# $j^2 = 1$, but $j \neq \pm 1$; what is $j$?

I was watching a Numberphile video about the favorite number of some mathematicians, and at one point, the creator of MinutePhysics said the following -

Similar to the way that $i$ is $\sqrt{-1}$, but what that actually means is that $i^2$ is $-1$, $j^2$ is $+1$, but $j$ is not $1$.

Here's the video with him saying all this - http://youtu.be/ygqIfLHGTu4?t=5m35s

I've searched the internet for anything about j, but this video seems to be the only place where $j$ is mentioned.

Does $j$ exist? If so, can someone explain what $j$ is?

Or is this whole $j$ thing just a joke?

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Yes, $j$ is a shorthand for "joke". Much like this comment is $j$! :-) –  Asaf Karagila Apr 21 '13 at 17:00
@Sim $(-1)^2=1$ –  Git Gud Apr 21 '13 at 17:00
@GitGud So $j$ doesn't represent anything else then? I did question if it was just equal to $-1$, but I thought there might be more to it. –  Xenon Apr 21 '13 at 17:02
@Stefan, then how would you define $i$? Isn't the positive value ($i$) the principal square root of $-1$? (with "principal" not meaning "better" IMHO, but simply the simplest - the positive, as the square root is a function and can't yield two values for the same $x$). –  JMCF125 Jul 15 '13 at 13:23

The number of the form $a+bj$ are split-complex numbers. The j is exactly like what is i for complexes to split-complex numbers, exept with the property $j^2=+1$. Using a matrix interpretation with these numbers, we have,

$i=\;\; \begin{pmatrix} 0 & -1\\ 1 & 0 \end{pmatrix}$

$i^2 =\;\; \begin{pmatrix} -1 & 0\\ 0 & -1 \end{pmatrix} = -I$

$I^2 = \;\; \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix}^2 = \;\; \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix}$ But also we have

$\begin{pmatrix} 0 & -1\\ -1 & 0 \end{pmatrix}^2 = \;\; \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} = I$

and $\begin{pmatrix} 0 & -1\\ -1 & 0\\ \end{pmatrix}$ is actually the definition of j as a matrix, so $j^2=I$ and $j≠\pm I$. This Wikipedia article explains it pretty well.

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How to prevent confusing with the 'j' in Quaternions? en.wikipedia.org/wiki/Quaternion –  user1095332 Apr 21 '13 at 18:07
Look to my updated answer using matrix definitions. A split complex is defined using 2x2 matrices whereas a quaternion is defined as a 4x4 matrix. Matrices are quite simple for not confusing them! –  moray95 Apr 21 '13 at 18:15
We can also define quaternions as $2\times 2$ matrices with (certain) complex entries. –  Pedro Tamaroff May 2 '13 at 0:58
@PeterTamaroff Yes you're right, I should have 2x2 real matrices. And we could also add the most important property : $j^2=1$ in slpix-complexes. But $j^2=-1$ in Quaternions. –  moray95 May 2 '13 at 15:07