{"id":568806,"date":"2026-07-27T09:18:18","date_gmt":"2026-07-27T02:18:18","guid":{"rendered":"https:\/\/hoanghamobile.com\/tin-tuc\/?p=568806"},"modified":"2026-07-27T09:18:18","modified_gmt":"2026-07-27T02:18:18","slug":"so-phuc-la-gi","status":"publish","type":"post","link":"https:\/\/hoanghamobile.com\/tin-tuc\/so-phuc-la-gi\/","title":{"rendered":"S\u1ed1 ph\u1ee9c l\u00e0 g\u00ec? Ph\u1ea7n th\u1ef1c, ph\u1ea7n \u1ea3o, m\u00f4\u0111un v\u00e0 c\u00e1c ph\u00e9p t\u00ednh"},"content":{"rendered":"

S\u1ed1 ph\u1ee9c l\u00e0 s\u1ed1 c\u00f3 d\u1ea1ng z = a + bi, trong \u0111\u00f3 a v\u00e0 b l\u00e0 c\u00e1c s\u1ed1 th\u1ef1c, c\u00f2n i l\u00e0 \u0111\u01a1n v\u1ecb \u1ea3o v\u1edbi t\u00ednh ch\u1ea5t i\u00b2 = \u22121; a g\u1ecdi l\u00e0 ph\u1ea7n th\u1ef1c, b g\u1ecdi l\u00e0 ph\u1ea7n \u1ea3o. S\u1ed1 ph\u1ee9c m\u1edf r\u1ed9ng t\u1eadp s\u1ed1 th\u1ef1c, gi\u00fap gi\u1ea3i \u0111\u01b0\u1ee3c c\u1ea3 nh\u1eefng ph\u01b0\u01a1ng tr\u00ecnh v\u00f4 nghi\u1ec7m tr\u00ean t\u1eadp s\u1ed1 th\u1ef1c. \u0110\u00e2y l\u00e0 ki\u1ebfn th\u1ee9c Gi\u1ea3i t\u00edch l\u1edbp 12. B\u00e0i vi\u1ebft n\u00e0y tr\u00ecnh b\u00e0y<\/span> s\u1ed1 ph\u1ee9c l\u00e0 g\u00ec<\/b>, c\u00e1c kh\u00e1i ni\u1ec7m c\u01a1 b\u1ea3n, s\u1ed1 ph\u1ee9c li\u00ean h\u1ee3p, m\u00f4\u0111un, c\u00e1c ph\u00e9p t\u00ednh v\u00e0 b\u00e0i t\u1eadp c\u00f3 l\u1eddi gi\u1ea3i.<\/span><\/p>\n

S\u1ed1 ph\u1ee9c l\u00e0 g\u00ec?<\/b><\/h2>\n

S\u1ed1 ph\u1ee9c l\u00e0 s\u1ed1 c\u00f3 d\u1ea1ng z = a + bi, v\u1edbi a, b l\u00e0 s\u1ed1 th\u1ef1c v\u00e0 i l\u00e0 \u0111\u01a1n v\u1ecb \u1ea3o th\u1ecfa m\u00e3n i\u00b2 = \u22121. Trong \u0111\u00f3 a l\u00e0 ph\u1ea7n th\u1ef1c (k\u00fd hi\u1ec7u Re(z)), b l\u00e0 ph\u1ea7n \u1ea3o (k\u00fd hi\u1ec7u Im(z)). Khi b = 0, s\u1ed1 ph\u1ee9c tr\u1edf th\u00e0nh s\u1ed1 th\u1ef1c; khi a = 0, s\u1ed1 ph\u1ee9c \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 s\u1ed1 thu\u1ea7n \u1ea3o. Nh\u1edd s\u1ed1 ph\u1ee9c, ph\u01b0\u01a1ng tr\u00ecnh nh\u01b0 x\u00b2 + 1 = 0 (v\u00f4 nghi\u1ec7m th\u1ef1c) l\u1ea1i c\u00f3 nghi\u1ec7m l\u00e0 x = \u00b1i.<\/span><\/p>\n

