  • 2,012,174 = Leyland number
  • 2,012,674 = Markov number[18]
  • 2,027,025 = double factorial of 15
  • 2,085,136 = 14442 = 384
  • 2,097,152 = 1283 = 87 = 221
  • 2,097,593 = Leyland prime[39] using 2 & 21 (221 + 212)
  • 2,118,107 = largest integer such that , where is the prime omega function for distinct prime factors. The corresponding sum for 2118107 is indeed 57.
  • 2,124,679 = largest known Wolstenholme prime[40]
  • 2,144,505 = number of trees with 21 unlabeled nodes[41]
  • 2,162,160 = 41st highly composite number,[12] 2079th triangular number
  • 2,177,399 = smallest pandigital number in base 8.[42]
  • 2,178,309 = Fibonacci number[28]
  • 2,222,222 = repdigit
  • 2,266,502 = number of signed trees with 13 nodes[43]
  • 2,274,205 = number of different ways of expressing 1,000,000,000 as the sum of two prime numbers[44]
  • 2,313,441 = 15212 = 394
  • 2,356,779 = Motzkin number[45]
  • 2,405,236 = Number of 28-bead necklaces (turning over is allowed) where complements are equivalent[23]
  • 2,423,525 = Markov number[18]
  • 2,476,099 = 195
  • 2,485,534 = number of 27-bead binary necklaces with beads of 2 colors where the colors may be swapped but turning over is not allowed[26]
  • 2,515,169 = number of reduced trees with 30 nodes[24]
  • 2,560,000 = 16002 = 404
  • 2,567,284 = number of partially ordered set with 10 unlabelled elements[46]
  • 2,598,560 = chances of getting a royal flush in a hand of poker (52!/5!47!) (n choose r)
  • 2,646,723 = little Schroeder number
  • 2,674,440 = Catalan number[47]
  • 2,692,537 = Leonardo prime
  • 2,704,900 = initial number of fourth century xx00 to xx99 containing seventeen prime numbers[48][a] {2,704,901, 2,704,903, 2,704,907, 2,704,909, 2,704,927, 2,704,931, 2,704,937, 2,704,939, 2,704,943, 2,704,957, 2,704,963, 2,704,969, 2,704,979, 2,704,981, 2,704,987, 2,704,993, 2,704,997}
  • 2,744,210 = Pell number[17]
  • 2,796,203 = Wagstaff prime,[51] Jacobsthal prime
  • 2,825,761 = 16812 = 414
  • 2,890,625 = 1-automorphic number[52]
  • 2,922,509 = Markov prime
  • 2,985,984 = 17282 = 1443 = 126 = 1,000,00012 AKA a great-great-gross

3,000,000 to 3,999,999

[edit]
  • 3,111,696 = 17642 = 424
  • 3,200,000 = 205
  • 3,263,442 = product of the first five terms of Sylvester's sequence
  • 3,263,443 = sixth term of Sylvester's sequence[53]
  • 3,276,509 = Markov prime
  • 3,294,172 = 22×77[54]
  • 3,301,819 = alternating factorial[55]
  • 3,333,333 = repdigit
  • 3,360,633 = palindromic in 3 consecutive bases: 62818269 = 336063310 = 199599111
  • 3,418,801 = 18492 = 434
  • 3,426,576 = number of free 15-ominoes
  • 3,524,578 = Fibonacci number,[28] Markov number[18]
  • 3,554,688 = 2-automorphic number[56]
  • 3,626,149 = Wedderburn–Etherington prime[32]
  • 3,628,800 = 10!
  • 3,748,096 = 19362 = 444
  • 3,880,899/2,744,210 ≈ √2

