100,000
100,000 (one hundred thousand; also known as one lakh in the Indian numbering system) is a natural number following 99,999 and preceding 100,001, equivalent to $10^5 in scientific notation and serving as a key power of ten in the decimal system.[1]It has the prime factorization $2^5 \times 5^5, making it an even composite number composed of two distinct prime factors.[1][2]
As a highly factorable integer, 100,000 possesses exactly 36 positive divisors.[3][2]
The sum of these divisors totals 246,078, which exceeds twice the number (200,000), classifying 100,000 as an abundant number in number theory.[2]
Additionally, it qualifies as a Hamming number (or regular number), expressible solely as a product of powers of 2, 3, and 5—here with the exponent of 3 being zero—due to its form as $10^5.[2]
The square root of 100,000 is approximately 316.227766, an irrational value, and the number itself is a perfect fifth power since $10^5 = (10)^5.[1] In broader contexts, 100,000 represents a significant scale in measurement systems, such as 100 kilounits in the metric system (e.g., 100 kilometers), and it often denotes large quantities in demographics, finance (e.g., the A$100,000 fine imposed on Crown Melbourne casino in November 2025 for allowing an excluded person to gamble)[4], and computing, though its precise decimal value contrasts with binary approximations like 100 KB equaling 102,400 bytes.[5]
Names and Notation
Etymology and Linguistic Terms
The term "hundred thousand" in modern English is a compound formed from "hundred," derived from Old English hund, ultimately tracing back to Proto-Indo-European \ḱm̥tóm, denoting a group of one hundred, and "thousand," from Old English þusend, rooted in Proto-Germanic þūsundī, signifying a large or great thousand.[6][7] This phrasing evolved as English adopted multiplicative naming conventions for large numbers in the short scale, where 100,000 represents $10^5. In other languages, equivalents reflect similar compounding or unique cultural units. French uses cent mille, combining cent (hundred) and mille (thousand); German employs hunderttausend, from hundert (hundred) and tausend (thousand); and Spanish denotes it as cien mil, with cien (hundred) and mil (thousand).[8][9][10] In Hindi and broader South Asian numbering systems, lakh precisely means 100,000, originating from Sanskrit lakṣa, which denoted a mark or sign for this quantity and symbolized abundance or a milestone in wealth. In Mandarin Chinese, it is expressed as shí wàn (十万), multiplying wàn (ten thousand) by ten, aligning with the traditional wan-based structure for numbers beyond 10,000.[11] The number 100,000 represents a significant scale in early human counting systems, with written records of large numbers appearing around 3000 BCE in Sumerian cuneiform tablets used for accounting in trade and administration.[12] These proto-cuneiform notations, precursors to full cuneiform, employed impressed tokens on clay to tally commodities.[13] In the Babylonian sexagesimal system, inherited from the Sumerians around 2000 BCE, base-60 notation facilitated calculations involving large values—such as $60^3 = 216,000—for astronomical and economic purposes, underscoring its role in managing vast scales without decimal precision.[14]Numeral Representations
In the decimal system, which is the most widely used positional numeral system today, 100,000 is represented as 100,000, utilizing Arabic numerals where the commas serve as thousands separators in Anglo-American conventions. In Roman numerals, a non-standard extension for large values, 100,000 is denoted by C with a vinculum (overline), symbolizing 100 multiplied by 1,000, or equivalently the Unicode character ↈ for one hundred thousand.[15] The binary representation of 100,000 in base-2 is 11000011010100000, consisting of seventeen bits to encode the value through powers of two.[16] In hexadecimal notation, base-16, 100,000 is expressed as 186A0, where digits beyond nine use letters A–F to represent values 10–15.[17] International variations in decimal formatting affect the display of 100,000, particularly for thousands separators; for instance, many European locales use a period as the separator (100.000) while employing a comma for decimals, whereas the United States and United Kingdom favor the comma (100,000); these conventions align with ISO standards for currency presentation under ISO 4217 to ensure clarity in financial contexts.[18] In non-positional systems like ancient Egyptian hieroglyphs, which relied on additive repetition of base-10 symbols, 100,000 is represented by a distinct hieroglyph, the tadpole (Gardiner sign M17).[19][20] In the Indian numbering system, 100,000 corresponds to one lakh, a unit that groups digits differently from the Western system, placing commas after three digits from the right and then every two digits thereafter (1,00,000).[21]Mathematical Properties
