square root: see root, in mathematics.

root crop, vegetable cultivated chiefly for its edible roots, e.g., the beet, turnip, mangel-wurzel, carrot, and parsnip. All root crops have a large water content and grow best in deeply cultivated soil in cool, overcast weather when the plant's loss of water through transpiration is lowest. Because they require thorough cultivating they are often desirable in a rotation of crops—beets and turnips being most frequently so used. Root crops, especially beets, turnips, and carrots, are also grown as food for livestock.

root, in botany, the descending axis of a plant, as contrasted with the stem, the ascending axis. In most plants the root is underground, but in epiphytes the roots grow in the air and in hydrophytes (e.g., cattails and water lilies) they grow in water or marshes. Roots function to absorb water and dissolved minerals from the soil, to anchor the plant, and often to store food. There are two main types of root system: the tap-root system, in which there is a main primary root larger than the other branching roots; and the diffuse (or fibrous) root system, in which there are many slender roots with numerous smaller root branches. Tap roots are characteristic of most trees and of many other plants, including the carrot, parsnip, radish, beet, and dandelion. The grasses (e.g., corn, rye, and alfalfa) have diffuse roots; in the sweet potato some of the larger fibrous roots swell to store food—although these should not be confused with the tuber of the Irish potato, which is a modified underground stem. Root systems often far exceed in mass the aboveground portions of the plant: alfalfa roots sometimes reach 40 ft (12 m) in length, and the combined length of all the roots of a mature rye plant has been measured at 380 mi (612 km). These ramified root systems are important agents in preventing soil erosion. Roots grow primarily in length; only the older roots may develop a cambium layer that increases their diameter. Protecting the constantly growing tip of the root is a cap of cells that break off as the root probes through the soil; they are replaced by new cells from a layer of meristematic tissue just behind them. In the center of the root the cells formed earlier by the embryonic cells of this layer differentiate into storage tissue and xylem and phloem vessels to conduct sap upward to the leaves and back down to nourish the root cells. On the surface of the epidermis of the growing portion of the root, tiny cellular projections called root hairs extend into the soil to absorb water and minerals. Although root hairs are less than 1/3 in. (.84 cm) long, their great number enables the plant to collect enormous quantities of water, most of which is promptly lost into the air by transpiration. In spite of their slenderness and delicate structure, the spiraling forward thrust of the root tips and the pressure of their expanding cells is sufficient to split solid rock.

root, in mathematics, number or quantity r for which an equation f(r)=0 holds true, where f is some function. If f is a polynomial, r is called a root of f; for example, r=3 and r=-4 are roots of the equation x²+x-12=0, because (3)²+(3)-12=0 and (-4)²+(-4)-12=0. In the special case where f(x)=xⁿ-a for some number a, a root of f is called an nth root of a, denoted by [root]n[sqr]a or a^1/n. For example, 2 is the third, or cube, root of 8 ([root]3[sqr]8=2), since it satisfies the equation x³-8=0. Every number has n different (real or complex) nth roots; e.g., there are two square roots of 9 (3 and -3) since (3)(3)=9 and (-3)(-3)=9.

pleurisy root: see milkweed.

coral-root: see orchid.

Root, John Wellborn, 1850-91, American architect, b. Lumpkin, Ga. He worked in New York City with James Renwick and became a partner of D. H. Burnham in Chicago. The firm created the modern type of highly organized architectural office suited to the planning of metropolitan buildings. Its partners were pioneers in the development of the steel-frame office building, and won international attention by their planning of the World's Columbian Exposition, Chicago, 1893. Root developed a type of ornament, based upon Romanesque design, that was later further developed by Louis Henry Sullivan.

Root, George Frederick, 1820-95, American composer, b. Sheffield, Mass. He taught at schools in Boston and New York City. He wrote gospel songs and composed sentimental ballads to Fanny Crosby's lyrics, but most famous were his Civil War songs "The Battle Cry of Freedom," "Tramp, Tramp, Tramp," and "Just before the Battle, Mother."