4,000,000 to 4,999,999

[edit]
  • 4,008,004 = 20022, palindromic square
  • 4,037,913 = sum of the first ten factorials
  • 4,084,101 = 215
  • 4,100,625 = 20252 = 454
  • 4,194,304 = 20482 = 411 = 222
  • 4,194,788 = Leyland number
  • 4,202,496 = number of primitive polynomials of degree 27 over GF(2)[27]
  • 4,208,945 = Leyland number
  • 4,210,818 = equal to the sum of the seventh powers of its digits
  • 4,213,597 = Bell number[57]
  • 4,260,282 = Fine number[19]
  • 4,297,512 = 12-th derivative of xx at x=1[58]
  • 4,324,320 = 12th colossally abundant number,[30] 12th superior highly composite number,[31] pronic number
  • 4,400,489 = Markov number[18]
  • 4,444,444 = repdigit
  • 4,477,456 = 21162 = 464
  • 4,636,390 = Number of 29-bead necklaces (turning over is allowed) where complements are equivalent[23]
  • 4,741,632 = number of primitive polynomials of degree 28 over GF(2)[27]
  • 4,782,969 = 21872 = 97 = 314
  • 4,782,974 = n such that n | (3n + 5)[59]
  • 4,785,713 = Leyland number
  • 4,794,088 = number of 28-bead binary necklaces with beads of 2 colors where the colors may be swapped but turning over is not allowed[26]
  • 4,805,595 = Riordan number
  • 4,826,809 = 21972 = 1693 = 136
  • 4,879,681 = 22092 = 474
  • 4,913,000 = 1703
  • 4,937,284 = 22222

5,000,000 to 5,999,999

[edit]
  • 5,049,816 = number of reduced trees with 31 nodes[24]
  • 5,096,876 = number of prime numbers having eight digits[60]
  • 5,134,240 = the largest number that cannot be expressed as the sum of distinct fourth powers
  • 5,153,632 = 225
  • 5,221,225 = 22852, palindromic square
  • 5,293,446 = Large Schröder number
  • 5,308,416 = 23042 = 484
  • 5,496,925 = first cyclic number in base 6
  • 5,555,555 = repdigit
  • 5,623,756 = number of trees with 22 unlabeled nodes[41]
  • 5,702,887 = Fibonacci number[28]
  • 5,761,455 = the number of primes under 100,000,000
  • 5,764,801 = 24012 = 494 = 78
  • 5,882,353 = 5882 + 23532

6,000,000 to 6,999,999

[edit]
  • 6,250,000 = 25002 = 504
  • 6,436,343 = 235
  • 6,536,382 = Motzkin number[45]
  • 6,625,109 = Pell number,[17] Markov number[18]
  • 6,666,666 = repdigit
  • 6,765,201 = 26012 = 514
  • 6,948,496 = 26362, palindromic square

7,000,000 to 7,999,999

[edit]
  • 7,109,376 = 1-automorphic number[52]
  • 7,311,616 = 27042 = 524
  • 7,453,378 = Markov number[18]
  • 7,529,536 = 27442 = 1963 = 146
  • 7,652,413 = Largest n-digit pandigital prime
  • 7,777,777 = repdigit
  • 7,779,311 = A hit song written by Prince and released in 1982 by The Time
  • 7,861,953 = Leyland number
  • 7,890,481 = 28092 = 534
  • 7,906,276 = pentagonal triangular number
  • 7,913,837 = Keith number[13]
  • 7,962,624 = 245

8,000,000 to 8,999,999

[edit]
  • 8,000,000 = 2003, Used to represent infinity in Japanese mythology
  • 8,053,393 = number of prime knots with 17 crossings
  • 8,108,731 = repunit prime in base 14
  • 8,388,607 = second composite Mersenne number with a prime exponent
  • 8,388,608 = 223
  • 8,389,137 = Leyland number
  • 8,399,329 = Markov number[18]
  • 8,436,379 = Wedderburn-Etherington number[32]
  • 8,503,056 = 29162 = 544
  • 8,675,309 = A hit song for Tommy Tutone (also a twin prime with 8,675,311)
  • 8,675,311 = Twin prime with 8,675,309
  • 8,877,691 = number of nonnegative integers with distinct decimal digits[61]
  • 8,888,888 = repdigit
  • 8,946,176 = self-descriptive number in base 8
  • 8,964,800 = Number of 30-bead necklaces (turning over is allowed) where complements are equivalent[23]

