{"id":2902,"date":"2025-06-02T03:23:47","date_gmt":"2025-06-02T03:23:47","guid":{"rendered":"https:\/\/diznr.com\/?p=2902"},"modified":"2025-06-02T03:23:47","modified_gmt":"2025-06-02T03:23:47","slug":"day-06part-08-discrete-mathematics-for-gate-in-hindi-cube-root-and-fourth-root-of-unity","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/day-06part-08-discrete-mathematics-for-gate-in-hindi-cube-root-and-fourth-root-of-unity\/","title":{"rendered":"Day 06Part 08- Discrete mathematics for gate in Hindi- Cube root and Fourth root of unity."},"content":{"rendered":"

Day 06Part 08- Discrete mathematics for gate in Hindi- Cube root and Fourth root of unity.<\/p>\n

[fvplayer id=”166″]<\/p>\n

\u00a0\u0921\u093f\u0938\u094d\u0915\u094d\u0930\u0940\u091f \u092e\u0948\u0925\u092e\u0947\u091f\u093f\u0915\u094d\u0938 (Discrete Mathematics) \u2013 Cube Root \u0914\u0930 Fourth Root of Unity (GATE \u0915\u0947 \u0932\u093f\u090f)<\/strong><\/h3>\n

\u00a0\u092f\u0942\u0928\u093f\u091f\u0940 \u0915\u0947 \u092e\u0942\u0932 (Roots of Unity) \u0915\u094d\u092f\u093e \u0939\u094b\u0924\u0947 \u0939\u0948\u0902?<\/strong><\/h3>\n

\u27a1\ufe0f Roots of Unity<\/strong> \u0935\u0947 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 \u0939\u094b\u0924\u0940 \u0939\u0948\u0902, \u091c\u093f\u0928\u0915\u093e \u0915\u094b\u0908 \u0918\u093e\u0924 (Power) \u0932\u0947\u0928\u0947 \u092a\u0930 \u092a\u0930\u093f\u0923\u093e\u092e 1<\/strong> \u0906\u0924\u093e \u0939\u0948\u0964
\u27a1\ufe0f \u0915\u093f\u0938\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u0947 n\u0935\u0947\u0902 \u092e\u0942\u0932 (n-th roots)<\/strong> \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0935\u093f\u092d\u093f\u0928\u094d\u0928 \u0917\u0923\u093f\u0924\u0940\u092f \u0914\u0930 \u0907\u0902\u091c\u0940\u0928\u093f\u092f\u0930\u093f\u0902\u0917 \u0905\u0928\u0941\u092a\u094d\u0930\u092f\u094b\u0917\u094b\u0902 \u092e\u0947\u0902 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964
\u27a1\ufe0f \u0938\u092c\u0938\u0947 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 Cube Root of Unity (\u0924\u0940\u0938\u0930\u093e \u092e\u0942\u0932) \u0914\u0930 Fourth Root of Unity (\u091a\u094c\u0925\u093e \u092e\u0942\u0932)<\/strong> \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n

\u00a0Cube Root of Unity (\u0924\u0940\u0938\u0930\u093e \u092e\u0942\u0932) \u0915\u094d\u092f\u093e \u0939\u094b\u0924\u093e \u0939\u0948?<\/strong><\/h3>\n

\u00a0\u092a\u0930\u093f\u092d\u093e\u0937\u093e (Definition):<\/strong><\/h3>\n

\u0924\u0940\u0938\u0930\u0947 \u0915\u094d\u0930\u092e \u0915\u093e \u092f\u0942\u0928\u093f\u091f\u0940 \u092e\u0942\u0932 \u0935\u0947 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 \u0939\u094b\u0924\u0940 \u0939\u0948\u0902, \u091c\u094b \u0924\u0940\u0938\u0930\u0940 \u0918\u093e\u0924 (Cube) \u0932\u0947\u0928\u0947 \u092a\u0930 1<\/strong> \u0926\u0947\u0924\u0940 \u0939\u0948\u0902\u0964<\/p>\n

\u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0930\u0942\u092a:<\/strong><\/p>\n

x3=1x^3 = 1<\/span>x<\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u0907\u0938 \u0938\u092e\u0940\u0915\u0930\u0923 \u0915\u094b \u0939\u0932 \u0915\u0930\u0928\u0947 \u092a\u0930 \u0939\u092e\u0947\u0902 Cube Roots of Unity<\/strong> \u092e\u093f\u0932\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n