\"so-phuc-la-gi-1\"<\/p>\n

C\u00e1c kh\u00e1i ni\u1ec7m c\u01a1 b\u1ea3n<\/b><\/h2>\n

V\u1edbi s\u1ed1 ph\u1ee9c c\u00f3 d\u1ea1ng: <\/span>z = a + bi<\/b>, ta c\u00f3 c\u00e1c kh\u00e1i ni\u1ec7m c\u01a1 b\u1ea3n sau:<\/span><\/p>\n

    \n
  • Ph\u1ea7n th\u1ef1c: L\u00e0 s\u1ed1 a trong s\u1ed1 ph\u1ee9c z = a + bi, \u0111\u01b0\u1ee3c k\u00fd hi\u1ec7u l\u00e0 Re(z).<\/span><\/li>\n
  • Ph\u1ea7n \u1ea3o: L\u00e0 s\u1ed1 b trong s\u1ed1 ph\u1ee9c z = a + bi, \u0111\u01b0\u1ee3c k\u00fd hi\u1ec7u l\u00e0 Im(z).<\/span><\/li>\n
  • (L\u01b0u \u00fd: Ph\u1ea7n \u1ea3o l\u00e0 h\u1ec7 s\u1ed1 c\u1ee7a i, kh\u00f4ng ph\u1ea3i to\u00e0n b\u1ed9 ph\u1ea7n bi.)<\/span><\/li>\n
  • \u0110\u01a1n v\u1ecb \u1ea3o: K\u00fd hi\u1ec7u l\u00e0 i, th\u1ecfa m\u00e3n:<\/span> i\u00b2 = \u22121<\/b><\/li>\n
  • Hai s\u1ed1 ph\u1ee9c b\u1eb1ng nhau: Hai s\u1ed1 ph\u1ee9c z\u2081 = a + bi v\u00e0 z\u2082 = c + di b\u1eb1ng nhau khi v\u00e0 ch\u1ec9 khi:<\/span>\n
      \n
    • a = c (ph\u1ea7n th\u1ef1c b\u1eb1ng nhau).<\/span><\/li>\n
    • b = d (ph\u1ea7n \u1ea3o b\u1eb1ng nhau).<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

      V\u00ed d\u1ee5:<\/b> Hai s\u1ed1 ph\u1ee9c z\u2081 = 3 + 2i v\u00e0 z\u2082 = 3 + 2i b\u1eb1ng nhau v\u00ec c\u00f3 c\u00f9ng ph\u1ea7n th\u1ef1c l\u00e0 3 v\u00e0 c\u00f9ng ph\u1ea7n \u1ea3o l\u00e0 2.<\/span><\/p>\n

      S\u1ed1 ph\u1ee9c li\u00ean h\u1ee3p v\u00e0 m\u00f4\u0111un<\/b><\/h2>\n

      S\u1ed1 ph\u1ee9c li\u00ean h\u1ee3p c\u1ee7a z = a + bi l\u00e0 z\u0304 = a \u2212 bi (gi\u1eef nguy\u00ean ph\u1ea7n th\u1ef1c, \u0111\u1ed5i d\u1ea5u ph\u1ea7n \u1ea3o). M\u00f4\u0111un c\u1ee7a s\u1ed1 ph\u1ee9c z = a + bi l\u00e0 kho\u1ea3ng c\u00e1ch t\u1eeb \u0111i\u1ec3m bi\u1ec3u di\u1ec5n z \u0111\u1ebfn g\u1ed1c t\u1ecda \u0111\u1ed9, t\u00ednh b\u1eb1ng: |z| = \u221a(a\u00b2 + b\u00b2). V\u00ed d\u1ee5, s\u1ed1 ph\u1ee9c z = 3 + 4i c\u00f3 li\u00ean h\u1ee3p z\u0304 = 3 \u2212 4i v\u00e0 m\u00f4\u0111un |z| = \u221a(9 + 16) = 5.<\/span><\/p>\n