9,000,000 to 9,999,999

[edit]
  • 9,000,000 = 30002
  • 9,150,625 = 30252 = 554
  • 9,227,465 = Fibonacci number,[28] Markov number[18]
  • 9,256,396 = number of 29-bead binary necklaces with beads of 2 colors where the colors may be swapped but turning over is not allowed[26]
  • 9,261,000 = 2103
  • 9,369,319 = Newman–Shanks–Williams prime[62]
  • 9,647,009 = Markov number[18]
  • 9,653,449 = square Stella octangula number
  • 9,581,014 = n such that n | (3n + 5)[59]
  • 9,663,500 = Initial number of first century xx00 to xx99 that possesses an identical prime pattern to any century with four or fewer digits: its prime pattern of {9663503, 9663523, 9663527, 9663539, 9663553, 9663581, 9663587} is identical to {5903, 5923, 5927, 5939, 5953, 5981, 5987}[63][64]
  • 9,694,845 = Catalan number[47]
  • 9,699,690 = eighth primorial
  • 9,765,625 = 31252 = 255 = 510
  • 9,800,817 = equal to the sum of the seventh powers of its digits
  • 9,834,496 = 31362 = 564
  • 9,865,625 = Leyland number
  • 9,926,315 = equal to the sum of the seventh powers of its digits
  • 9,938,375 = 2153, the largest 7-digit cube
  • 9,997,156 = largest triangular number with 7 digits and the 4,471st triangular number
  • 9,998,244 = 31622, the largest 7-digit square
  • 9,999,991 = Largest 7-digit prime number
  • 9,999,999 = repdigit

Prime numbers

[edit]

There are 78,498 primes less than 106, where 999,983 is the largest prime number smaller than 1,000,000.

Increments of 106 from 1 million through a 10 million have the following prime counts:

  • 70,435 primes between 1,000,000 and 2,000,000.
  • 67,883 primes between 2,000,000 and 3,000,000.
  • 66,330 primes between 3,000,000 and 4,000,000.
  • 65,367 primes between 4,000,000 and 5,000,000.
  • 64,336 primes between 5,000,000 and 6,000,000.
  • 63,799 primes between 6,000,000 and 7,000,000.
  • 63,129 primes between 7,000,000 and 8,000,000.
  • 62,712 primes between 8,000,000 and 9,000,000.
  • 62,090 primes between 9,000,000 and 10,000,000.

In total, there are 586,081 prime numbers between 1,000,000 and 10,000,000.[65]

See also

[edit]

Notes

[edit]

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

1,000,000

View on Grokipedia
from Grokipedia
One million (1,000,000), also expressed as 10610^6, is the natural number that follows 999,999 and precedes 1,000,001 in the sequence of positive integers. Mathematically, one million is a composite number with the prime factorization 26×562^6 \times 5^6, making it both a perfect square (100021000^2) and a perfect cube (1003100^3). It has exactly 49 positive divisors, and the sum of these divisors is 2,480,437, classifying it as an abundant number since this sum exceeds twice the number itself (2,000,000). In other bases, it is represented as 11110100001001000000 in binary (with a Hamming weight of 7) and F4240 in hexadecimal. Its square root is precisely 1,000, and Euler's totient function ϕ(1,000,000)\phi(1,000,000) equals 400,000, indicating the count of integers up to one million that are coprime to it. The term "million" originated in the early Italian language as millione, derived from mille ("thousand") augmented by the suffix -one to denote a large quantity, and it entered English usage around the late 14th century to describe this specific power of ten. In the Roman numeral system, one million is denoted by M‾\overline{M} (a barred M, signifying 1,000 repeated 1,000 times). Beyond pure mathematics, one million serves as a fundamental unit in scientific notation, the International System of Units (SI) prefix "mega-" (denoting 10610^6), and everyday contexts like population counts or financial metrics, where it represents a thousand thousands.

Definition and Notation

Numerical Definition

One million, denoted as 1,000,000, is the natural number that follows 999,999 and precedes 1,000,001 in the sequence of positive integers. In standard English nomenclature, it is expressed in words as "one million" or equivalently as "one thousand thousand." This number can be represented through basic arithmetic equivalences, such as 1,000,000=1,000×1,000=1003=1061,000,000 = 1{,}000 \times 1{,}000 = 100^3 = 10^6. In scientific notation, 1,000,000 is written as 1×1061 \times 10^6.

Etymology and Naming

The term "million" derives from the Italian word millione (modern Italian milione), an augmentative form of mille meaning "thousand," which itself originates from the Latin mille. This augmentative suffix -one emphasized a "great thousand," reflecting its initial use to denote 1,000 thousands or 10610^6, and the word was coined in Italy during the 14th century amid growing commercial needs for expressing large quantities in trade and finance. The word entered the English language in the late 14th century, borrowed through Old French million (attested around the late 13th century), appearing in Middle English as milioun around 1390, as evidenced in early mathematical and visionary texts. Initially employed by mathematicians for precise enumeration, it gradually permeated broader usage by the 16th century, supplanting earlier indefinite expressions for vast numbers. Naming conventions for multiples of the million have historically varied between the long scale and short scale systems. The long scale, originating in 15th-century France with mathematician Nicolas Chuquet's nomenclature in Le Triparty en la science des nombres (1484), defines higher terms by powers of the million: thus, a billion is a million squared (101210^{12}) and a trillion is a million cubed (101810^{18}). In contrast, the short scale, emerging in France during the 17th century, increments by thousands: a billion is a thousand millions (10910^9) and a trillion is a thousand billions (101210^{12}). The short scale gained prominence in the United States by the 19th century and was later adopted in the United Kingdom in 1974, while the long scale persists in much of continental Europe and French-speaking regions.