\u00a0Cube Root of Unity \u0915\u0947 \u092e\u093e\u0928<\/strong><\/h3>\n

\u0924\u0940\u0938\u0930\u0947 \u092e\u0942\u0932 \u0915\u0940 \u0924\u0940\u0928 \u092e\u093e\u0928 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902:<\/p>\n

1,\u03c9,\u03c921, \\omega, \\omega^2<\/span>1<\/span>,<\/span>\u03c9<\/span>,<\/span>\u03c9<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u091c\u0939\u093e\u0901 \u03c9<\/strong> \u090f\u0915 \u092e\u0942\u0932\u092d\u0942\u0924 Cube Root of Unity \u0939\u0948 \u0914\u0930 \u0907\u0938\u0915\u093e \u092e\u093e\u0928 \u0939\u094b\u0924\u093e \u0939\u0948:<\/p>\n

\u03c9=\u22121+3i2,\u03c92=\u22121\u22123i2\\omega = \\frac{-1 + \\sqrt{3}i}{2}, \\quad \\omega^2 = \\frac{-1 – \\sqrt{3}i}{2}<\/span>\u03c9<\/span>=<\/span><\/span>2\u22121+<\/span>3<\/span>\u200b<\/span><\/span>i<\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>,<\/span>\u03c9<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>2\u22121\u2212<\/span>3<\/span>\u200b<\/span><\/span>i<\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u091c\u0939\u093e\u0901 i<\/strong> \u090f\u0915 \u0915\u0932\u094d\u092a\u0928\u093e\u0924\u094d\u092e\u0915 \u0938\u0902\u0916\u094d\u092f\u093e (Imaginary Number) \u0939\u0948 \u0914\u0930 i2=\u22121i^2 = -1<\/span>i<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span> \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/p>\n

\u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0917\u0941\u0923:<\/strong>
1\ufe0f\u20e3 \u03c93=1\\omega^3 = 1<\/span>\u03c9<\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>1<\/span><\/span><\/span><\/span>
2\ufe0f\u20e3 1+\u03c9+\u03c92=01 + \\omega + \\omega^2 = 0<\/span>1<\/span>+<\/span><\/span>\u03c9<\/span>+<\/span><\/span>\u03c9<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>0<\/span><\/span><\/span><\/span>
3\ufe0f\u20e3 \u03c92=1\u03c9\\omega^2 = \\frac{1}{\\omega}<\/span>\u03c9<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u03c9<\/span><\/span><\/span>1<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u0909\u0926\u093e\u0939\u0930\u0923:<\/strong>
\u092f\u0926\u093f \u0939\u092e\u0947\u0902 2+3\u03c9+4\u03c922 + 3\\omega + 4\\omega^2<\/span>2<\/span>+<\/span><\/span>3<\/span>\u03c9<\/span>+<\/span><\/span>4<\/span>\u03c9<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u0915\u093e \u092e\u093e\u0928 \u0928\u093f\u0915\u093e\u0932\u0928\u093e \u0939\u094b, \u0924\u094b \u0939\u092e \u090a\u092a\u0930 \u0926\u093f\u090f \u0917\u090f \u0917\u0941\u0923\u094b\u0902 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0915\u0947 \u0939\u0932 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n

\u00a0Fourth Root of Unity (\u091a\u094c\u0925\u093e \u092e\u0942\u0932) \u0915\u094d\u092f\u093e \u0939\u094b\u0924\u093e \u0939\u0948?<\/strong><\/h3>\n

\u00a0\u092a\u0930\u093f\u092d\u093e\u0937\u093e (Definition):<\/strong><\/h3>\n

\u091a\u094c\u0925\u0947 \u0915\u094d\u0930\u092e \u0915\u0947 \u092f\u0942\u0928\u093f\u091f\u0940 \u092e\u0942\u0932 \u0935\u0947 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 \u0939\u094b\u0924\u0940 \u0939\u0948\u0902, \u091c\u094b \u091a\u094c\u0925\u0940 \u0918\u093e\u0924 (Fourth Power) \u0932\u0947\u0928\u0947 \u092a\u0930 1<\/strong> \u0926\u0947\u0924\u0940 \u0939\u0948\u0902\u0964<\/p>\n

