Let's Invent Two-Dimensional Multiplication

Comments · 205

• WR

  @WrathofMath · 1 year ago (edited) · pinned

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• KE

  @kent631420 · 1 year ago

  bro discovered the dot product 🔥🔥🔥

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• GL

  @glassman96 · 1 year ago

  Why is the math saying lul at me stop laughing at me math

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• CA

  @calebdupree3564 · 1 year ago

  i turned on the video thinking “wouldn’t it just be like multiplying matrices?” and then you started talking and i was like “i guess not” and then you kept talking and i was like “it is just like multiplying matrices”

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• VI

  @viniciusares321 · 1 year ago

  Good dot product explanation but, a real 2D multiplication should result in a 2D result, and everything be still contained in a 2D plane without third dimension

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• DO

  @douglasmagowan4918 · 1 year ago

  Alternatively, you can define two dimensional multiplication such that we multiply the magnitudes and add the arguements. &nbsp;This is isomorphic to the multiplication of the complex numbers, and is a natural extension of multiplication of the real numbers.

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• AN

  @angeldude101 · 1 year ago (edited)

  Multiplying 2D numbers is easy: (ax + by)(cx+ dy) = acx² + adxy + bcyx + bdy². The problem is that the result we got is 4D, but we&apos;d probably want the result to stay in 2D. For starters, let&apos;s define x as 1, because a multiplicative identity is useful, which gives us ac + (ad + bc)y + bdy². That shaved off a dimension of the result, but we still ended up with a trinomial rather than a binomial, so we just have to pick something for y² to be. Since scalar factors are mostly irrelevant, that really only leaves 5 options: y² ∈ { 0, 1, -1, y, -y }. Choosing y² = -1 gives the well-known complex numbers. Choosing y² = 1 gives the less well-known hyperbolic numbers (more often called the worse name of &quot;split-complex numbers&quot;). Choosing y² = 0 gives the dual numbers. I&apos;m actually not sure what would happen when choosing y² = ±y, but it&apos;s possible that it&apos;d actually just result in a shifted or skewed copy of one of the prior three.

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• SL

  @slushpupp1 · 1 year ago

  <a href="https://www.youtube.com/watch?v=04IAZCONqro&amp;t=385">6:25</a> OH YEAH, VECTOR!!!

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• BL

  @Blox-Beam · 1 year ago

  <a href="https://www.youtube.com/watch?v=04IAZCONqro&amp;t=214">3:34</a> thats basically what modulus of a complex number is

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• NO

  @notthecheatr23 · 1 year ago

  This is important stuff but where it really gets beautiful (and starts to seem much more systematic) is when you get to geometric algebra... perhaps that one&apos;s considered a higher barrier to entry, but it does a great job of tying together a lot of &quot;weirdness&quot; like dot and cross products, complex numbers and generalizing to higher dimensions, etc.

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• DA

  @Dani0x1B · 1 year ago

  Freya Holmer made a video about vector multiplication a while ago that&apos;s very well explain and I recommend it to anyone who enjoyed this one

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• NI

  @nicholasleclerc1583 · 6 months ago

  Basically <b>the &quot;Geometric Product&quot;,</b> or an introduction to the intuition &amp; the &quot;main idea&quot; (&quot;)behind(&quot;)

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