What is the exact definition of the modulus of a number? As far as I know, it is the distance between the origin and the point associated with this number. So if $z=a+bi \in \Bbb C,|z|=OM=\sqrt{a^2+b^2}$ where M is the point with affix $z$. By extending this definition to split-complex numbers, we should have $z=a+bj$ and $|z|=\sqrt{a^2+b^2}$. But the modulus of a split-complex number is given by $|z|=\sqrt{a^2-b^2}$. Where this formula come from and what does it represents?
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$\begingroup$The modulus can also be written as
$$|z|^2 = z\overline{z}.$$
In the split-complex numbers, if we have $z = a+bj$, then
$$|z|^2 = (a+bj)(a-bj) = a^2+abj-abj-b^2j^2 = a^2-b^2$$
since $j^2 = +1$.
Thus, $|z| = \sqrt{a^2-b^2}.$
For complex numbers, you can see how $-b^2i^2$ becomes $+b^2$.
For complex numbers, the notion between "modulus" and "size" goes through the notion that the modulus is a norm, and norms and sizes are intuitively linked.
For the split-complex numbers, the modulus is not a norm, so the link between modulus and size does not exist.
Rather, think of modulus as something that has a handy property that $|zw| = |z||w|$.
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