https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:complex/x2ec2f6f830c9fb89:complex-add-sub/v/adding-complex-numbers 2023-08-25T17:15:34.811723966Z Adding complex numbers Learn how to break down the process of adding complex numbers into simple steps. First, add the real parts. Then, do the same with the imaginary parts. The result? A new complex number! Learn and try some examples with Sal. Sal Khan video https://cdn.kastatic.org/googleusercontent/Wd8kN-oLGMV0Foadohf0YD4Z9K86EmTTsF0pzc5LUQU9u8Vqa5ztDLQWo_CFeeUFoNqrvHqHfyKGB65QBT3Ds_BK Adding complex numbers Learn how to break down the process of adding complex numbers into simple steps. First, add the real parts. Then, do the same with the imaginary parts. The result? A new complex number! Learn and try some examples with Sal. https://cdn.kastatic.org/ka-youtube-converted/SfbjqVyQljk.mp4/SfbjqVyQljk.mp4 71 Algebra of complex numbers https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:complex/x2ec2f6f830c9fb89:complex-add-sub/v/subtracting-complex-numbers 2023-08-25T17:15:34.811723966Z Subtracting complex numbers Learn how to break down the process of subtracting complex numbers into simple steps. Distribute the negative sign to get rid of parentheses, then add up the real and imaginary parts separately. Voila! You've got your answer in the form of a + bi. Sal Khan video https://cdn.kastatic.org/googleusercontent/qNpLikDP9UKJlyiBToKSd1KwFsd7YGm9xnsNSZTvYI8POpXSwN2I-fgFKXoceCwe_wVZx9R-9IAF0FFHvb9llUSJ Subtracting complex numbers Learn how to break down the process of subtracting complex numbers into simple steps. Distribute the negative sign to get rid of parentheses, then add up the real and imaginary parts separately. Voila! You've got your answer in the form of a + bi. https://cdn.kastatic.org/ka-youtube-converted/tvXRaZbIjO8.mp4/tvXRaZbIjO8.mp4 113 Algebra of complex numbers https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:complex/x2ec2f6f830c9fb89:complex-add-sub/e/adding_and_subtracting_complex_numbers 2025-03-01T15:52:37.13987578Z Add & subtract complex numbers Practice addition and subtraction with complex numbers. Sulinya Ramanan exercise https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:complex/x2ec2f6f830c9fb89:complex-mul/v/multiplying-complex-numbers 2023-08-25T17:33:50.463554978Z Multiplying complex numbers Discover how to multiply complex numbers! Just like multiplying regular numbers, you can use the distributive property or FOIL method. Remember, the imaginary unit 'i' squared equals -1. So, when you multiply complex numbers like 1-3i and 2+5i, you get a new complex number: 17-i. Sal Khan video https://cdn.kastatic.org/googleusercontent/pZb74VD7tUGU_ni69fUn6kHlXN04LuJ1R9gmcoVkzkTNsbb7Ju_fzSA6lxeBH8YPZIq7b-oGxhpF3Cit5dV7L2k Multiplying complex numbers Discover how to multiply complex numbers! Just like multiplying regular numbers, you can use the distributive property or FOIL method. Remember, the imaginary unit 'i' squared equals -1. So, when you multiply complex numbers like 1-3i and 2+5i, you get a new complex number: 17-i. https://cdn.kastatic.org/ka-youtube-converted/cWn6g8Qqvs4.mp4/cWn6g8Qqvs4.mp4 332 Algebra of complex numbers https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:complex/x2ec2f6f830c9fb89:complex-mul/e/multiply-complex-numbers-by-real-or-imaginary-numbers 2021-08-12T16:00:39.880213098Z Multiply complex numbers (basic) Multiply complex numbers by single terms that are either real or pure imaginary. Sulinya Ramanan exercise https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:complex/x2ec2f6f830c9fb89:complex-mul/e/multiplying_complex_numbers 2025-03-01T15:52:37.13987578Z Multiply complex numbers Given two complex numbers, find their product. Michal Pokorny exercise https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:complex/x9e81a4f98389efdf:complex-div/v/dividing-complex-numbers 2021-06-10T13:39:36Z Dividing complex numbers Sal divides (6+3i) by (7-5i). Sal Khan video https://cdn.kastatic.org/googleusercontent/v6x3LjgscQXXRr3vMbXZz2VTQ6zEco0MLDv2D-Ub-f9Kxw6SH_HZhnVR5Wm6tCGNCnOp0N-xEW4XlF8Y_rEaKkpU Dividing complex numbers Sal divides (6+3i) by (7-5i). https://cdn.kastatic.org/ka-youtube-converted/Z8j5RDOibV4.mp4/Z8j5RDOibV4.mp4 298 Algebra of complex numbers https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:complex/x9e81a4f98389efdf:complex-div/e/dividing_complex_numbers 2025-03-01T15:52:37.13987578Z Divide complex numbers Given a complex number division, express the result as a complex number of the form a+bi. Michal Pokorny exercise https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:complex-numbers-and-quadratic-equations-ncert-new/x0419e5b3b578592a:algebra-of-complex-numbers/v/algebra-of-complex-numbers 2025-03-01T15:52:37.13987578Z Algebra of complex numbers In this video, we talk about how we use operators in the complex world. We look at the properties of addition of complex numbers first. We then look which of these properties also hold true for subtraction. Then, we look at multiplication of complex numbers. Finally, we learn how to find the multiplicative inverse of a complex number. