Copyright © Philip M. Parker, INSEAD. Terms of Use.
Definition: Cubic |
CubicAdjective1. Having three dimensions. Source: WordNet 1.7.1 Copyright © 2001 by Princeton University. All rights reserved. |
Date "cubic" was first used in popular English literature: sometime before 1663. (references) |
| Domain | Definition |
Mining | A. Having the form of a cube, as a cubic crystal; or referring to directions parallel to the faces of a cube, as cubic cleavage. See also:cubic system b. Crystal cleavage with three planes at mutual right angles. c. An alternate name for the isometric crystal system. See also:isometricd. In crystal structures, a cation coordination of six equidistant anions. (references) |
Source: compiled by the editor from various references; see credits. | |
(From Wikipedia, the free Encyclopedia)
In math, to the power of 3 (i.e. x^3 means x to the cube).See Cube.
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Cubic."
(From Wikipedia, the free Encyclopedia)
A cubic equation is a polynomial equation in which the highest occurring power of the unknown x is the third power. An example is the equationand the general form may be written as
- 2x3 - 4x2 + 3x - 4 = 0
where we assume that the coefficients a0,...,a3 are real numbers with a3 being non-zero.
- a3x3 + a2x2 + a1x + a0 = 0
Solving a cubic equation amounts to finding the rootss of a cubic function. Every cubic equation has at least one solution x among the real numbers. The following qualitatively different cases are possible:
The discriminant can be used to quickly decide whether the equation has multiple solutions.
- Three different real solutions
- Two real solutions, one of them is a double solution
- A single real solution which is a triple solution
- A single real solution and a pair of complex conjugate solutions which are complex numbers.
The solutions can be found with the following method due to Tartaglia and published by Gerolamo Cardano in 1545.
We first divide the given equation by a3 to arrive at an equation of the form
The substitution x = t - a/3 eliminates the quadratic term and we get at a cubic equation of the form
- x3 + ax2 + bx + c = 0
To solve this equation, find two numbers u and v such that
- t3 + pt + q = 0. (1)
A solution to our equation is then given by
- u3 - v3 = q
- uv = p/3
as can be checked by directly substituting this value for t in (1).
- t = v - u
The above system for u and v can always be solved: solve the second equation for v, substitute into the first equation, solve the resulting quadratic equation for u3, then take the cube root to find u. In some cases the quadratic equation will give complex solutions, even though at least one solution t of (1) will be real. This was already noticed by Cardano and is a strong argument for the usefulness (if not the existence) of complex numbers.
Once the values for t are known, the substitution x = t - a/3 can be undone to find the values of x solving the original equation.
So, if we have an equation
we set
and have
- and
So that u3 - v3 = q, and uv = p/3, we find
and since x + b/3 = v - u then
- and
If the square root is of a negative number, then the cubic root will be of a complex number. A way of taking the cubic root of a complex number is to convert the complex number to polar coordinates with the angle 0 along the real axis, divide the angle by 3, and take the cubic root of the modulus. There might be an easier way.
Note that in finding u, there were 6 possibilities, since there are two solutions to the square root, and three complex solutions to the cubic root. However, which solution to the square root is chosen does not affect the final resulting x.
See also: linear equation, quadratic equation, quartic equation, quintic equation, Omar Khayyam
To do:
- Example
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Cubic equation."
