Hubbry Logo
700 (number)700 (number)Main
Open search
700 (number)
Community hub
700 (number)
logo
7 pages, 0 posts
0 subscribers
Be the first to start a discussion here.
Be the first to start a discussion here.
700 (number)
700 (number)
from Wikipedia

← 699 700 701 →
Cardinalseven hundred
Ordinal700th
(seven hundredth)
Factorization22 × 52 × 7
Greek numeralΨ´
Roman numeralDCC, dcc
Binary10101111002
Ternary2212213
Senary31246
Octal12748
Duodecimal4A412
Hexadecimal2BC16
ArmenianՉ
Hebrewת"ש / ן
Babylonian cuneiform𒌋𒐕𒐏
Egyptian hieroglyph𓍨

700 (seven hundred) is the natural number following 699 and preceding 701.

It is the sum of four consecutive primes (167 + 173 + 179 + 181), the perimeter of a Pythagorean triangle (75 + 308 + 317)[1] and a Harshad number.

Integers from 701 to 799

[edit]

Nearly all of the palindromic integers between 700 and 800 (i.e. nearly all numbers in this range that have both the hundreds and units digit be 7) are used as model numbers for Boeing Commercial Airplanes.

700s

[edit]

710s

[edit]

720s

[edit]

730s

[edit]
  • 730 = 2 × 5 × 73, sphenic number, nontotient, Harshad number, number of generalized weak orders on 5 points [30]
  • 731 = 17 × 43, sum of three consecutive primes (239 + 241 + 251), number of Euler trees with total weight 7 [31]
  • 732 = 22 × 3 × 61, sum of eight consecutive primes (73 + 79 + 83 + 89 + 97 + 101 + 103 + 107), sum of ten consecutive primes (53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97), Harshad number, number of collections of subsets of {1, 2, 3, 4} that are closed under union and intersection [32]
  • 733 = prime number, emirp, balanced prime,[33] permutable prime, sum of five consecutive primes (137 + 139 + 149 + 151 + 157)
  • 734 = 2 × 367, nontotient, number of traceable graphs on 7 nodes [34]
  • 735 = 3 × 5 × 72, Harshad number, Zuckerman number, smallest number such that uses same digits as its distinct prime factors
  • 736 = 25 × 23, centered heptagonal number,[35] happy number, nice Friedman number since 736 = 7 + 36, Harshad number
  • 737 = 11 × 67, palindromic number, blum integer.
  • 738 = 2 × 32 × 41, Harshad number.
  • 739 = prime number, strictly non-palindromic number,[36] lucky prime,[25] happy number, prime index prime

740s

[edit]
  • 740 = 22 × 5 × 37, nontotient, number of connected squarefree graphs on 9 nodes [37]
  • 741 = 3 × 13 × 19, sphenic number, 38th triangular number[3]
  • 742 = 2 × 7 × 53, sphenic number, decagonal number,[38] icosahedral number. It is the smallest number that is one more than triple its reverse. Lazy caterer number (sequence A000124 in the OEIS). Number of partitions of 30 into divisors of 30.[39]
  • 743 = prime number, Sophie Germain prime, Chen prime, Eisenstein prime with no imaginary part
  • 744 = 23 × 3 × 31, sum of four consecutive primes (179 + 181 + 191 + 193). It is the coefficient of the first degree term of the expansion of Klein's j-invariant, and the zeroth degree term of the Laurent series of the J-invariant. Furthermore, 744 = 3 × 248 where 248 is the dimension of the Lie algebra E8.
  • 745 = 5 × 149 = 24 + 36, number of non-connected simple labeled graphs covering 6 vertices[40]
  • 746 = 2 × 373 = 15 + 24 + 36 = 17 + 24 + 36, nontotient, number of non-normal semi-magic squares with sum of entries equal to 6[41]
  • 747 = 32 × 83 = ,[42] palindromic number.
  • 748 = 22 × 11 × 17, nontotient, happy number, primitive abundant number[43]
  • 749 = 7 × 107, sum of three consecutive primes (241 + 251 + 257), blum integer

