My terminology is off at some points. I say "complex quadratics" a few times but I actually mean "Solving Quadratics Over the Complex Numbers". Reach out if you need further clarification. wyneducation.com/contact Prior Knowledge https://youtu.be/NUWP0KHf-CM?si=JMLD_8hr0wPLTZ5M https://youtu.be/R-jnUeoWZhg?si=5D1be_hChM7AfAS8 https://youtu.be/l1WY1EnrV08?si=X_Kci6UeqB0I1_U0 https://youtu.be/P53egv5KfOY?si=M7C2oeV_2q8YoRSx 00:00 Solving Complex Quadratics Overview 01:06 Factorising Quadratics Over Complex Numbers (Sum of Two Squares) 02:43 Completing the Square Over Complex Numbers 04:14 Quadratic Formula for Quadratics Over Complex Numbers 09:13 Visualising Solutions to Quadratics Over Complex Numbers

What am I talking about?! It has everything to do with 'pi' - so many circular functions properties! Prior Knowledge https://youtu.be/NUWP0KHf-CM?si=mVR98CWGbI9mAMvV https://youtu.be/R-jnUeoWZhg?si=aSGLB52kgo-i0Erh https://youtu.be/lnsO9itNz74?si=NTcPuNbYGV2OueIO https://youtu.be/99ahQcLVsuA?si=a9El_5IQOREoSU1t https://youtu.be/RIwQBKSJoME?si=_FLhPz1LbthTZG7q 00:00 Operations on Complex Numbers in Polar Form 01:44 Addition/Subtraction of Numbers in Polar Form 05:02 Multiplication by a Positive Scalar 05:54 Multiplication by a Negative Scalar 07:56 Multiplication of Complex Numbers in Polar Form 10:56 Polar Form Multiplication Theorem 13:07 Division of Complex Numbers in Polar Form 14:33 Polar Form Division Theorem 17:42 Proof of De'Moivre's Theorem

Correction: I said that the argument is what is "inside the brackets" but that's not right...the argument is the entire thing. i.e. tan^-1(b/a). The Principal Argument is when that value is between (-pi,pi] Prior Knowledge https://youtu.be/NUWP0KHf-CM https://youtu.be/R-jnUeoWZhg https://youtu.be/FwGRZQXwXUQ 00:00 Complex Numbers in Polar Form Overview 01:00 Definition of Complex Number in Polar Form 03:36 Argand Diagrams in Polar Form 07:06 Non-uniqueness of Polar Form 08:55 Principal Argument of a Complex Number 10:10 Complex Conjugate in Polar Form

Prior Knowledge https://youtu.be/NUWP0KHf-CM?si=SWijYIWqym7kTxmS https://youtu.be/Ti4IHPiEYqY?si=MQkJ9ZHtYMyyThYQ Converse of a Mathematical Statement: https://youtu.be/xXNA72j_vhw?t=906 Triangle Inequality: https://youtu.be/SNHZY645hdA?t=215 00:00 Overview of the Modulus, Conjugate and Division 01:46 Modulus of a Complex Number 03:27 Properties of the Modulus 05:23 Definition of a Complex Number 06:54 Complex Conjugate of a Sum 09:22 Complex Conjugate of the Product 12:00 Scalar Multiple of the Complex Conjugate 13:37 Product of Complex Number and It's Conjugate 15:23 Sum of a Complex Number and its Conjugate 17:18 Proof of the Partial Converse of Complex Conjugate 24:27 Division of Complex Numbers

Year 12 Methods 3&4: Re-expression Video: https://youtu.be/9pyF71bgKY4?si=7ibbzZWoBSjQBllY

Prior Knowledge: https://youtu.be/Ti4IHPiEYqY?si=gL_vNlcwU1SQ_zc5 https://youtu.be/dyxJTl6DV-s?si=KHtd25Gs68XHDijG Key Concepts: 00:00 Introduction to Complex Numbers Overview 01:35 What is the Imaginary Unit? 02:38 Set of Complex Numbers 04:00 Real and Imaginary Components of Complex Numbers 05:25 Powers of i 07:20 Equality of Complex Numbers 08:37 Operations on Complex Numbers 11:35 Argand Diagrams Explained 12:59 Geometric Interpretation of Operations on Complex Numbers 15:25 Multiplication of Complex Numbers 17:33 Geometric Interpretation of Multiplication of Complex Numbers

Prior Knowledge https://youtu.be/N4UwHZdJkho?si=ondn5pHj1NhSaREc

A better explanation of the distance between skew lines here: https://youtu.be/FmoU4NcI-54?si=4dNMA4Vuo7_EzaXt Prior Knowledge https://youtu.be/Ti4IHPiEYqY?si=pbu2hmXt9fpI4mF6 https://youtu.be/Jft5KgQwh1A?si=tYV38BhVrA8krxsy https://youtu.be/ja_yb51BdME?si=DC6qbh1aaz9GarqR https://youtu.be/tRBSENpi5gE?si=WQQczfe-AIMf5hA2 https://youtu.be/rSfJg7oq9BU?si=9TSmC_1DzAvXvZ_z https://youtu.be/SwtRWA3TBnE?si=3_c8KDRiHWbv1XCX https://youtu.be/N4UwHZdJkho?si=5XcjWTt_oB7ecxaK https://youtu.be/KGHoQhT1MRw?si=0IdKkqVmkQgZ7wBe 00:00 Angles, Distance and Intersections Overview 01:08 Distance from a Point to a Plane 04:24 Distance of a Plane from the Origin 08:54 Distance between Parallel Planes 11:10 Intersection between Planes 17:06 Intersection of a Line and a Plane 20:58 Distance between Skew Lines

Prior Knowledge https://youtu.be/Ti4IHPiEYqY?si=pbu2hmXt9fpI4mF6 https://youtu.be/Jft5KgQwh1A?si=tYV38BhVrA8krxsy https://youtu.be/ja_yb51BdME?si=DC6qbh1aaz9GarqR https://youtu.be/tRBSENpi5gE?si=WQQczfe-AIMf5hA2 https://youtu.be/rSfJg7oq9BU?si=9TSmC_1DzAvXvZ_z https://youtu.be/SwtRWA3TBnE?si=3_c8KDRiHWbv1XCX https://youtu.be/N4UwHZdJkho?si=5XcjWTt_oB7ecxaK 00:00 Equations of Planes Overview 01:04 Normal Vectors to Planes 04:02 Equation of a Plane 10:43 Equation of a Plane by Point and Normal 13:23 Equation of a Plane by Three Points 18:15 Equation of a Plane by Two Intersecting Lines