\u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0930\u0942\u092a:<\/strong><\/p>\n

x4=1x^4 = 1<\/span>x<\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u0907\u0938 \u0938\u092e\u0940\u0915\u0930\u0923 \u0915\u094b \u0939\u0932 \u0915\u0930\u0928\u0947 \u092a\u0930 \u0939\u092e\u0947\u0902 Fourth Roots of Unity<\/strong> \u092e\u093f\u0932\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n

\u00a0Fourth Root of Unity \u0915\u0947 \u092e\u093e\u0928<\/strong><\/h3>\n

\u091a\u094c\u0925\u0947 \u092e\u0942\u0932 \u0915\u0947 \u091a\u093e\u0930 \u092e\u093e\u0928 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902:<\/p>\n

1,\u22121,i,\u2212i1, -1, i, -i<\/span>1<\/span>,<\/span>\u2212<\/span>1<\/span>,<\/span>i<\/span>,<\/span>\u2212<\/span>i<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0917\u0941\u0923:<\/strong>
1\ufe0f\u20e3 i2=\u22121i^2 = -1<\/span>i<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span> \u0914\u0930 i4=1i^4 = 1<\/span>i<\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>1<\/span><\/span><\/span><\/span>
2\ufe0f\u20e3 \u0907\u0928 \u092e\u0942\u0932\u094b\u0902 \u0915\u094b Argand Plane (\u091c\u094d\u092f\u093e\u092e\u093f\u0924\u0940\u092f \u0930\u0942\u092a) \u092e\u0947\u0902 \u0926\u0930\u094d\u0936\u093e\u0928\u0947 \u092a\u0930, \u092f\u0947 90\u00b0 \u0915\u0947 \u0905\u0902\u0924\u0930\u093e\u0932 \u092a\u0930 \u0938\u094d\u0925\u093f\u0924 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964<\/strong>
3\ufe0f\u20e3 \u091a\u094c\u0925\u0947 \u092e\u0942\u0932 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0938\u093f\u0917\u094d\u0928\u0932 \u092a\u094d\u0930\u094b\u0938\u0947\u0938\u093f\u0902\u0917 \u0914\u0930 \u091f\u094d\u0930\u093e\u0902\u0938\u092b\u0949\u0930\u094d\u092e\u094d\u0938 (Fourier Transform) \u092e\u0947\u0902 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/strong><\/p>\n

\u0909\u0926\u093e\u0939\u0930\u0923:<\/strong>
\u092f\u0926\u093f \u0939\u092e\u0947\u0902 1+i\u22121\u2212i1 + i – 1 – i<\/span>1<\/span>+<\/span><\/span>i<\/span>\u2212<\/span><\/span>1<\/span>\u2212<\/span><\/span>i<\/span><\/span><\/span><\/span> \u0928\u093f\u0915\u093e\u0932\u0928\u093e \u0939\u094b, \u0924\u094b \u0909\u0924\u094d\u0924\u0930 0<\/strong> \u0906\u090f\u0917\u093e \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u0938\u092d\u0940 \u092e\u0942\u0932 \u090f\u0915-\u0926\u0942\u0938\u0930\u0947 \u0915\u094b \u0938\u0902\u0924\u0941\u0932\u093f\u0924 \u0915\u0930 \u0926\u0947\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n

\u00a0\u0928\u093f\u0937\u094d\u0915\u0930\u094d\u0937 (Conclusion)<\/strong><\/h3>\n