      S\u1ed1 ph\u1ee9c li\u00ean h\u1ee3p c\u00f3 m\u1ed9t s\u1ed1 t\u00ednh ch\u1ea5t quan tr\u1ecdng th\u01b0\u1eddng d\u00f9ng khi gi\u1ea3i b\u00e0i t\u1eadp: li\u00ean h\u1ee3p c\u1ee7a t\u1ed5ng b\u1eb1ng t\u1ed5ng c\u00e1c li\u00ean h\u1ee3p, li\u00ean h\u1ee3p c\u1ee7a t\u00edch b\u1eb1ng t\u00edch c\u00e1c li\u00ean h\u1ee3p, v\u00e0 t\u00edch c\u1ee7a m\u1ed9t s\u1ed1 ph\u1ee9c v\u1edbi li\u00ean h\u1ee3p c\u1ee7a n\u00f3 lu\u00f4n l\u00e0 m\u1ed9t s\u1ed1 th\u1ef1c kh\u00f4ng \u00e2m: z \u00b7 z\u0304 = a\u00b2 + b\u00b2 = |z|\u00b2. \u0110\u00e2y ch\u00ednh l\u00e0 m\u1eb9o gi\u00fap kh\u1eed ph\u1ea7n \u1ea3o \u1edf m\u1eabu khi th\u1ef1c hi\u1ec7n ph\u00e9p chia.<\/span><\/p>\n

      \"so-phuc-la-gi-2\"<\/p>\n

      C\u00e1c ph\u00e9p t\u00ednh v\u1edbi s\u1ed1 ph\u1ee9c l\u00e0 g\u00ec<\/b><\/h2>\n

      V\u1edbi c\u00e1c s\u1ed1 ph\u1ee9c, ta th\u1ef1c hi\u1ec7n c\u00e1c ph\u00e9p t\u00ednh t\u01b0\u01a1ng t\u1ef1 nh\u01b0 v\u1edbi s\u1ed1 th\u1ef1c, \u0111\u1ed3ng th\u1eddi s\u1eed d\u1ee5ng t\u00ednh ch\u1ea5t: <\/span>i\u00b2 = \u22121<\/b>. B\u1ea3ng d\u01b0\u1edbi \u0111\u00e2y t\u00f3m t\u1eaft b\u1ed1n ph\u00e9p t\u00ednh c\u01a1 b\u1ea3n \u0111\u1ec3 b\u1ea1n d\u1ec5 tra c\u1ee9u.<\/span><\/p>\n\n\n\n\n\n\n\n\n
      Ph\u00e9p t\u00ednh<\/th>\nC\u00f4ng th\u1ee9c t\u1ed5ng qu\u00e1t (z\u2081 = a + bi, z\u2082 = c + di)<\/th>\n<\/tr>\n<\/thead>\n
      C\u1ed9ng<\/td>\n(a + c) + (b + d)i<\/td>\n<\/tr>\n
      Tr\u1eeb<\/td>\n(a \u2212 c) + (b \u2212 d)i<\/td>\n<\/tr>\n
      Nh\u00e2n<\/td>\n(ac \u2212 bd) + (ad + bc)i<\/td>\n<\/tr>\n
      Chia<\/td>\n[(ac + bd) + (bc \u2212 ad)i] \/ (c\u00b2 + d\u00b2)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Ph\u00e9p c\u1ed9ng v\u00e0 ph\u00e9p tr\u1eeb:<\/span><\/p>\n

        \n
      • C\u1ed9ng (ho\u1eb7c tr\u1eeb) c\u00e1c ph\u1ea7n th\u1ef1c v\u1edbi nhau v\u00e0 c\u00e1c ph\u1ea7n \u1ea3o v\u1edbi nhau.<\/span><\/li>\n
      • V\u00ed d\u1ee5: (2 + 3i) + (1 \u2212 i) = (2 + 1) + (3i \u2212 i) = 3 + 2i<\/span><\/li>\n<\/ul>\n