750s

[edit]
  • 750 = 2 × 3 × 53, enneagonal number.[44]
  • 751 = prime number with a prime number of prime digits,[45] Chen prime, emirp,
  • 752 = 24 × 47, nontotient, number of partitions of 11 into parts of 2 kinds[46]
  • 753 = 3 × 251, blum integer
  • 754 = 2 × 13 × 29, sphenic number, nontotient, totient sum for first 49 integers, number of different ways to divide a 10 × 10 square into sub-squares [47]
  • 755 = 5 × 151, number of vertices in a regular drawing of the complete bipartite graph K9,9.[48]
  • 756 = 22 × 33 × 7, sum of six consecutive primes (109 + 113 + 127 + 131 + 137 + 139), pronic number,[2] Harshad number
  • 757 = prime number, palindromic prime, sum of seven consecutive primes (97 + 101 + 103 + 107 + 109 + 113 + 127), happy number.
  • 758 = 2 × 379, nontotient, prime number of measurement [49]
  • 759 = 3 × 11 × 23, sphenic number, sum of five consecutive primes (139 + 149 + 151 + 157 + 163), a q-Fibonacci number for q=3 [50]

760s

[edit]

770s

[edit]

780s

[edit]
  • 780 = 22 × 3 × 5 × 13, sum of four consecutive primes in a quadruplet (191, 193, 197, and 199); sum of ten consecutive primes (59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101), 39th triangular number,[3]a hexagonal number,[4] Harshad number
    • 780 and 990 are the fourth smallest pair of triangular numbers whose sum and difference (1770 and 210) are also triangular.
  • 781 = 11 × 71. 781 is the sum of powers of 5/repdigit in base 5 (11111), Mertens function(781) = 0, lazy caterer number (sequence A000124 in the OEIS)
  • 782 = 2 × 17 × 23, sphenic number, nontotient, pentagonal number,[13] Harshad number, also, 782 gear used by U.S. Marines
  • 783 = 33 × 29, heptagonal number
  • 784 = 24 × 72 = 282 = , the sum of the cubes of the first seven positive integers, happy number
  • 785 = 5 × 157, Mertens function(785) = 0, number of series-reduced planted trees with 6 leaves of 2 colors [68]

790s

[edit]

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia
from Grokipedia
700 is the natural number following 699 and preceding 701, an even composite integer with the prime factorization 22×52×72^2 \times 5^2 \times 7. It has exactly 18 positive divisors (1, 2, 4, 5, 7, 10, 14, 20, 25, 28, 35, 50, 70, 100, 140, 175, 350, and 700), whose sum is 1736, and is a Harshad number because it is divisible by the sum of its digits (7 + 0 + 0 = 7). In the Roman numeral system, 700 is denoted as DCC, representing 500 + 100 + 100.

Mathematical Properties

700 belongs to several number-theoretic categories due to its and other attributes. As a , it is the product of three distinct primes each raised to a power. Its divisors are 1, 2, 4, 5, 7, 10, 14, 20, 25, 28, 35, 50, 70, 100, 140, 175, 350, and 700. Notably, 700 is the number of symmetric octacubes (polycubes composed of 8 cubes with ) in . It is not a perfect square (√700 ≈ 26.457), perfect cube, or number.

Numeral Representations

Beyond , 700 in binary is 1010111100, in 2BC, and in base-10 it is simply 700. In ancient numeral systems, such as Greek, it is denoted as ψʹ in the alphabetic system.

Historical and Cultural Contexts

The number 700 has appeared in historical timelines, such as the year 700 AD marking the flourishing of the kingdom in . In modern culture, it denotes metrics like the 700 MHz frequency band in standards or 700-calorie diets in nutrition, though these are contextual uses rather than inherent properties.