Cube Root of Unity<\/strong> \u0915\u0947 \u0924\u0940\u0928 \u092e\u093e\u0928 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902: 1,\u03c9,\u03c921, \\omega, \\omega^2<\/span>1<\/span>,<\/span>\u03c9<\/span>,<\/span>\u03c9<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, \u091c\u093f\u0928\u0915\u093e \u092f\u094b\u0917 0<\/strong> \u0939\u094b\u0924\u093e \u0939\u0948\u0964
Fourth Root of Unity<\/strong> \u0915\u0947 \u091a\u093e\u0930 \u092e\u093e\u0928 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902: 1,\u22121,i,\u2212i1, -1, i, -i<\/span>1<\/span>,<\/span>\u2212<\/span>1<\/span>,<\/span>i<\/span>,<\/span>\u2212<\/span>i<\/span><\/span><\/span><\/span>, \u091c\u094b 90\u00b0 \u092a\u0930 \u0938\u094d\u0925\u093f\u0924 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964<\/strong>
\u00a0\u092f\u0947 \u091c\u094d\u092f\u093e\u092e\u093f\u0924\u0940\u092f \u0914\u0930 \u0917\u0923\u093f\u0924\u0940\u092f \u0905\u0928\u0941\u092a\u094d\u0930\u092f\u094b\u0917\u094b\u0902<\/strong> \u092e\u0947\u0902 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u092d\u0942\u092e\u093f\u0915\u093e \u0928\u093f\u092d\u093e\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n

\u00a0\u0915\u094d\u092f\u093e \u0906\u092a\u0915\u094b \u0915\u094b\u0908 \u0935\u093f\u0936\u0947\u0937 \u0909\u0926\u093e\u0939\u0930\u0923 \u092f\u093e \u0935\u093f\u0938\u094d\u0924\u0943\u0924 \u0938\u094d\u092a\u0937\u094d\u091f\u0940\u0915\u0930\u0923 \u091a\u093e\u0939\u093f\u090f?<\/strong><\/p>\n

Day 06Part 08- Discrete mathematics for gate in Hindi- Cube root and Fourth root of unity.<\/a><\/h3>\n

Mathematics (Discrete Structure).pdf<\/a><\/h3>\n

SK Mondal’s – GATE Mathematics<\/a><\/h3>\n

Cubes and Cube Roots<\/a><\/h3>\n

\u092c\u093f\u0932\u0915\u0941\u0932! \u0928\u0940\u091a\u0947 \u0926\u093f\u092f\u093e \u0917\u092f\u093e \u0939\u0948 Discrete Mathematics \u2013 GATE \u0915\u0947 \u0932\u093f\u090f (Day 06, Part 08)<\/strong> \u0915\u093e \u090f\u0915 \u0938\u0930\u0932 \u0939\u093f\u0902\u0926\u0940 \u0928\u094b\u091f\u094d\u0938 \u091c\u093f\u0938\u092e\u0947\u0902 \u0936\u093e\u092e\u093f\u0932 \u0939\u0948:<\/p>\n

\ud83e\uddee Cube Root \u0914\u0930 Fourth Root of Unity (\u0939\u093f\u0928\u094d\u0926\u0940 \u092e\u0947\u0902)<\/strong><\/h2>\n
\n

\ud83d\udcd8 Roots of Unity \u0915\u094d\u092f\u093e \u0939\u094b\u0924\u0947 \u0939\u0948\u0902?<\/strong><\/h2>\n

Roots of Unity<\/strong> \u0935\u0947 \u0938\u092d\u0940 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 \u0939\u094b\u0924\u0940 \u0939\u0948\u0902 \u091c\u094b \u0915\u093f\u0938\u0940 \u0928\u093f\u0936\u094d\u091a\u093f\u0924 \u0918\u093e\u0924 \u092a\u0930 \u091c\u093e\u0915\u0930 1<\/strong> \u092c\u0928 \u091c\u093e\u0924\u0940 \u0939\u0948\u0902\u0964<\/p><\/blockquote>\n

\u092e\u0924\u0932\u092c:<\/p>\n

    \n
  • \u0905\u0917\u0930 zn=1z^n = 1<\/span>, \u0924\u094b zz<\/span> \u0915\u094b n-th root of unity<\/strong> \u0915\u0939\u0924\u0947 \u0939\u0948\u0902\u0964<\/li>\n<\/ul>\n
    \n

    \ud83d\udd3a 1. Cube Root of Unity (\u0918\u0928\u092e\u0942\u0932)<\/strong><\/h2>\n

    \u0939\u092e z3=1z^3 = 1<\/span> \u0915\u094b \u0939\u0932 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n

    \u2705 \u0907\u0938\u0915\u093e \u092e\u0924\u0932\u092c:<\/h3>\n

    \u0939\u092e\u0947\u0902 \u0909\u0928 \u0938\u092d\u0940 complex numbers \u0915\u094b \u0922\u0942\u0902\u0922\u0928\u093e \u0939\u0948 \u091c\u094b z\u00d7z\u00d7z=1z \\times z \\times z = 1<\/span> \u092c\u0928\u093e\u090f\u0902\u0964<\/p>\n