        Ph\u00e9p nh\u00e2n:<\/span><\/p>\n

          \n
        • Th\u1ef1c hi\u1ec7n ph\u00e9p nh\u00e2n nh\u01b0 nh\u00e2n c\u00e1c \u0111a th\u1ee9c, sau \u0111\u00f3 thay i\u00b2 = \u22121 \u0111\u1ec3 thu g\u1ecdn k\u1ebft qu\u1ea3.<\/span><\/li>\n
        • V\u00ed d\u1ee5: (1 + i)(1 \u2212 i) = 1 \u2212 i\u00b2 = 1 \u2212 (\u22121) = 2<\/span><\/li>\n<\/ul>\n

          Ph\u00e9p chia:<\/span><\/p>\n

            \n
          • \u0110\u1ec3 chia hai s\u1ed1 ph\u1ee9c, ta nh\u00e2n c\u1ea3 t\u1eed s\u1ed1 v\u00e0 m\u1eabu s\u1ed1 v\u1edbi s\u1ed1 ph\u1ee9c li\u00ean h\u1ee3p c\u1ee7a m\u1eabu s\u1ed1. Khi \u0111\u00f3, m\u1eabu s\u1ed1 tr\u1edf th\u00e0nh m\u1ed9t s\u1ed1 th\u1ef1c, gi\u00fap vi\u1ec7c t\u00ednh to\u00e1n v\u00e0 r\u00fat g\u1ecdn d\u1ec5 d\u00e0ng h\u01a1n.<\/span><\/li>\n
          • V\u00ed d\u1ee5: Khi chia (a + bi) cho (c + di), ta nh\u00e2n c\u1ea3 t\u1eed v\u00e0 m\u1eabu v\u1edbi (c \u2212 di) \u0111\u1ec3 lo\u1ea1i b\u1ecf ph\u1ea7n \u1ea3o \u1edf m\u1eabu.<\/span><\/li>\n<\/ul>\n

            V\u00ed d\u1ee5 c\u00f3 l\u1eddi gi\u1ea3i chi ti\u1ebft (ph\u00e9p chia):<\/b> T\u00ednh z = (3 + 2i) \/ (1 \u2212 i).<\/span><\/p>\n

              \n
            • B\u01b0\u1edbc 1: Nh\u00e2n c\u1ea3 t\u1eed v\u00e0 m\u1eabu v\u1edbi li\u00ean h\u1ee3p c\u1ee7a m\u1eabu l\u00e0 (1 + i): z = [(3 + 2i)(1 + i)] \/ [(1 \u2212 i)(1 + i)].<\/span><\/li>\n
            • B\u01b0\u1edbc 2: Khai tri\u1ec3n t\u1eed s\u1ed1: (3 + 2i)(1 + i) = 3 + 3i + 2i + 2i\u00b2 = 3 + 5i \u2212 2 = 1 + 5i.<\/span><\/li>\n
            • B\u01b0\u1edbc 3: T\u00ednh m\u1eabu s\u1ed1: (1 \u2212 i)(1 + i) = 1 \u2212 i\u00b2 = 1 + 1 = 2.<\/span><\/li>\n
            • B\u01b0\u1edbc 4: R\u00fat g\u1ecdn: z = (1 + 5i) \/ 2 = 1\/2 + (5\/2)i. V\u1eady ph\u1ea7n th\u1ef1c l\u00e0 1\/2, ph\u1ea7n \u1ea3o l\u00e0 5\/2.<\/span><\/li>\n<\/ul>\n

              Bi\u1ec3u di\u1ec5n h\u00ecnh h\u1ecdc c\u1ee7a s\u1ed1 ph\u1ee9c<\/b><\/h2>\n

              M\u1ed7i s\u1ed1 ph\u1ee9c z = a + bi \u0111\u01b0\u1ee3c bi\u1ec3u di\u1ec5n b\u1edfi m\u1ed9t \u0111i\u1ec3m M(a; b) tr\u00ean m\u1eb7t ph\u1eb3ng t\u1ecda \u0111\u1ed9, g\u1ecdi l\u00e0 m\u1eb7t ph\u1eb3ng ph\u1ee9c. Tr\u1ee5c ho\u00e0nh bi\u1ec3u di\u1ec5n ph\u1ea7n th\u1ef1c (tr\u1ee5c th\u1ef1c), tr\u1ee5c tung bi\u1ec3u di\u1ec5n ph\u1ea7n \u1ea3o (tr\u1ee5c \u1ea3o). Nh\u1edd c\u00e1ch bi\u1ec3u di\u1ec5n n\u00e0y, m\u00f4\u0111un |z| ch\u00ednh l\u00e0 kho\u1ea3ng c\u00e1ch t\u1eeb \u0111i\u1ec3m M \u0111\u1ebfn g\u1ed1c t\u1ecda \u0111\u1ed9.<\/span><\/p>\n