Mathematical Properties

Prime Factorization

The prime of 700 is obtained by repeatedly dividing by the smallest prime numbers until only primes remain. Starting with 700, divide by 2 to get 350; divide 350 by 2 to get 175; divide 175 by 5 to get 35; divide 35 by 5 to get 7; and 7 is prime. Thus, 700=22×52×7700 = 2^2 \times 5^2 \times 7. From this , ϕ(700)\phi(700) counts the integers up to 700 that are coprime to 700. The is ϕ(n)=npn(11/p)\phi(n) = n \prod_{p \mid n} (1 - 1/p), where pp are the distinct prime factors. Substituting the values: ϕ(700)=700×(11/2)×(11/5)×(11/7)=700×1/2×4/5×6/7\phi(700) = 700 \times (1 - 1/2) \times (1 - 1/5) \times (1 - 1/7) = 700 \times 1/2 \times 4/5 \times 6/7. First, 700×1/2=350700 \times 1/2 = 350; then 350×4/5=280350 \times 4/5 = 280; finally, 280×6/7=240280 \times 6/7 = 240. Thus, ϕ(700)=240\phi(700) = 240. The number of distinct prime factors, denoted ω(700)\omega(700), is 3 (corresponding to 2, 5, and 7). The total number of prime factors counting multiplicity, denoted Ω(700)\Omega(700), is 5 (two 2's, two 5's, and one 7).

Divisors and Multiples

The positive divisors of 700 are 1, 2, 4, 5, 7, 10, 14, 20, 25, 28, 35, 50, 70, 100, 140, 175, 350, and 700. These divisors arise from the prime factorization of 700 as 22×52×712^2 \times 5^2 \times 7^1, where every combination of the exponents (0 to 2 for 2 and 5, 0 to 1 for 7) yields a . The number of divisors, denoted d(700)d(700) or τ(700)\tau(700), is calculated using the for with prime p1e1p2e2pkekp_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} as d(n)=(e1+1)(e2+1)(ek+1)d(n) = (e_1 + 1)(e_2 + 1) \cdots (e_k + 1). Thus, d(700)=(2+1)(2+1)(1+1)=3×3×2=18d(700) = (2+1)(2+1)(1+1) = 3 \times 3 \times 2 = 18. The sum of the divisors, denoted σ(700)\sigma(700), is the total 1+2+4+5+7+10+14+20+25+28+35+50+70+100+140+175+350+700=17361 + 2 + 4 + 5 + 7 + 10 + 14 + 20 + 25 + 28 + 35 + 50 + 70 + 100 + 140 + 175 + 350 + 700 = 1736. The sum of the proper divisors (excluding 700) is 1036. The multiples of 700 follow the pattern 700×k700 \times k for positive integers kk. The first ten multiples are presented below for reference:
kkMultiple
1
21400
32100
42800
53500
64200
74900
85600
96300
107000
satisfies standard divisibility rules for its prime factors: it is divisible by 2 as an even number, by 5 as it ends in 00, and by 7 since [700](/page/700)÷7=100[700](/page/700) \div 7 = 100. These properties extend to checking divisibility by itself for larger numbers.

Other Characteristics

700 is an even number, as it is divisible by 2. It is also a , possessing divisors other than 1 and itself. 700 qualifies as an , where the sum of its proper divisors totals 1036, exceeding the number itself; this contrasts with perfect numbers, where the sum equals the number, and deficient numbers, where it falls short. The full sum of divisors, σ(700)\sigma(700), is 1736, which is greater than twice 700 (1400), confirming its abundant status. Additionally, 700 is a Harshad number, being divisible by the sum of its digits: 7 + 0 + 0 = 7, and 700 ÷ 7 = 100. Under the Collatz conjecture, the sequence starting from 700 reaches 1 after 82 steps. As the 700th positive integer, 700 falls between the squares of 26 (676) and 27 (729).