Representations in Numeral Systems

In scientific notation, 1,000,000 is expressed as 1×1061 \times 10^6, a compact form used in mathematics and sciences to represent large numbers by factoring out powers of ten. The binary representation of 1,000,000 is 11110100001001000000211110100001001000000_2, requiring 20 bits to encode the value in base-2, where each bit corresponds to a power of 2 from 2192^{19} down to 202^0. In hexadecimal, or base-16, it is written as F424016_{16}, utilizing digits 0-9 and letters A-F to represent values up to 15, which simplifies handling in computing contexts compared to binary. The Roman numeral for 1,000,000 employs the vinculum (overline) as M‾\overline{\rm M}, denoting multiplication by 1,000 of the base numeral M (1,000), an extension of classical Roman notation for numbers beyond standard limits. Common abbreviations include M, derived from the Latin mille for thousand but conventionally denoting one million in financial and scientific writing, or MM to explicitly indicate "thousand thousands," though mm is used in some non-ambiguous technical contexts to avoid confusion with millimeter.

Mathematical Properties

As a Power of Ten

One million, denoted as 1,000,000, is precisely equal to ten raised to the sixth power, or 10610^6, in the decimal system. This positions it as the sixth entry in the ascending sequence of positive powers of ten, immediately following 105=100,00010^5 = 100,000. As a fundamental exponentiation in base-10 arithmetic, 10610^6 exemplifies how powers of ten scale magnitudes by adding zeros to the right of 1, serving as a cornerstone for expressing large quantities succinctly in scientific and everyday contexts. In the decimal place value system, 1,000,000 corresponds to the millions place, which is the seventh position from the right in a whole number (counting the units place as the first). A digit occupying this place is multiplied by 10610^6 to determine its contribution to the total value of the number, enabling the representation of numbers up to billions and beyond without ambiguity. For instance, in the number 2,345,678, the digit 3 in the millions place contributes 3×1,000,000=3,000,0003 \times 1,000,000 = 3,000,000. This positional notation, rooted in the base-10 structure, relies on 10610^6 to define the scale for millions. Additionally, 1,000,000 arises as the square of 1,000 through basic multiplication. To derive 100021000^2, compute 1000×10001000 \times 1000 step by step: first, 1000×0=01000 \times 0 = 0 (units place), shifted by zero positions; then 1000×0=01000 \times 0 = 0 (tens place), shifted by one position; then 1000×0=01000 \times 0 = 0 (hundreds place), shifted by two positions; and finally 1000×1=10001000 \times 1 = 1000 (thousands place), shifted by three positions, yielding $1,000,000whenalignedandadded.Alternatively,recognizingwhen aligned and added. Alternatively, recognizing1000 = 10^3,theexponentrulegives, the exponent rule gives (10^3)^2 = 10^{6} = 1,000,000$. Similarly, 1,000,000 is the cube of 100. To derive 1003100^3, multiply 100×100×100100 \times 100 \times 100: first, 100×100=10,000100 \times 100 = 10,000; then 10,000×100=1,000,00010,000 \times 100 = 1,000,000 by shifting the decimal two places right (equivalent to multiplying by 10210^2). Using exponents, since 100=102100 = 10^2, (102)3=106=1,000,000(10^2)^3 = 10^{6} = 1,000,000. These relations highlight 1,000,000's interconnected role in exponentiation and place value within the powers of ten framework.