Basic Arithmetic and Factorization
100,000 is a composite number with the prime factorization $100,000 = 2^5 \times 5^5, derived from its representation as $10^5 where $10 = 2 \times 5.[1][22] This factorization consists of the primes 2 and 5, each raised to the fifth power, confirming it is not prime as it has divisors other than 1 and itself.[23] The positive divisors of 100,000 total 36, calculated as (5+1)(5+1) = 36 from the exponents in its prime factorization.[1][2] These divisors are all products of the form $2^a \times 5^b where $0 \leq a \leq 5 and $0 \leq b \leq 5, including 1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 80, 100, 125, 160, 200, 250, 400, 500, 625, 800, 1,000, 1,250, 2,000, 2,500, 3,125, 4,000, 5,000, 6,250, 10,000, 12,500, 20,000, 25,000, 50,000, and 100,000.[24][2] Basic arithmetic operations with 100,000 include addition, such as $100,000 + 100,000 = 200,000, and multiplication, such as $100,000 \times 2 = 200,000.[25] The square root of 100,000 is exactly $100\sqrt{10}, which approximates to 316.227766.[26][27] As an even number ending in zero, 100,000 has parity even and $100,000 \mod 10 = 0.[28] For example, the greatest common divisor \gcd(100,000, 50,000) = 50,000, as 50,000 divides 100,000 evenly.Powers and Exponents
100,000 can be expressed as the fifth power of 10, denoted as $10^5.[29] It also equals $100^{2.5}, as $100 = 10^2 implies (10^2)^{2.5} = 10^5. Primarily, however, 100,000 arises from its prime factorization of $2^5 \times 5^5, which underpins these exponential forms.[30] Raising 100,000 to higher powers yields larger magnitudes: $100,000^2 = (10^5)^2 = 10^{10}, equivalent to 10 billion. Similarly, $100,000^3 = 10^{15}, which represents one quadrillion in the short scale numbering system.[31] These computations highlight the rapid growth in exponential contexts. The base-10 logarithm of 100,000 is precisely 5, reflecting its position as $10^5.[32] The natural logarithm, \ln(100,000), approximates 11.5129, calculated as $5 \ln(10). In exponential equations, such as solving $10^x = 100,000, the result is x = 5.[32] In scientific notation, 100,000 is compactly written as $1.0 \times 10^5, emphasizing its order of magnitude.[33] This form facilitates computations in fields like physics and engineering where large numbers are common.Approximations and Scale
Orders of Magnitude
In the base-10 logarithmic scale, orders of magnitude provide a framework for comparing quantities by powers of ten, where the order of magnitude of a number is the exponent when expressed in scientific notation. The number 100,000 equals exactly $10^5, positioning it firmly in the fifth order of magnitude, between $10^4 (10,000) and $10^6 (1,000,000).[34] This scale is widely used across scientific disciplines to convey relative sizes without precise values, emphasizing conceptual scale over exact measurements.[35] In acoustics, the decibel (dB) scale measures sound intensity logarithmically, with each 10 dB increment representing a tenfold increase in intensity relative to a reference level. Consequently, a level of 50 dB corresponds to an intensity ratio of $10^5, serving as a threshold for moderate sound pressures in environments like quiet offices.[36] While the Richter scale for earthquakes is also base-10 logarithmic—each unit increase denoting about 31.6 times more energy—it does not map directly to $10^5 as a simple intensity ratio, though magnitude 5 events release energy on the order of $10^7 compared to baseline tremors. Astronomically, large distances are quantified in parsecs (pc), where 1 pc approximates 3.26 light-years (ly). Thus, $10^5 pc equates to about 326,000 ly, a scale relevant for mapping structures like the Local Group of galaxies.[37] In computing and data storage, non-metric (binary) prefixes distinguish powers of 2 from decimal SI units, addressing ambiguities in byte measurements. For instance, 100,000 bytes equals approximately 97.65625 kibibytes (KiB), since 1 KiB = 1,024 bytes, in contrast to 100 kilobytes (kB) = 100,000 bytes under SI decimal conventions. This binary-decimal distinction, formalized by the International Electrotechnical Commission in 1998 and endorsed in NIST guidelines, persists amid ongoing standardization efforts to clarify large-scale data metrics.[38] To illustrate $10^5 in temporal contexts, the following table compares adjacent orders using seconds as a unit:| Order of Magnitude | Seconds | Approximate Equivalent |
|---|---|---|
| $10^4 | 10,000 s | 2.78 hours |
| $10^5 | 100,000 s | 1.16 days |
| $10^6 | 1,000,000 s | 11.57 days |