Representations in Numeral Systems

Decimal and Roman Numerals

In the decimal numeral system, 700 is written as 700, a three-digit number where the leftmost digit represents 7 hundreds (700), the middle digit 0 tens (0), and the rightmost digit 0 units (0), following the base-10 positional notation that assigns powers of 10 to each place value from right to left. This Hindu-Arabic numeral system, originating in India and transmitted through the Islamic world, was introduced to Europe by the Italian mathematician Leonardo Fibonacci in his 1202 book Liber Abaci, which popularized its use for arithmetic and commerce over the more cumbersome Roman system. In English, the number is pronounced "seven hundred," combining the cardinal numeral "seven"—derived from Old English seofon, from Proto-Germanic sebun, ultimately tracing to Proto-Indo-European septḿ̥ meaning "seven"—with "hundred," from Old English hundred, a compound of hund ("a hundred," akin to the root for "hund-redth") and rad ("count" or "reckoning"), denoting a base unit of 100 in early Germanic reckoning. The Roman numeral representation of 700 is DCC, constructed additively as D (500) plus CC (two instances of C, each worth 100, totaling 200), adhering to the standard rules of Roman notation where values are summed from left to right without subtractive principles for this quantity, as subtractive notation (e.g., ) applies only to specific cases like 4, 9, 40, 90, , and to avoid excessive repetition. Across languages, the verbal form of 700 varies: in French, it is "sept cents," literally "seven hundreds," reflecting the language's tendency to pluralize "cent" (hundred) for multiples beyond one. In , it is "qī bǎi" (七百), combining "qī" for seven and "bǎi" for hundred, written with the characters 七 (seven) and 百 (hundred) in the Sino-Xenic numbering system.

Binary, Octal, and Hexadecimal

In binary, the number 700 is represented as 101011110021010111100_2, requiring 10 bits to express its value. This uses powers of 2, where the leftmost bit represents 29=5122^9 = 512 and the rightmost is 20=12^0 = 1. To convert 700 to binary, repeatedly divide by 2 and record the s from bottom to top: 700÷2=350700 \div 2 = 350 remainder 0, 350÷2=175350 \div 2 = 175 r 0, 175÷2=87175 \div 2 = 87 r 1, 87÷2=4387 \div 2 = 43 r 1, 43÷2=2143 \div 2 = 21 r 1, 21÷2=1021 \div 2 = 10 r 1, 10÷2=510 \div 2 = 5 r 0, 5÷2=25 \div 2 = 2 r 1, 2÷2=12 \div 2 = 1 r 0, 1÷2=01 \div 2 = 0 r 1, yielding 101011110021010111100_2. The binary system forms the foundation of digital , as it aligns directly with the two-state logic of electronic circuits (0 for off, 1 for on). In octal (base-8), 700 is represented as 127481274_8. This can be derived from the binary form by grouping bits into sets of three from the right (padding with a leading zero if needed): 101011110021010111100_2 becomes 0010101111002001|010|111|100_2, which converts to 127481|2|7|4_8 since 0012=1001_2=1, 0102=2010_2=2, 1112=7111_2=7, and 1002=4100_2=4. Alternatively, divide repeatedly by 8: 700÷8=87700 \div 8 = 87 remainder 4, 87÷8=1087 \div 8 = 10 r 7, 10÷8=110 \div 8 = 1 r 2, 1÷8=01 \div 8 = 0 r 1, reading remainders upward as 127481274_8. Octal representations compactly group three binary digits each, and were commonly used in early computing for file permissions and assembly language addressing. In hexadecimal (base-16), 700 is represented as 2BC162BC_{16}. Digits beyond 9 use letters A-F (where A=10, B=11, C=12, etc.). Conversion involves repeated division by 16: 700÷16=43700 \div 16 = 43 remainder 12 (C), 43÷16=243 \div 16 = 2 r 11 (B), 2÷16=02 \div 16 = 0 r 2, yielding 2BC162BC_{16} from bottom to top. Hexadecimal efficiently encodes four binary digits per symbol, making it prevalent in modern computing for displaying memory addresses, debugging low-level code, and specifying colors in web development (e.g., RGB values). The following table summarizes these representations for quick comparison:
Numeral SystemRepresentationEquivalent Decimal Value
Binary1010111100700
Octal1274700
Hexadecimal2BC700