| The following table is compiled from various sources, across various languages. When English abbreviations or acronyms come from a non-English source, this is noted. | |||
| Entry | Source | Expression | Field |
CUBIC | English | Common User Baseline for Intelligence Community | Military & Defense |
| cu | English | Cubic | Meteorology & Standards |
Source: compiled by the editor, based on several corpora (additional references). | |||
Synonym: CubicSynonym: three-dimensional (adj). (additional references) |
| Antonyms: linear (adj), planar (adj). (additional references) |
| Context | Synonyms within Context (source: adapted from Roget's Thesaurus). |
Jewelry | Diamond, brilliant, rock; beryl, emerald; chalcedony, agate, heliotrope; girasol, girasole; onyx, plasma; sard, sardonyx; garnet, lapis lazuli, opal, peridot, tourmaline, chrysolite; sapphire, ruby, synthetic ruby; spinel, spinelle; balais; oriental, oriental topaz; turquois, turquoise; zircon, cubic zirconia; jacinth, hyacinth, carbuncle, amethyst; alexandrite, cat's eye, bloodstone,diamond, brilliant, rock; beryl, emerald; chalcedony, agate, heliotrope; girasol, girasole; onyx, plasma; sard, sardonyx; garnet, lapis lazuli, opal, peridot, tourmaline, chrysolite; sapphire, ruby, synthetic ruby; spinel, spinelle; balais; oriental, oriental topaz; turquois, turquoise; zircon, cubic zirconia; jacinth, hyacinth, carbuncle, amethyst; alexandrite, cat's eye, bloodstone, hematite, jasper, moonstone, sunstone. |
| Source: adapted from Roget's Thesaurus. | |
| Domain | Usage | |
Screenplays | It's got a cop motor, a 440 cubic inch plant, it's got cop tires, cop suspensions, cop shocks (The Blues Brothers; writing credit: Dan Aykroyd ; John Landis) I know everything about you, Colonel Stok from the size of your refrigerator to the cubic capacity of your mistress (Funeral in Berlin; writing credit: Len Deighton; Evan Jones) Well, you'd look cheerful too with fifty cubic centimeters throbbing away between your legs (The Killing of Sister George; writing credit: Lukas Heller; Frank Marcus) | |
Source: compiled by the editor from various references; see credits. | ||
| Domain | Title | ||
References | |||
Books |
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High Tech |
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Consumer Goods |
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Source: compiled by the editor from various references; see credits. | |||
| Thumbnail | Description & Credit | Thumbnail | Description & Credit |
 | West Fork Blackbird Creek, a tailing facility. The structure to the right is a spillway. Several million cubic yards contaminated tailings are behind the dam. Water is diverted into a concrete channel so it does not come into contact with the tailings. Credit: NOAA Restoration Center. |  | Figure 44. Thoulet total immersion hydrometer, designed by Professor Julien Thoulet for the purpose of studying the density of water from depth for the purpose of studying sub-surface currents. Professor Thoulet produced this relatively small sampling device of 200 cubic centimeter volume with Professor Henry Adolpe Chevalier of the University of Nancy. Credit: Sailing for Science - the NOAA Fleet Then and Now. |
 | NRCS employee Stan Janowitz oversaw the removal of more than 142 cubic feet of rock to help restore salmon habitat. Colville Indian Reservation,Omak Washington. Credit: Gary Wilson. |  | 142 Cubic feet were removed from the Omak Creek watershed in order to restore fish habitat. Omak, Washington. Credit: Gary Wilson. |
 | Ten-million cubic foot "Winzen" research balloon on the carrier's flight deck just prior to launching, during Operation "Skyhook", Refly "B", 30 January 1960. The balloon carried scientific devices to measure and record primary cosmic rays at 18-to-22 miles altitude. Credit: NAVY. |  | Strip mining operations with a thirty-two cubic yard steam shovel. Cherokee County, Kansas. Credit: Library of Congress. |
 | Moving van, American Storage & Transfer. 1000 cubic ft. capacity. Credit: Library of Congress. | ||
Source: pictures compiled by the editor from various references; see picture credits. | |||
 |
| "Cubic Feet" by Ariel C. Commentary: "Cubic Feet." |
Source: photographs selected by the editor, with permission from the photographers. |
| Author | Quotation |
Walt Whitman | To me every hour of the light and dark is a miracle. Every cubic inch of space is a miracle. |
