This One Idea Explains i, j and ε

Comments · 213

• RA

  @ramoncalvoloco · 5 months ago

  How good is starting the day this way

  115

• P3

  @P3R10T · 5 months ago

  The reason that R[x]/&lt;x²+1&gt; is a field whearas R[x]/&lt;x²&gt; or R[x]/&lt;x²-1&gt; has a zero divisor is because x²+1 is irreducible on R[x] while x² and x²-1 aren&apos;t. <br>In general, for arbitrary ring and its ideal R/I is integral (having no zero divisor) if and only if I is a prime ideal. For R = R[x], as it has unique factorization property, every ideal is generated by a single element and being prime is equivalent to be generated by an irreducible element.

  109

• AR

  @arericarnau · 5 months ago

  I don&apos;t really know what i&apos;m watching, but hey! 0=0 i understand that!

  19

• CY

  @cycygamingfrenglish · 5 months ago (edited)

  this is how I made my own number set!! (x^3=0, called the trual)

  85

• NA

  @nagarythe5216 · 5 months ago

  gigachad pvz music

  42

• GI

  @gimikER · 5 months ago

  Fun fact, some other interesting algebras can be seen as quotients of polynomial. For instance the quaternions can be seen as the algebra: R⟨x,y⟩/(x²+1)(y²+1)(xy+yx)<br><br>What exactly are the triangular brackets? They are free variables. That means they are not obligated to be commutative, i.e. xy≠yx and whatnot.<br><br>So you can have free polynomials like 1+x+4xyx which will be different than 1+x+4x²y and so on.

  18

• US

  @USSR-MATH · 5 months ago

  1,i,ε,j

  15

• YA

  @YasmeenSafdar-j4i · 5 months ago

  The split complex numbers are to Lorentzian relativity as the dual numbers are to Galilean relativity

  20

• DA

  @davidebic · 5 months ago

  Okay I understood like 70%, I&apos;ll watch it again in a few days

  7

• BR

  @BrianSpurrier · 5 months ago (edited)

  The idea of zero divisors is a lot more intuitive if you introduce it in normal modulo arithmetic. In regular arithmetic, if two numbers multiply to zero, one or both of them have to be zero. But for certain moduluses this breaks down.<br><br>The modulo almost everyone is familiar with even if they don’t know it is mod 10. Basically, just do the math as normal, but only care about the 1’s digit. If a number ends in zero, you know it has to be divisible by 10, but 10 is also 2*5. So you can have two numbers, say 8 and 15, neither of which are divisible by 10, but their product 120 is.<br>In mod 10, 8 is the same, 15 is equivalent to 5, and 120 is 0. So the equation would be<br>8×5 = 0 (mod 10)

  3

• JO

  @johnpayne7873 · 5 months ago

  Great stuff! &nbsp;Never before have I grasped the rotational geometric nature of the complex plane <br>Like a bird circling high above only to land on its buried nest without effort

• HE

  @hedytang177 · 5 months ago

  such clear explanations!

  1