Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 -

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?

share|cite|improve this question
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
up vote 12 down vote accepted

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.

share|cite|improve this answer
How to prevent confusing with the 'j' in Quaternions? – 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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.