Prime Factorization and Divisors

The prime factorization of 1,000,000 is 26×562^6 \times 5^6. This decomposition arises from expressing 1,000,000 as 10610^6, where 10=2×510 = 2 \times 5. Given this prime factorization, the number of positive divisors is calculated as (6+1)(6+1)=49(6+1)(6+1) = 49. These divisors consist of all integers of the form 2a×5b2^a \times 5^b, where aa and bb are integers satisfying 0≤a≤60 \leq a \leq 6 and 0≤b≤60 \leq b \leq 6. The complete list of positive divisors is: 1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 125, 160, 200, 250, 320, 400, 500, 625, 800, 1000, 1250, 1600, 2000, 2500, 3125, 4000, 5000, 6250, 8000, 10000, 12500, 15625, 20000, 25000, 31250, 40000, 50000, 62500, 100000, 125000, 250000, 500000, 1000000. The sum of these divisors, denoted σ(1,000,000)\sigma(1,000,000), equals 2,480,437.

Other Arithmetic Properties

1,000,000 is an even integer, divisible by 2 as its prime factorization includes the factor 262^6. Due to the exponents of 6 in this factorization, 26×562^6 \times 5^6, it qualifies as a powerful number, where for every prime pp dividing the number, p2p^2 also divides it. The number is not perfect, as the sum of its proper divisors exceeds the number itself; specifically, it is abundant with the divisor function value σ(1,000,000)=2,480,437\sigma(1,000,000) = 2,480,437, yielding an abundance of 2,480,437−2×1,000,000=480,4372,480,437 - 2 \times 1,000,000 = 480,437. In modular arithmetic, 1,000,000 is congruent to 0 modulo 10, reflecting its divisibility by 10, and congruent to 1 modulo 999,999, since 1,000,000=999,999+11,000,000 = 999,999 + 1. Additionally, 1,000,000 is a Harshad number in base 10, divisible by the sum of its digits, which equals 1.

Magnitude and Visualization

Spatial and Physical Analogies

To visualize the magnitude of 1,000,000 in linear terms, consider that 1,000,000 millimeters equals exactly 1 kilometer, a distance comparable to the length of a short urban street or a typical neighborhood block. Another analogy involves human hair strands: if each has an average length of 100 millimeters (a typical measurement for short-cut scalp hair), then 1,000,000 such strands laid end-to-end would span approximately 100 kilometers, roughly the driving distance from London to Cambridge. In terms of area, 1,000,000 square centimeters equals 100 square meters, an expanse equivalent to the floor space of a modest two-bedroom apartment. For volume, 1,000,000 cubic millimeters of water occupies precisely 1 liter, the capacity of a standard beverage bottle. Extending this to granular materials, 1,000,000 grains of uncooked rice approximate 25 liters in bulk volume, filling a container about the size of a large backpack. A physical stacking analogy highlights the scale in height: each U.S. one-dollar bill measures approximately 0.11 millimeters thick, so a stack of 1,000,000 bills reaches about 110 meters tall—comparable to the height of a 30-story building. To arrive at this: multiply the number of bills (1,000,000) by the thickness per bill (0.11 mm), yielding 110,000 mm, which converts to 110 meters (since 1 m = 1,000 mm).

Temporal and Volumetric Scales

One million seconds equates to approximately 11.57 days, providing a tangible sense of this duration in everyday terms. This calculation derives from dividing 1,000,000 by the 86,400 seconds in a standard day (24 hours × 60 minutes × 60 seconds). Similarly, one million years spans exactly 1,000 millennia, as a millennium is defined as a period of 1,000 years. In biological contexts, one million heartbeats at an average resting rate of 70 beats per minute corresponds to about 238 hours, or roughly 9.92 days. This estimate assumes a consistent rate, though actual heartbeats vary with activity; at this pace, a person would accumulate one million beats in just under 10 days of continuous resting. For volumetric scales, one million cubic centimeters precisely equals one cubic meter, illustrating the unit's role in defining standard volumes in metric systems. In human physiology, one million red blood cells represent a minuscule fraction of the average adult's total, which comprises around 25 trillion such cells circulating to transport oxygen. One million characters of text approximates 125 pages in the Encyclopædia Britannica's 15th edition, based on its 32,640 pages containing about 44 million words at an average length of 6.3 characters per word (including spaces and punctuation). This comparison underscores the informational density of encyclopedic content, where such a volume captures substantial knowledge.

