MTH 461 Spring 2020

MTH 461

Spring 2020

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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.]

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