# Quaternion

Mathematicians create entire worlds from a small set of rules. This can be seen with the advent of real numbers. The world around the real numbers is still growing even today. However, there are always roadblocks that stop mathematicians dead in their tracks. For example, in the real numbers, there is no solution to $x^2 + 1 = 0$ because we would have to take the square root of $-1$. As it turns out, very few mathematicians are fazed by these roadblocks. Instead, we actually just define new rules and keep rolling on. This is exactly what happened in this case.

Many of us have seen the complex numbers. These are numbers of the form $a + bi$ where $a$ and $b$ are real numbers and $i = \sqrt{-1}$. Perhaps more important to this story, a complex number can be interpreted as points in a plane. Instead of the usual Cartesian plane with $x$ and $y$ axes, we have a Complex plane with a real and imaginary axis. So, for example, we can plot the complex number $3 + 4i$ as the point $(3,4)$.

This plane has many interesting geometrical concepts. But it is still just a plane. So yet again, we’ve hit a wall. What if we wanted a number system that works nicely, but applies to our 3-space? Enter William Hamilton.

William Hamilton was an Irish mathematician and physicist concerned with this exact problem. In 1843 Hamilton, likely in an attempt to clear his mind, went for a stroll with his wife. While walking on that October day, they decided to cross Broome Bridge. But little did Hamilton know, mathematics inspiration waits for no one. It suddenly dawned on him!

$i^2 = j^2 = k^2 = ijk = -1$

William, beside himself, was absolutely not prepared for writing down such a discovery. So, he did what any self-respecting mathematician would do. Gone was the care for public property as he carved hard these rules into the stone of the bridge, his face grinning gleefully as he did so. Unfortunately, no trace of the carving remains today. Instead, there is just a plaque commemorating the event.

The Quaternion, as he called them, found their use in mechanics and other areas of Physics. Multiplication in this group can be interpreted as a rotation in 4-dimensional space. For more on the algebraic side of things, consult popular Abstract algebra textbooks such as Dummit and Foote.

Hamilton also introduced the Bi-quaternion. This is an 8-tuple with a multiplication rule. One might be led to ask the question, is there a 16-tuple that works nicely? How high can we go? There are a ton of methods for extending the real numbers. Check out the Cayley-Dickson construction and go from there. However, if we are talking about finite dimensional division algebras, we can only go to the Quaternion and no farther. This is Frobenius’ theorem. Hyper-complex numbers is an ever interesting field of mathematics. These not only extend the real numbers, but also extend our understanding of the inherent nature of that we call number.