Source: compiled by the editor from various references. | |
| Subject | Topic | Quote |
Health | The number of platelets (the component of the blood that helps stop bleeding in case of injury) per cubic millimeter of blood is counted. (references) | |
The white blood cells, those components of the blood that fight infection, are counted, and the number of those cells per cubic millimeter of blood is established. (references) | ||
Business | Shipping charges from the U.S. are approximately US$80 per cubic meter. (references) | |
On average, each German pays approximately DM 3.60 ($2.3) per cubic meter. (references) | ||
In 1998 and 1999, total production was 930,000 and 700,000 cubic meters respectively. (references) | ||
Economic History | Netherlands | Per year, a solar boiler saves about 200 cubic meters of natural gas. (references) |
Poland | Annual shipments will amount to 14 billion cubic meters of natural gas. (references) | |
Vietnam | Vietnam has an estimated 6 trillion cubic feet (Tcf) of proven gas reserves. (references) | |
Political Economy | BAHRAIN | Bahrain's natural gas production is about 1.11 billion cubic feet per day. (references) |
ALGERIA | It is one of the world's largest producers of natural gas with reserves of 4.52 trillion cubic meters. (references) | |
Trade | Argentina | Storage of merchandise in the Tucuman Free Zone's general depots costs $6.00 per cubic meter, per month. (references) |
Travel | Chad | The rate is fixed at 20 cubic meters at USD 0.20. If water installation is required, the rate is based on the location and importance of the site. (references) |
Nicaragua | A 3,500 cubic foot cold storage facility at the airport opened in 1996. The airport is currently building an expansion to double its passenger traffic capacity to handle 1.4 million passengers per year by the end of 2002. (references) | |
Thailand | Water quality ranges widely . Many Thai people drink bottled water which is inexpensive and readily available . In Bangkok and major cities, tap water is safe enough for bathing . Water is priced from $0.15 to $0.43 per cubic meter in accordance with the usage. (references) | |
Worker Rights | Nepal | To date, approximately 1,000 heads of household have been provided with up to .335 acres of land and 75 cubic feet of timber to build houses. (references) |
Source: compiled by the editor from ICON Group International, Inc.; see credits. | ||
| "Cubic" is generally used as an adjective (general or positive) -- approximately 100.00% of the time. "Cubic" is used about 437 times out of a sample of 100 million words spoken or written in English. Its rank is based on over 700,000 words used in the English language. Some parts-of-speech are not covered due to the samples used by the British National Corpus. (note: percents less than one-hundredth of one percent have been omitted) |
| Parts of Speech | Percent | Usage per 100 Million Words | Rank in English |
| Adjective (general or positive) | 100% | 437 | 13,188 |
Source: compiled by the editor from several corpora; see credits.
| Country | Name |
| USA | Cubic Corporation |
| (more examples...) |
Source: compiled by the editor from Icon Group International, Inc.
Expressions using "cubic": bale cubic capacity ♦ Circular cubic ♦ cubic capacity ♦ cubic centimeter ♦ cubic centimetre ♦ Cubic cleavage ♦ cubic content ♦ cubic content unit ♦ cubic decimeter ♦ cubic decimetre ♦ cubic equation ♦ cubic foot ♦ cubic inch ♦ cubic kilometer ♦ cubic kilometre ♦ cubic measure ♦ cubic meter ♦ cubic metre ♦ cubic millimeter ♦ cubic millimetre ♦ cubic niter ♦ Cubic number ♦ cubic ton ♦ cubic yard ♦ cubic zirconia. Additional references. | |
| Hyphenated Usage | |
Beginning with "cubic": cubic-foot. | |
Containing "cubic": eight-cubic-foot. | |
| Source: compiled by the editor from various references; see credits. | |
| The following statistics estimate the number of searches per day across the major English-language search engines as identified by various trade publications. Hyperlinks lead to commercial use of the expression at Amazon.com. |
| Language | Translations for "cubic"; alternative meanings/domain in parentheses. | |
Afrikaans | Kubaans (Cuban). (various references) | |
Albanian | në trajtë kubi (cubical), kubik (cubical). (various references) | |
Arabic | مكعب الشكل, مكعب (cube). (various references) | |
Bulgarian | кубичен (cubical, solid). (various references) | |
