I read wiki topic about history of Quaternion, and it confuses me on why, according to Hamilton, there's a problem with multiplication of triple (i.e. 1+i+j), and somehow the quadruple (or Quaternion) solves that problem.
What was the trouble he faced when he tried to define multiplication for triple? and why did introduction of the 4th element solve the problem?
Let's say you've just invented a new square root of $-1$, called $j$, to augment the complex numbers. That's what Hamilton did to describe rotations in $3$ dimensions, the way complex numbers work for rotations in $2$. Now, you need your number system to be closed under multiplication, but is $ij$ of the form $a+bi+cj$ with $a,\,b,\,c\in\Bbb R$? Sadly not. Indeed, Hurwitz's theorem shows with hindsight what the problem was: Hamilton was trying to do in dimension $3$ what can only happen in dimension $1$, $2$, $4$ or $8$.