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Algebra of complex numbers In this video, we talk about how we use operators in the complex world. We look at the properties of addition of complex numbers first. We then look which of these properties also hold true for subtraction. Then, we look at multiplication of complex numbers. Finally, we learn how to find the multiplicative inverse of a complex number. https://cdn.kastatic.org/ka-youtube-converted/GkgwbsFs2TE.mp4/GkgwbsFs2TE.mp4 710 Algebra of complex numbers https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:complex-numbers-and-quadratic-equations-ncert-new/x0419e5b3b578592a:algebra-of-complex-numbers/v/identities-involving-two-complex-numbers 2025-03-01T15:52:37.13987578Z Identities involving two complex numbers In this video, we look at a few algebraic identities involving squares and cubes of additon and subtraction of numbers. We realise that these identities work in a similar way in the complex world. We then prove one of these identities using laws of complex numbers (discussed in the previous video). We finally apply these identities to find cubes and other powers of addition and subtraction of two complex numbers. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Identities involving two complex numbers In this video, we look at a few algebraic identities involving squares and cubes of additon and subtraction of numbers. We realise that these identities work in a similar way in the complex world. We then prove one of these identities using laws of complex numbers (discussed in the previous video). We finally apply these identities to find cubes and other powers of addition and subtraction of two complex numbers. https://cdn.kastatic.org/ka-youtube-converted/2BNV-tm6eYI.mp4/2BNV-tm6eYI.mp4 453 Algebra of complex numbers https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:complex/x2ec2f6f830c9fb89:imaginary/v/calculating-i-raised-to-arbitrary-exponents 2023-08-11T18:51:22.662895381Z Powers of the imaginary unit The imaginary unit i is defined such that i²=-1. So what's i³? i³=i²⋅i=-i. What's i⁴? i⁴=i²⋅i²=(-1)²=1. What's i⁵? i⁵=i⁴⋅i=1⋅i=i. Discover how the powers of 'i' cycle through values, making it possible to calculate high exponents of 'i' easily. Sal Khan video https://cdn.kastatic.org/googleusercontent/iChCSFx4onWi8z32VteE3IuNrkNwD_Z65oEG5ixHAbFM5VZ2YlBKMON_WfRVontY13ohV4hbv8ZFToSUu52lI0GE Powers of the imaginary unit The imaginary unit i is defined such that i²=-1. So what's i³? i³=i²⋅i=-i. What's i⁴? i⁴=i²⋅i²=(-1)²=1. What's i⁵? i⁵=i⁴⋅i=1⋅i=i. Discover how the powers of 'i' cycle through values, making it possible to calculate high exponents of 'i' easily. https://cdn.kastatic.org/ka-youtube-converted/QiwfF83NWNA.mp4/QiwfF83NWNA.mp4 380 Algebra of complex numbers https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:complex/x2ec2f6f830c9fb89:imaginary/e/imaginary_unit_powers 2025-03-01T15:52:37.13987578Z Powers of the imaginary unit Simplify expressions with base i (the imaginary unit) raised to a positive exponent. Sulinya Ramanan exercise https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:complex/x2ec2f6f830c9fb89:imaginary/v/imaginary-roots-of-negative-numbers 2023-08-11T18:51:22.662895381Z Simplifying roots of negative numbers Discover the magic of the imaginary unit 'i'! This lesson dives into simplifying the square root of negative numbers using 'i', the principal square root of -1. We'll explore how to rewrite negative numbers as products, and use prime factorization to simplify roots. Sal Khan video https://cdn.kastatic.org/googleusercontent/ujHKobKRkpjNT4qHc8SHdhh_FhLy6vU3H8H7yIUyC0SpxGOHmyHfCWBTFM8fhtqUX92jop5dQFAgzkYMUzE_Bsll Simplifying roots of negative numbers Discover the magic of the imaginary unit 'i'! This lesson dives into simplifying the square root of negative numbers using 'i', the principal square root of -1. We'll explore how to rewrite negative numbers as products, and use prime factorization to simplify roots. https://cdn.kastatic.org/ka-youtube-converted/s03qez-6JMA.mp4/s03qez-6JMA.mp4 244 Algebra of complex numbers https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:complex/x2ec2f6f830c9fb89:imaginary/e/simplify-square-roots-of-negative-numbers 2025-03-01T15:52:37.13987578Z Simplify roots of negative numbers Rewrite square roots of negative numbers as imaginary numbers. Sulinya Ramanan exercise