Applications and Cultural Significance

In Measurement Systems

The prefix "mega-" (symbol M) in the International System of Units (SI) represents a factor of 10⁶, or one million, facilitating the expression of large-scale measurements across scientific disciplines. This standardization aligns with the decimal nature of the metric system, where 1,000,000 serves as the foundational multiplier for the prefix. The adoption of the "mega-" prefix occurred as part of the formal establishment of the SI in 1960 by the 11th General Conference on Weights and Measures (CGPM), which unified existing metric conventions into a coherent international framework. Prior to this, similar prefixes were in informal use, but the 1960 resolution ensured their consistent application to SI base units like the meter, kilogram, and second, as well as derived units. In practice, the prefix is integral to fields such as physics and engineering; for instance, megahertz (MHz) measures frequency, where 1 MHz equals 1,000,000 cycles per second, often applied in electromagnetism and acoustics. Similarly, megawatt (MW) quantifies power, with 1 MW equivalent to 1,000,000 watts, a scale used for assessing the capacity of power plants and industrial machinery. For length, 1 megameter (Mm) corresponds to 1,000 kilometers, providing a convenient unit for geophysical distances despite its rarer usage. In energy terms, 1 megajoule (MJ) approximates the thermal content of 239 kilocalories, illustrating its relevance in nutritional and thermodynamic contexts.

In Computing and Data Storage

In computing, the number 1,000,000 serves as the basis for the decimal definition of a megabyte (MB), which equals exactly 1,000,000 bytes, aligning with the SI prefix "mega-" denoting a factor of 10^6. However, in binary-based systems common to digital storage and memory, a megabyte has historically been interpreted as 1,048,576 bytes (2^20), reflecting the power-of-two architecture of computers where memory is allocated in binary increments. This discrepancy arose in the early days of computing, as engineers approximated 2^10 (1,024) to the decimal "kilo" (1,000) for convenience, leading to widespread use of binary prefixes like 1,024^2 for megabytes in hardware specifications such as RAM. To resolve confusion, the International Electrotechnical Commission (IEC) introduced binary prefixes in 1998, defining the mebibyte (MiB) as precisely 1,048,576 bytes while reserving the megabyte for the decimal 1,000,000 bytes, particularly in contexts like hard drive capacities reported by operating systems. For example, early personal computers in the 1980s and 1990s advertised 1 MB of RAM as 1,048,576 bytes to match actual addressable memory, but storage vendors shifted toward decimal megabytes to align with marketing and SI standards, creating a persistent dual convention. Beyond storage, 1,000,000 pixels approximate one megapixel (MP), a unit used to describe the resolution of digital images and camera sensors, where total pixel count is often rounded to the nearest million for simplicity. This decimal interpretation avoids binary complications in imaging, as pixel arrays are typically specified in exact dimensions (e.g., 1,000 by 1,000 pixels yielding 1,000,000 total), emphasizing visual detail over computational powers of two.

Economic and Linguistic Uses

In economics, the number 1,000,000 serves as a benchmark for substantial wealth, most notably in the term "millionaire," which refers to an individual whose net worth equals or exceeds one million units of currency, such as dollars or pounds. The word originated in French as "millionnaire" around 1762, entering English usage by 1821 to describe those with such fortunes, often arising from trade, speculation, or industry during the Industrial Revolution. This threshold has symbolized financial elite status, influencing concepts like wealth distribution and economic inequality in modern societies. Linguistically, 1,000,000 frequently appears in idioms that convey rarity, value, or exaggeration in everyday English. The phrase "not one in a million" describes something or someone exceptionally rare or unique, emphasizing improbability on a vast scale. Similarly, a "million-dollar idea" denotes a concept with immense potential to generate significant profit or success, often used in business and entrepreneurial contexts to highlight innovative opportunities. These expressions underscore the number's cultural role as a metaphor for abundance or scarcity beyond literal arithmetic. The milestone of reaching a population of 1,000,000 has marked key economic turning points for urban centers, reflecting growth in commerce, labor, and infrastructure. For instance, London became the first modern European city to surpass this figure around 1800, with its 1801 census recording 1,096,784 residents, driven by industrialization and migration that fueled the British Empire's economic expansion. Such achievements highlighted 1,000,000 as a symbol of urban prosperity and the challenges of scaling economies in burgeoning metropolises.