    \ud83d\udccc Cube roots of unity \u0939\u094b\u0924\u0947 \u0939\u0948\u0902:<\/h3>\n

    1,\u00a0\u03c9,\u03c92\\text{1, } \\omega, \\omega^2<\/span><\/p>\n

    \u091c\u0939\u093e\u0901:<\/p>\n

      \n
    • \u03c9=\u221212+32i\\omega = -\\frac{1}{2} + \\frac{\\sqrt{3}}{2}i<\/span><\/li>\n
    • \u03c92=\u221212\u221232i\\omega^2 = -\\frac{1}{2} – \\frac{\\sqrt{3}}{2}i<\/span><\/li>\n<\/ul>\n

      \ud83d\udd01 Important Properties:<\/h3>\n
        \n
      1. \u03c93=1\\omega^3 = 1<\/span><\/li>\n
      2. 1+\u03c9+\u03c92=01 + \\omega + \\omega^2 = 0<\/span><\/li>\n
      3. \u092f\u0947 \u0924\u0940\u0928\u094b\u0902 \u091c\u0921\u093c\u0947\u0902 \u0907\u0915\u093e\u0908 \u0935\u0943\u0924\u094d\u0924 (unit circle) \u092a\u0930 \u0938\u094d\u0925\u093f\u0924 \u0939\u094b\u0924\u0940 \u0939\u0948\u0902 \u0914\u0930 120\u00b0 \u092a\u0930 \u090f\u0915-\u0926\u0942\u0938\u0930\u0947 \u0938\u0947 \u0905\u0932\u0917 \u0939\u094b\u0924\u0940 \u0939\u0948\u0902\u0964<\/li>\n<\/ol>\n
        \n

        \ud83d\udd37 2. Fourth Root of Unity (\u091a\u0924\u0941\u0930\u094d\u0925\u092e\u0942\u0932)<\/strong><\/h2>\n

        \u0939\u092e z4=1z^4 = 1<\/span> \u0915\u094b \u0939\u0932 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n

        \ud83d\udccc Fourth roots of unity \u0939\u094b\u0924\u0947 \u0939\u0948\u0902:<\/h3>\n

        1,\u00a0\u22121,i,\u2212i\\text{1, } -1, i, -i<\/span><\/p>\n

        \u091c\u0939\u093e\u0901:<\/p>\n

          \n
        • i=\u22121i = \\sqrt{-1}<\/span><\/li>\n<\/ul>\n

          \ud83d\udd01 Important Properties:<\/h3>\n
            \n
          1. i2=\u22121i^2 = -1<\/span>, i4=1i^4 = 1<\/span><\/li>\n
          2. \u091a\u093e\u0930\u094b\u0902 root \u092c\u0930\u093e\u092c\u0930 \u0915\u094b\u0923 (90\u00b0) \u092a\u0930 unit circle \u092a\u0930 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964<\/li>\n
          3. Complex Plane \u092e\u0947\u0902 \u092f\u0939 4 \u0926\u093f\u0936\u093e\u0913\u0902 \u092e\u0947\u0902 \u092c\u0902\u091f\u0947 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902 \u2014 East (1), West (-1), North (i), South (-i)<\/li>\n<\/ol>\n
            \n

            \ud83e\udde0 GATE \u0915\u0947 \u0932\u093f\u090f \u0915\u094d\u092f\u094b\u0902 \u091c\u093c\u0930\u0942\u0930\u0940 \u0939\u0948?<\/strong><\/h2>\n
              \n
            • \u092f\u0947 \u0938\u0935\u093e\u0932 \u0905\u0915\u094d\u0938\u0930 complex numbers<\/strong>, graph theory<\/strong>, \u0914\u0930 group theory<\/strong> \u0915\u0947 concepts \u092e\u0947\u0902 \u092a\u0942\u091b\u0947 \u091c\u093e\u0924\u0947 \u0939\u0948\u0902\u0964<\/li>\n
            • nthn^{th}<\/span> roots of unity \u090f\u0915 cyclic group<\/strong> \u092c\u0928\u093e\u0924\u0947 \u0939\u0948\u0902 \u2014 \u092f\u0947 Discrete Maths \u092e\u0947\u0902 group theory<\/strong> \u0915\u093e practical \u0909\u0926\u093e\u0939\u0930\u0923 \u0939\u0948\u0964<\/li>\n<\/ul>\n
              \n

