Also written as conjugate. It means to be in a pair or to join in a pair, but in mathematics, it is used with a special meaning in each case. (1) Conjugate diameters c.diameters The locus of the midpoints of chords parallel to one diameter of an ellipse or hyperbola is also the other diameter of the ellipse or hyperbola. These two diameters are said to be conjugate, and one is called the conjugate diameter of the other, or simply the conjugate diameter. (2) Conjugate axis c.axis In the hyperbola x 2 / a 2 - y 2 / b 2 = 1, the y axis does not intersect with the curve. In this way, of the two symmetric axes of a hyperbola, the axes that do not intersect with the curve are called conjugate axes. (3) Conjugate hyperbola c.hyperbola The two hyperbolae x 2 / a 2 - y 2 / b 2 = 1 and x 2 / a 2 - y 2 / b 2 = -1 are said to be conjugate hyperbolas. Their asymptote is the same. (4) Conjugate poles c.poles A quadratic curve (or quadratic surface) has the property that if the polar line (or polar plane) of point P passes through point P', then the polar line (or plane) of P' also passes through P (→ polar lines and poles). Two points P and P' that have this relationship are said to be conjugate with respect to this quadratic curve (or quadratic surface). (5) Conjugate polar lines (polar planes) c.polars A quadratic curve (or quadratic surface) has the property that if the pole P of a line (plane) p is on p ', then the pole P' of p ' is also on p . Two lines (planes) p and p ' that have this relationship are said to be conjugate with respect to this quadratic curve (quadratic surface). (6) Conjugate complex numbers For z = x + yi ( x and y are real numbers), the complex number = x - yi is called the conjugate complex number. Geometrically, on a Gaussian plane, z and are symmetric with respect to the x- axis. The only numbers for which z = 1 are real numbers. (7) Conjugate quaternion c.quateernion Two quaternions q = x0 + x1i + x2j + x3k and = x0 - x1i - x2j - x3k ( i , j , k are imaginary numbers ) are said to be conjugate to each other. (8) Conjugate roots c.roots If an equation with real coefficients has an imaginary root a + bi ( a and b are real numbers, b ≠ 0), then its conjugate complex number a - bi is also a root of the equation. Two roots that are conjugate complex numbers to each other like this are called conjugate roots. (9) Conjugate subgroup c.subgroup Let H be a subgroup of a group G , and let a be any element of G. If we consider the set a -1 Ha = { a -1 ha | h ∈ H }, then a -1 Ha is also a subgroup of G. This is called a conjugate subgroup of H. If a -1 Ha = H holds for any element a of G , then H is said to be a normal subgroup or an invariant subgroup of G ( a -1 is the inverse of a ). (10) Conjugate fields c.fields Let K 1 and K 2 be two extension fields of field K. If there exists a correspondence σ that maps K 1 onto K 2 , and σ(α) = α holds for any element α of K , then K 2 is called a conjugate field of K 1 with respect to K. (11) Conjugate elements c.elements If two elements α 1 and α 2 of an extension field K ' of field K are both algebraic elements with respect to K and are roots of the same irreducible polynomial f ( x ) in K , then α 1 and α 2 are called conjugate elements with respect to K. Conjugation |
共軛とも書く。対になっていること,あるいは対になって結合することをいうが,数学では,それぞれの場合に特別の意味をもって使用される。(1) 共役直径 c.diameters 楕円あるいは双曲線の,一つの直径に平行な弦の中点の軌跡はまた,これらの楕円あるいは双曲線の,他の直径となる。この二つの直径は共役であるといい,一方を他方の共役直径,あるいは単に共役径という。(2) 共役軸 c.axis 双曲線 x2/a2-y2/b2=1 において,y 軸はこの曲線と交わらない。このように双曲線の二つの対称軸のうち,曲線と交わらない軸を共役軸という。(3) 共役双曲線 c.hyperbola 二つの双曲線 x2/a2-y2/b2=1 と x2/a2-y2/b2=-1 とは,互いに共役な双曲線であるという。これらの漸近線は一致する。(4) 共役な極 c.poles 二次曲線(または二次曲面)には,点 Pの極線(または極平面)が点 P'を通れば,P'の極線(または平面)も Pを通るという性質がある(→極線と極点)。このような関係にある 2点 P,P'はこの二次曲線(または二次曲面)に関して共役であるという。(5) 共役な極線(極平面) c.polars 二次曲線(または二次曲面)には,直線(平面)p の極 Pが p'の上にあれば,p'の極 P'も p の上にあるという性質がある。このような関係にある 2直線(平面)p,p'は,この二次曲線(二次曲面)に関して共役であるという。(6) 共役複素数 z=x+yi(x,y は実数)に対して,複素数 =x-yi を共役複素数という。幾何学的には,ガウス平面上で,z と とは x軸に関して,対称である。z= となる数は実数だけである。(7) 共役四元数 c.quateernion 二つの四元数 q=x0+x1i+x2j+x3k と =x0-x1i-x2j-x3k(i,j,k は虚数)は,互いに共役であるという。(8) 共役根 c.roots 実係数の方程式が虚根 a+bi(a,b は実数,b≠0)をもてば,必ずその共役複素数 a-bi もこの方程式の根となる。このように互いに共役複素数となっているような 2根を共役根という。(9) 共役部分群 c.subgroup 群 G の部分群を H とし,a を G の任意の元とするとき,集合 a-1Ha={a-1ha|h∈H}を考えれば,a-1Ha もまた G の部分群である。これを H の共役部分群という。G の任意の元 a に対して,a-1Ha=H が成り立てば,H は G の正規部分群,あるいは不変部分群といわれる(a-1 は a の逆元)。(10) 共役体 c.fields 体 K の二つの拡大体を K1,K2 とするとき,K1 を K2 の上へ写像する対応σが存在し,K の任意の元αに対して,σ(α)=αが成り立てば,K2 を K に関する K1 の共役体という。(11) 共役元 c.elements 体 K の拡大体 K'の 2元α1,α2が,ともに K に関する代数的元であって,K における同一の既約多項式 f(x)の根となっているとき,このα1,α2を K に関する共役元という。
共役
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