              \"so-phuc-la-gi-3\"<\/p>\n

              \u1ee8ng d\u1ee5ng c\u1ee7a s\u1ed1 ph\u1ee9c<\/b><\/h2>\n

              Ngo\u00e0i vai tr\u00f2 gi\u1ea3i c\u00e1c ph\u01b0\u01a1ng tr\u00ecnh v\u00f4 nghi\u1ec7m tr\u00ean t\u1eadp s\u1ed1 th\u1ef1c, s\u1ed1 ph\u1ee9c c\u00f2n c\u00f3 nhi\u1ec1u \u1ee9ng d\u1ee5ng th\u1ef1c t\u1ebf. Trong k\u1ef9 thu\u1eadt \u0111i\u1ec7n v\u00e0 \u0111i\u1ec7n t\u1eed, s\u1ed1 ph\u1ee9c \u0111\u01b0\u1ee3c d\u00f9ng \u0111\u1ec3 m\u00f4 t\u1ea3 d\u00f2ng \u0111i\u1ec7n xoay chi\u1ec1u, t\u1ed5ng tr\u1edf v\u00e0 \u0111\u1ed9 l\u1ec7ch pha m\u1ed9t c\u00e1ch g\u1ecdn g\u00e0ng. Trong v\u1eadt l\u00fd, s\u1ed1 ph\u1ee9c xu\u1ea5t hi\u1ec7n trong l\u00fd thuy\u1ebft dao \u0111\u1ed9ng, s\u00f3ng v\u00e0 c\u01a1 h\u1ecdc l\u01b0\u1ee3ng t\u1eed. Trong to\u00e1n h\u1ecdc, s\u1ed1 ph\u1ee9c l\u00e0 c\u00f4ng c\u1ee5 c\u1ed1t l\u00f5i \u0111\u1ec3 ch\u1ee9ng minh \u0110\u1ecbnh l\u00fd c\u01a1 b\u1ea3n c\u1ee7a \u0111\u1ea1i s\u1ed1 v\u00e0 gi\u1ea3i c\u00e1c b\u00e0i to\u00e1n l\u01b0\u1ee3ng gi\u00e1c th\u00f4ng qua c\u00f4ng th\u1ee9c Euler. V\u00ec v\u1eady, n\u1eafm v\u1eefng s\u1ed1 ph\u1ee9c kh\u00f4ng ch\u1ec9 ph\u1ee5c v\u1ee5 k\u1ef3 thi m\u00e0 c\u00f2n l\u00e0 n\u1ec1n t\u1ea3ng cho nhi\u1ec1u m\u00f4n h\u1ecdc n\u00e2ng cao.<\/span><\/p>\n

              B\u00e0i t\u1eadp v\u1eadn d\u1ee5ng c\u00f3 l\u1eddi gi\u1ea3i v\u1ec1 s\u1ed1 ph\u1ee9c l\u00e0 g\u00ec<\/b><\/h2>\n

              B\u00e0i 1: T\u00ecm ph\u1ea7n th\u1ef1c v\u00e0 ph\u1ea7n \u1ea3o c\u1ee7a z = 5 \u2212 2i. L\u1eddi gi\u1ea3i: ph\u1ea7n th\u1ef1c l\u00e0 5, ph\u1ea7n \u1ea3o l\u00e0 \u22122.<\/span><\/p>\n

              B\u00e0i 2: T\u00ednh m\u00f4\u0111un c\u1ee7a z = 6 + 8i. L\u1eddi gi\u1ea3i: |z| = \u221a(36 + 64) = \u221a100 = 10.<\/span><\/p>\n