Root, Elihu, 1845-1937, American cabinet member and diplomat, b. Clinton, N.Y. Admitted to the bar in 1867, he practiced law in New York City, became prominent in Republican politics, and was appointed (1883) U.S. attorney of the southern district of New York. He soon returned (1885) to his private practice, in which he gained distinction as a corporation lawyer. As U.S. Secretary of War (1899-1904) under Presidents William McKinley and Theodore Roosevelt, Root improved the efficiency of the War Dept., made drastic reforms in the organization of the army, introduced the principle of the general staff, and established the Army War College. He helped direct U.S. policy in the areas acquired as a result of the Spanish-American War and was largely responsible for the Platt Amendment (see under Platt, Orville Hitchcock) regarding Cuba. He also fostered the establishment of civilian governments in Puerto Rico and the Philippines. Root became Secretary of State under Roosevelt in 1905, serving until 1909. He improved relations with Latin America somewhat, after much criticism had been leveled at U.S. activities in Panama, and he concluded (1908) the Root-Takahira agreement with Japan, by which both nations agreed to maintain the status quo in the Pacific and to uphold the Open Door Policy in China. He also negotiated a series of arbitration treaties. Although reluctant to run for public office—partly because his opponents made much of his having been defense attorney for William M. Tweed in 1873—he accepted appointment in 1909 as U.S. Senator from New York and served until 1915. In 1912 he was chairman of the Republican national convention, and in the break between Roosevelt and William Howard Taft he adhered to the conservative Taft faction. He was a member of the Hague Tribunal (Permanent Court of Arbitration) and was prominent (1910) in the North Atlantic Coast Fisheries Arbitration. Root received the Nobel Peace Prize in 1912 in recognition of his efforts toward international peace. He advocated U.S. entry into the League of Nations and helped to bring the World Court (Permanent Court of International Justice) into existence.

Culver's root: see figwort.

In botany, the underground anchoring part of a plant. It grows downward in response to gravity, absorbs water and dissolved minerals, and stores reserve food. Primary root systems have a deep sturdy taproot (in gymnosperms and dicots; see cotyledon) plus secondary or lateral smaller roots, and root hairs. Grasses and other monocots produce a shallow diffuse mass of fibrous secondary roots. Additional support (e.g., in corn and orchids) comes from stem offshoots called adventitious, or prop, roots. Fleshy roots that store food may be modified taproots (e.g., carrots, turnips, and beets) or modified adventitious roots (e.g. cassava). Tubers such as the potato are modified, fleshy, underground stems, or rhizomes. Aerial roots arise from the stem and either pass for some distance through the air before reaching the soil or remain hanging in the air.

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North American plant (Asclepias tuberosa) of the milkweed family, a stout, rough-haired perennial with long horizontal roots. The leafy, erect, somewhat branching stem is about 1–3 ft (0.3–0.9 m) tall. In midsummer it bears numerous clusters of bright orange flowers. Unlike most milkweeds, it has a scanty milky juice. It is native to dry fields and is often planted in wild gardens or grown as a border plant.

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In mathematical analysis, one of a set of discrete values of a parameter, math.k, in an equation of the form math.Lx = math.kx. Such characteristic equations are particularly useful in solving differential equations, integral equations, and systems of equations. In the equation, math.L is a linear transformation such as a matrix or a differential operator, and x can be a vector or a function (called an eigenvector or eigenfunction). The totality of eigenvalues for a given characteristic equation is a set. In quantum mechanics, where math.L is an energy operator, the eigenvalues are energy values.

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(born Feb. 15, 1845, Clinton, N.Y., U.S.—died Feb. 7, 1937, New York, N.Y.) U.S. lawyer and diplomat. He became a U.S. attorney for the southern district of New York in 1883. He served as secretary of war from 1899 to 1904. After the Spanish-American War, he set up a civil government in Puerto Rico and organized U.S. control of the Philippines. As secretary of state (1905–09) under Theodore Roosevelt, he concluded treaties with Japan and persuaded Latin American states to participate in the second Hague conference in 1907 (see Hague Conventions). He was awarded the Nobel Prize for Peace in 1912. He served in the U.S. Senate from 1909 to 1915. A supporter of the League of Nations, he helped frame the statute that established the International Court of Justice.

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(born Feb. 15, 1845, Clinton, N.Y., U.S.—died Feb. 7, 1937, New York, N.Y.) U.S. lawyer and diplomat. He became a U.S. attorney for the southern district of New York in 1883. He served as secretary of war from 1899 to 1904. After the Spanish-American War, he set up a civil government in Puerto Rico and organized U.S. control of the Philippines. As secretary of state (1905–09) under Theodore Roosevelt, he concluded treaties with Japan and persuaded Latin American states to participate in the second Hague conference in 1907 (see Hague Conventions). He was awarded the Nobel Prize for Peace in 1912. He served in the U.S. Senate from 1909 to 1915. A supporter of the League of Nations, he helped frame the statute that established the International Court of Justice.

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