Selected Nearby Numbers

Numbers from 1,000,001 to 4,999,999

The range from 1,000,001 to 4,999,999 encompasses several notable seven-digit numbers with connections to mathematics, computing, and popular culture. 1,000,003 holds distinction as the smallest prime number with seven digits. In computing, 1,024,000 bytes serves as an approximation for a megabyte in contexts blending binary (where kilobyte equals 1,024 bytes) and decimal (where megabyte equals 1,000 kilobytes) conventions, though standards vary between 1,000,000 bytes (SI) and 1,048,576 bytes (binary). The number 1,771,561 appears in the Star Trek episode "The Trouble with Tribbles" (1967), where Spock calculates it as the total progeny from a single tribble reproducing over three days at a rate of 10 to 11 offspring per litter every 12 hours, illustrating exponential growth: 1×116=1,771,5611 \times 11^6 = 1,771,561. In mathematics, 3,141,592 represents the integer part of π×1,000,000\pi \times 1,000,000, capturing the first seven digits of the constant π≈3.1415926535…\pi \approx 3.1415926535\ldots.

Numbers from 5,000,000 to 9,999,999

The number 5,000,000 represents five million and serves as an approximation for the population of the United States according to the 1800 federal census, which recorded a total of 5,308,483 residents. This milestone reflects early American demographic growth following independence, with the census conducted under the direction of the first U.S. Secretary of State, Timothy Pickering. The integer 6,674,300 captures the leading digits of the Newtonian gravitational constant GG, which has a measured value of 6.67430×10−116.67430 \times 10^{-11} m³ kg⁻¹ s⁻² as determined by the Committee on Data for Science and Technology (CODATA). This constant quantifies the strength of gravitational attraction between masses and is fundamental to classical mechanics, appearing in Newton's law of universal gravitation; its precise value was refined through experiments like the Cavendish torsion balance. 8,675,309 is a notable seven-digit prime number, famously embedded in the 1981 hit song "867-5309/Jenny" by Tommy Tutone, where it appears as a fictional phone number that became culturally iconic. Mathematically, it forms the smaller member of a twin prime pair with 8,675,311 and serves as the hypotenuse of a primitive Pythagorean triple: 8,675,3092=2,460,2602+8,319,14128,675,309^2 = 2,460,260^2 + 8,319,141^2. Its primality and geometric properties highlight its significance in number theory and recreational mathematics. 9,999,991 stands as the largest prime number with exactly seven digits, positioned just below the eight-digit threshold. This primality has been verified through probabilistic tests and factorization algorithms, confirming it has no divisors other than 1 and itself. While detailed exploration of seven-digit primes occurs elsewhere, this number exemplifies the upper boundary of primality in the range. The number 9,999,999 is the largest seven-digit integer, equivalent to 107−110^7 - 1, and consists of seven repeated 9's, making it a repdigit in base 10. Its digits sum to 63 (calculated as 9×79 \times 7), which is divisible by 9, confirming its divisibility by 9 as per the divisibility rule; factorizations yield 9,999,999=32×239×4,6499,999,999 = 3^2 \times 239 \times 4,649. This structure often arises in discussions of repunits and powers of 10, underscoring its role in illustrating numerical patterns and arithmetic properties.

Prime Numbers in the 7-Digit Range

The seven-digit prime numbers range from 1,000,000 to 9,999,999, encompassing a total of 586,081 such primes, calculated as the difference between the prime-counting function values π(10^7) = 664,579 and π(10^6) = 78,498. This count reflects the increasing sparsity of primes as numbers grow larger, with approximately 5.86% of integers in this interval being prime. The distribution aligns with the prime number theorem, which predicts a density of primes around 1/ln(n) for numbers near n in this range. The smallest seven-digit prime is 1,000,003, which immediately follows the composite 1,000,000 and 1,000,001 through 1,000,002. At the upper end, the largest prime below 10,000,000 is 9,999,991, marking the final prime in this digit length before transitioning to eight digits. These boundary examples illustrate the irregular yet asymptotically predictable placement of primes, with 1,000,003 differing from 1,000,000 by just 3 and 9,999,991 by 9 from the decade's end. Prime gaps in the seven-digit range—the differences between consecutive primes—exhibit variability but follow trends predicted by the prime number theorem, where the average gap is approximately ln(n). For n ≈ 10^6, this yields an average gap of about 13.8, increasing to roughly 16.1 near 10^7 as the local density decreases. While most gaps are small (often 2 or 4, as in twin or cousin primes), larger gaps occur, such as the maximal known gap of 282 between 1,693,109 and 1,693,391 in this range, highlighting the non-uniform distribution despite the overall logarithmic growth. Representative examples include the gap of 14 between 1,000,003 and 1,000,019, and a 20-gap near 5,000,000, underscoring how gaps fluctuate around the expected average without exceeding bounds proportional to (ln n)^2 under conjectures like Cramér's.

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