              B\u00e0i 3: T\u00ednh (2 + i) + (3 \u2212 4i). L\u1eddi gi\u1ea3i: = (2 + 3) + (1 \u2212 4)i = 5 \u2212 3i.<\/span><\/p>\n

              B\u00e0i 4: T\u00ednh (1 + 2i)(2 \u2212 i). L\u1eddi gi\u1ea3i: = 2 \u2212 i + 4i \u2212 2i\u00b2 = 2 + 3i + 2 = 4 + 3i.<\/span><\/p>\n

              B\u00e0i 5: T\u00ecm s\u1ed1 ph\u1ee9c li\u00ean h\u1ee3p v\u00e0 m\u00f4\u0111un c\u1ee7a z = \u22124 + 3i. L\u1eddi gi\u1ea3i: li\u00ean h\u1ee3p z\u0304 = \u22124 \u2212 3i; m\u00f4\u0111un |z| = \u221a((\u22124)\u00b2 + 3\u00b2) = \u221a(16 + 9) = \u221a25 = 5.<\/span><\/p>\n

              B\u00e0i 6 (tr\u1eafc nghi\u1ec7m): S\u1ed1 ph\u1ee9c li\u00ean h\u1ee3p c\u1ee7a z = 3 + 5i l\u00e0 g\u00ec? A. 3 \u2212 5i B. \u22123 + 5i C. 5 + 3i D. \u22123 \u2212 5i. \u0110\u00e1p \u00e1n: A.<\/span><\/p>\n

              C\u00e2u h\u1ecfi th\u01b0\u1eddng g\u1eb7p v\u1ec1 s\u1ed1 ph\u1ee9c<\/b><\/h2>\n

              S\u1ed1 ph\u1ee9c l\u00e0 g\u00ec?<\/b><\/h3>\n

              L\u00e0 s\u1ed1 c\u00f3 d\u1ea1ng z = a + bi, v\u1edbi a, b l\u00e0 s\u1ed1 th\u1ef1c v\u00e0 i\u00b2 = \u22121.<\/span><\/p>\n

              \u0110\u01a1n v\u1ecb \u1ea3o i c\u00f3 t\u00ednh ch\u1ea5t g\u00ec?<\/b><\/h3>\n

              i\u00b2 = \u22121.<\/span><\/p>\n

              M\u00f4\u0111un c\u1ee7a s\u1ed1 ph\u1ee9c t\u00ednh th\u1ebf n\u00e0o?<\/b><\/h3>\n

              |z| = \u221a(a\u00b2 + b\u00b2), v\u1edbi z = a + bi.<\/span><\/p>\n

              S\u1ed1 ph\u1ee9c li\u00ean h\u1ee3p l\u00e0 g\u00ec?<\/b><\/h3>\n

              Li\u00ean h\u1ee3p c\u1ee7a a + bi l\u00e0 a \u2212 bi (\u0111\u1ed5i d\u1ea5u ph\u1ea7n \u1ea3o).<\/span><\/p>\n

              S\u1ed1 ph\u1ee9c \u0111\u01b0\u1ee3c bi\u1ec3u di\u1ec5n th\u1ebf n\u00e0o?<\/b><\/h3>\n

              B\u1eb1ng \u0111i\u1ec3m M(a; b) tr\u00ean m\u1eb7t ph\u1eb3ng ph\u1ee9c, tr\u1ee5c ho\u00e0nh l\u00e0 ph\u1ea7n th\u1ef1c, tr\u1ee5c tung l\u00e0 ph\u1ea7n \u1ea3o.<\/span><\/p>\n

              Chia hai s\u1ed1 ph\u1ee9c th\u1ef1c hi\u1ec7n ra sao?<\/b><\/h3>\n

              Nh\u00e2n c\u1ea3 t\u1eed v\u00e0 m\u1eabu v\u1edbi s\u1ed1 ph\u1ee9c li\u00ean h\u1ee3p c\u1ee7a m\u1eabu \u0111\u1ec3 m\u1eabu tr\u1edf th\u00e0nh s\u1ed1 th\u1ef1c, r\u1ed3i r\u00fat g\u1ecdn.<\/span><\/p>\n