Chinese | 立方體 (cube), 立方体 (Cube, Cubical, Cubically). (various references) | |
Czech | kubický (solid), krychlový (cubed, cubical, quadratic). (various references) | |
Danish | kubisk (cubical, cuboid). (various references) | |
Dutch | kubusvormig (Cuban). (various references) | |
Farsi | مکعبی . (various references) | |
Finnish | kuutiomainen. (various references) | |
French | cubique (cubical). (various references) | |
German | kubisch (cubical, cubiform). (various references) | |
Greek | κυβικόσ, κυβικός (cubical). (various references) | |
Hebrew | מעוקב (cube). (various references) | |
Hungarian | kocka alakú (cubical, cubiform, cuboid). (various references) | |
Indonesian | kubik. (various references) | |
Italian | cubico (cubical). (various references) | |
Japanese Kanji | キャンプ村 (campaign, campaign sale, campsite, Canberra, Canon, Cuba, Cuban heel, cubic type, cubism, cue, culotte, culotte skirt, Cupid, curacao, curator, cure, curie, curiosity, curium, cute, cuticle, cuticle cream, cuticle remover, cutie, killer, killer dust, kiosk, queue), 三次 (three times). (various references) | |
Japanese Katakana | さんじ (3 o'clock snack, childbirth, compliment, councillor, disaster, eulogy, horrible accident, newborn baby, praise, secretary, silkworm, three times), キュービック . (various references) | |
Korean | 입방 (Cubical). (various references) | |
Manx | kioobagh (cubical, cubist), kern-thummidagh (cubical). (various references) | |
Papiamen | kubano (Cuban), kúbiko (Cuban). (various references) | |
Pig Latin | ubiccay.(various references) | |
Portuguese | cúbico (cubical, solid). (various references) | |
Romanian | cubic (cubical), cub (bricks, cube), de gradul trei. (various references) | |
Russian | кубический (cubical). (various references) | |
Serbo-Croatian | kubni (cubical). (various references) | |
Spanish | cúbico (cubical), cubano (Cuban). (various references) | |
Swedish | kubiska, kubisk, kubik-, kubik (yardage). (various references) | |
Turkish | küp biçiminde (cubically), küp (cruse, cube, earthenware jar), kübik (cubiform, solid). (various references) | |
Ukrainian | кубічний. (various references) | |
Vietnamese | đường cubic (cubical). (various references) | |
| Source: compiled by the editor from various translation references. | ||
Derivations | |
Words beginning with "cubic": cubical, cubically, cubicities, cubicity, cubicle, cubicles, cubicly, cubics, cubicula, cubiculum. (additional references) | |
| |
"Cubic" is suggested in spellcheckers for the following: Caboc, cbi, Cbis, Ccbi, Cebaco, Cibc, Cibucu, cobi, conbac, cubia, cubica, cubics, cubik, cubin, cubip, cubo, cucib, cuib, cunic, Cunico, curic, Cuzick, Duboc, Dubuc, kebick, Kubach, kubic, kubicki, Kubik, Kubin, Kubis, kufic, rubic, sabic, scotbic, Ubik, Ucbc. (additional references) | |
| Source: compiled by the editor, based on several corpora (additional references). | |
| # of Phoneme Matches | Pronunciation | Word(s) rhyming with "cubic" (pronounced kyuw"bik) |
| 5 | -y uw" b i k | pubic. |
| 4 | -uw" b i k | cherubic. |
| 3 | -b i k | acerbic, aerobic, amoebic, anaerobic, arabic, ascorbic, claustrophobic, homophobic, phobic, syllabic, xenophobic. |
Source: compiled by the editor (additional references); see credits. | ||
Scrabble® Enable2K-Verified Anagrams | |
| Words within the letters "b-c-c-i-u" | |
-2 letters: cub. | |
-3 letters: bi. | |
| Words containing the letters "b-c-c-i-u" | |
+1 letter: cubics. | |
+2 letters: bucolic, cubical, cubicle, cubicly, succubi. | |
+3 letters: bacchius, bucolics, cherubic, columbic, crucible, cubicity, cubicles, cubicula, cubistic, cucurbit. | |
+4 letters: concubine, corbicula, coulombic, crucibles, cubically, cubiculum, cucurbits, scorbutic, subarctic. | |
+5 letters: buccinator, concubines, corbiculae, cubicities, incumbency, subarctics, suboceanic, subscience, succumbing. | |
| Source: compiled by the editor from various references; see credits. SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro. | |
| 1. Definition 2. Synonyms 3. Crosswords 4. Usage: Modern | 5. Usage: Commercial 6. Images: Slideshow 7. Images: Photo Album 8. Images: Digital Art | 9. Quotations: Familiar 10. Quotations: Non-fiction 11. Usage Frequency 12. Names: Company Usage | 13. Expressions 14. Expressions: Internet 15. Translations: Modern 16. Abbreviations | 17. Acronyms 18. Derivations 19. Rhymes 20. Anagrams | 21. Bibliography |
Copyright © Philip M. Parker, INSEAD. Terms of Use.
