The system ofquaternionswas discovered by William Rowan Hamilton on the afternoon ofMonday, October 16, 1843, as he walked along Royal Canal in Dublin. A quaternion is anabstract symbol of the forma`bi`cj`dk, wherea,b,c,dPRare real numbers:

MTH 461

Spring 2020

Writing Project

Drew Armstrong

The system ofquaternionswas discovered by William Rowan Hamilton on the afternoon ofMonday, October 16, 1843, as he walked along Royal Canal in Dublin. A quaternion is anabstract symbol of the forma`bi`cj`dk, wherea,b,c,dPRare real numbers:

H“ ta`bi`cj`dk:a,b,c,dPRu.

The “imaginary units”i,j,kare abstract symbols satisfying the following multiplication rules:

i2“j2“k2“ijk“ ́1.

One can check thatpH,`, ̈,0,1qis a ring. However, it is not a commutative ring because (forexample) we haveij“ ́kandji“ ́k‰k. Your assignment is to write a mathematicalpaper about the quaternions, including some of exposition and history, but focusing mainlyon mathematical results. Here are some ideas:

•Given α“a`bk`cj`dkPH, we define the quaternion conjugate byα ̊“a ́bi ́cj ́dk.Show thatαα ̊“α ̊α“ |α| “ pa2`b2`c2`d2q PRand use this to show that everynonzero quaternion has a two-sided inverse:αα ́1“α ́1α“1.

•For any α,βPHshow thatpαβq ̊“β ̊α ̊. Then it follows from the previous remarkthat|αβ| “ |α||β|. Use this to show that ifm,nPZcan each be expressed as a sum offour integer squares, thenmncan also be expressed as a sum of four integer squares.

•Explain how quaternions can be represented as 2ˆ2 matrices with complex entries.

•Quaternions of the formu“ui`vj`wkare called imaginary. We can also viewuasthe vectorpu,v,wqinR3.

Explain how the product of imaginary quaternions is relatedto the dot product and the cross product of vectors.

•If u is imaginary of length 1, show thatu2“ ́1. It follows that the polynomialx2`1PHrxsof degree 2 has infinitely many roots inH. Why does this not contradictDescartes’ Factor Theorem? [Hint:His not commutative.]

•Show that every quaternionαPHcan be written in polar form asα“ |α|pcosθ`usinθq,whereuis imaginary of length 1.

•For any α,xPHwithα‰0 andximaginary, show thatα ́1xαis imaginary.

•Suppose thatα“cosθ`usin θ where u is imaginary of length 1, and let x be any imaginary quaternion. Recall that we can also think of u and x as vectors inR3.Explain why α ́1xαcorresponds to the rotation of x around the axis u by angle 2θ.

•Let u P H be imaginary of length 1 and letθPRbe real. Explain why it makes sense to define the exponential notationeθu“cosθ`usinθ.•Consider the following set of 24 quaternion s:t ̆1, ̆i, ̆j, ̆k,p ̆1 ̆i ̆j ̆kq{2u. Explain how this set is related to a regular tetrahedron. [Hint: A regular tetrahedron has 12rotational symmetries.]