Significance in Science and Technology

In Physics and Chemistry

In physics, the number 700 is notably associated with the electromagnetic spectrum, where 700 nanometers (nm) marks the approximate upper limit of visible light, corresponding to the deepest red hues perceivable by the human eye. The visible spectrum spans roughly from 400 nm (violet) to 700 nm (red), encompassing the wavelengths that stimulate human color vision. This range is fundamental in optics and photometry, as it defines the portion of the electromagnetic spectrum interacting directly with biological photoreceptors. The energy of a photon at this wavelength can be calculated using the formula E=hcλE = \frac{hc}{\lambda}, where hh is Planck's constant (4.135667696×10154.135667696 \times 10^{-15} eV·s), cc is the speed of light (2.99792458×1082.99792458 \times 10^8 m/s), and λ=700\lambda = 700 nm (7×1077 \times 10^{-7} m). Substituting these values yields E1.77E \approx 1.77 eV, a low-energy photon typical of red light used in applications like photosynthesis studies, where chlorophyll absorbs near 700 nm. This energy level underscores the transition from visible to near-infrared radiation, influencing fields such as spectroscopy and photobiology. In chemistry and , 700°C serves as a critical threshold in processes like of ceramics, where powders are heated to achieve densification without full . For instance, certain ceramics form desired phases when processed at 700°C in methods like sol-gel synthesis, aiding in the development of materials with controlled microstructures. Similarly, in , 700°C is employed in process annealing of low-alloy steels to relieve internal stresses and refine grain structure while avoiding excessive phase transformations. These applications highlight 700's role in controlling material properties through controlled thermal treatments. Particle physics experiments frequently reference energies around 700 MeV (mega-electronvolts), such as in studies on nuclei like , where incident electron energies of 700 MeV probe proton spectral functions and nuclear structure via . Accelerators like the JINR Phasotron, which operates at 680 MeV, have historically enabled high-intensity proton beams for fragmentation cross-section measurements up to around 680 MeV/A, aiding understanding of interactions and . These examples illustrate 700 MeV as a benchmark energy scale in probing subatomic interactions.

In Computing and Engineering

In computing, the number 700 appears in non-standard extensions to HTTP status codes, where it is used by certain proxy servers like to indicate soft errors, such as unexpected responses during or proxy handling, beyond the standard range of codes up to 599 defined in RFC 9110. These extensions allow proxies to signal internal issues without disrupting standard client-server interactions, though they are not part of the official IETF specifications. The 700 series, introduced in the , represented a pivotal line of vacuum-tube-based mainframe computers that advanced commercial and scientific . The series debuted with the in 1953, capable of up to 16,000 additions per second and featuring up to 4,096 words of electrostatic () storage memory, marking IBM's entry into high-speed electronic for engineering applications. Subsequent models like the 704 and 709 built on this foundation, influencing transistor-based systems in the 1960s. In mobile communications engineering, the 700 MHz band is allocated for LTE networks, particularly Band 12, which operates in the lower 700 MHz spectrum (uplink: –716 MHz, downlink: –746 MHz) to provide extended coverage in rural and indoor environments due to its favorable propagation characteristics. This band supports up to 18 MHz of channel bandwidth per direction, enhancing signal penetration and range compared to higher frequencies. The bandwidth BB for such a paired frequency-division duplex (FDD) band is approximated as B2(fcfl)B \approx 2(f_c - f_l), where fcf_c is the center and flf_l is the lower edge, yielding about 34 MHz total for Band 12 to accommodate multiple carriers. Compact discs (CDs) commonly feature a capacity tied to 700 in storage contexts, where standard 74-minute audio CDs hold approximately 650–700 MB of data, depending on error correction and overhead. This equates to about 650 MB (650 × 10^6 bytes) for data modes under ISO/IEC 10149, but manufacturers often label 80-minute variants as 700 MB using notation (700 × 10^6 bytes), which operating systems may display as roughly 668 MiB in binary (mebibytes) due to the distinction between base-10 and base-2 byte measurements. In representation, 700 is 0x2BC, occasionally referenced in low-level addressing for legacy systems.