              \ud83d\udcdd Quick Revision Table:<\/strong><\/h2>\n\n\n\n\n\n\n
              Root Type<\/th>\nEquation<\/th>\nRoots<\/th>\n<\/tr>\n<\/thead>\n
              Cube Root<\/td>\nz3=1z^3 = 1<\/span><\/td>\n1,\u03c9,\u03c921, \\omega, \\omega^2<\/span><\/td>\n<\/tr>\n
              Fourth Root<\/td>\nz4=1z^4 = 1<\/span><\/td>\n1,\u22121,i,\u2212i1, -1, i, -i<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
              \n

              \ud83e\udde0 \u092e\u094b\u091f\u093f\u0935\u0947\u0936\u0928\u0932 \u0932\u093e\u0907\u0928:<\/strong><\/h3>\n

              “Complex \u0938\u0902\u0916\u094d\u092f\u093e \u092e\u0941\u0936\u094d\u0915\u093f\u0932 \u0928\u0939\u0940\u0902 \u0939\u0948, \u0935\u094b \u0938\u093f\u0930\u094d\u092b \u0906\u092a\u0915\u0947 imagination \u0915\u094b \u090f\u0915 \u0928\u0908 \u0926\u093f\u0936\u093e \u0926\u0947\u0924\u0940 \u0939\u0948!”<\/strong><\/p><\/blockquote>\n

              \n

              \u092f\u0926\u093f \u0906\u092a \u091a\u093e\u0939\u0947\u0902, \u092e\u0948\u0902 \u0907\u0938 \u091f\u0949\u092a\u093f\u0915 \u0915\u093e:<\/p>\n

                \n
              • PDF Handout<\/strong><\/li>\n
              • Visual Chart (Complex Plane with Roots)<\/strong><\/li>\n
              • \u092f\u093e \u090f\u0915 GATE \u0915\u0947 \u0938\u0935\u093e\u0932\u094b\u0902 \u0915\u093e \u0905\u092d\u094d\u092f\u093e\u0938 \u0938\u0947\u091f<\/strong> \u092d\u0940 \u0924\u0948\u092f\u093e\u0930 \u0915\u0930 \u0938\u0915\u0924\u093e \u0939\u0942\u0901\u0964<\/li>\n<\/ul>\n

                \u092c\u0924\u093e\u0907\u090f \u0915\u0948\u0938\u0947 \u091a\u093e\u0939\u093f\u090f?<\/p>\n","protected":false},"excerpt":{"rendered":"

                Day 06Part 08- Discrete mathematics for gate in Hindi- Cube root and Fourth root of unity. [fvplayer id=”166″] \u00a0\u0921\u093f\u0938\u094d\u0915\u094d\u0930\u0940\u091f \u092e\u0948\u0925\u092e\u0947\u091f\u093f\u0915\u094d\u0938 (Discrete Mathematics) \u2013 Cube Root \u0914\u0930 Fourth Root of Unity (GATE \u0915\u0947 \u0932\u093f\u090f) \u00a0\u092f\u0942\u0928\u093f\u091f\u0940 \u0915\u0947 \u092e\u0942\u0932 (Roots of Unity) \u0915\u094d\u092f\u093e \u0939\u094b\u0924\u0947 \u0939\u0948\u0902? \u27a1\ufe0f Roots of Unity \u0935\u0947 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 \u0939\u094b\u0924\u0940 \u0939\u0948\u0902, \u091c\u093f\u0928\u0915\u093e \u0915\u094b\u0908 \u0918\u093e\u0924 (Power) \u0932\u0947\u0928\u0947 […]<\/p>\n","protected":false},"author":71,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[76],"tags":[],"class_list":["post-2902","post","type-post","status-publish","format-standard","hentry","category-discrete-mathematics"],"_links":{"self":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/2902","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/users\/71"}],"replies":[{"embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/comments?post=2902"}],"version-history":[{"count":0,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/2902\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/media?parent=2902"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/categories?post=2902"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/tags?post=2902"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}