Historical and Cultural Contexts

In History and Chronology

The year 700 marks a pivotal point in as a signifying the conclusion of the AD, a era defined by profound geopolitical shifts, including the consolidation of Islamic rule in the , the zenith of Chinese imperial expansion under the , and the reconfiguration of power in and , ushering in deeper medieval transitions. This period's end underscored the transition from to the , with the rise of new empires disrupting longstanding Roman and Persian influences. In the Anno Domini dating system, 700 AD fell during the reign of Umayyad Caliph Abd al-Malik (r. 685–705), whose administration stabilized the after civil strife through key reforms, such as standardizing the Arab-Sassanian silver coinage in 696–697 and mandating for official documents by 700, enhancing imperial unity across diverse territories from to . Concurrently, in , the under Empress (r. 690–705) reached a high point of prosperity, with advancements in poetry, sculpture, and trade along the , as exemplified by the completion of grand Buddhist monuments like the . These developments highlighted 700 AD as a nexus of cultural and administrative innovation in both Islamic and Chinese spheres. The date 700 AD was calculated under the , introduced by in 45 BC and widely adopted in the Christian world by the AD, featuring a that began on a Wednesday. The subsequent Gregorian reform in 1582 AD, promulgated by to correct the Julian calendar's overestimation of the solar year by about 11 minutes annually, did not retroactively alter 700 AD dates, as the reform addressed accumulated drift over centuries without impacting earlier chronology. Archaeological findings associate circa 700 BC with the onset of decline for the kingdom in the , where Assyrian campaigns under (r. 721–705 BC) in the late 8th and early 7th centuries BC eroded its fortifications and economy, culminating in its collapse by the mid-6th century BC amid invasions by and . Likewise, around 700 BC, the Olmec civilization in —centered at sites like San Lorenzo and —was transitioning from its formative phase (c. 1400–900 BC) to its apogee (c. 900–600 BC), marked by monumental architecture and jade carvings, before environmental factors and internal shifts initiated a gradual depopulation and cultural transformation by 400 BC.
PeriodKey Events
700 AD under Abd al-Malik implements currency reforms, unifying the economy across conquered lands from Iberia to .
in , ruled by Empress , fosters artistic and religious , including expansions in and literature.
700 BC, under kings such as (r. 681–669 BC) and (r. 669–627 BC), dominates the , subjugating and while pressuring .
Greek city-states expand colonization efforts, founding settlements like in (c. 750 BC) and Cyrene in (c. 631 BC), spreading Hellenic culture.
Olmec society in produces iconic colossal heads and develops early writing systems during its peak phase at [La Venta](/page/La Venta).

In Culture and Symbolism

In the Bible, the number 700 frequently appears in narratives highlighting military prowess and royal excess. For instance, Judges 20:16 describes 700 elite left-handed slingers from the , each capable of accurately hitting a target the width of a with a stone, underscoring themes of precision and divine selection in battle. Similarly, 1 Kings 11:3 recounts King maintaining 700 wives of royal birth alongside 300 concubines, a vast that symbolized political alliances but ultimately contributed to his spiritual downfall by turning his heart from . While not assigned particular luck in , where numbers like 8 denote prosperity, 700's components (qī for 7, evoking rise or energy, and bǎi for hundred) evoke a sense of balanced progression without standout auspiciousness. In , 700 holds notable presence through "," a flagship program launched in 1966 and hosted by figures like , named for the 700 founding members who committed $10 monthly pledges to support its mission of faith-based news and ministry. In sports, the 700-home-run milestone in represents an elite achievement, reached by only four players: (714), (755), (762), and (703), with Pujols crossing the mark on September 23, 2022, against the , cementing his legacy amid widespread celebration.

References

Add your contribution
Related Hubs
User Avatar
No comments yet.