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The system of quaternions was discovered by William Rowan Hamilton on the afternoon of Monday, October 16, 1843, as he walked along Royal Canal in Dublin. A quaternion is an abstract symbol of the form a + bi + cj + dk, where a, b, c, d R are real numbers: H = {a + bi + cj + dk : a, b, c, d in R. The “imaginary units” i,j, k are abstract symbols satisfying the following multiplication rules: i^2+j^2+ k^2 = ijk=1. One can check that {H, +, *, 0, 1} is a ring. However, it is not a commutative ring because (for example) we have ij =-k and ji=-k not eaqual k. Your assignment is to write a mathematical paper about the quaternions, including some of exposition and history, but focusing mainly on mathematical results. Here are some ideas:

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The system of quaternions was discovered by William Rowan Hamilton on the afternoon of

Monday, October 16, 1843, as he walked along Royal Canal in Dublin. A quaternion is an

abstract symbol of the form a + bi + cj + dk, where a, b, c, d e R are real numbers:

H = {a + bi + cj + dk: a,b,c,d R}.

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

ip = ju = k = ijk = -1.

One can check that (H, +, -,0,1) is a ring. However, it is not a commutative ring because (for

example) we have ij = -k and ji = -k + k. Your assignment is to write a mathematical

paper about the quaternions, including some of exposition and history, but focusing mainly

on mathematical results. Here are some ideas:

• Given a = a+bk+ci+dk e H, we define the quaternion conjugate by a* = a-bi-cj-dk.

Show that aa* = a*a = lal = (a? + b2 +e? + d) € R and use this to show that every

nonzero quaternion has a two-sided inverse: aa-1 = a-‘a = 1.

For any a, B e H show that (a3)* = B*a*. Then it follows from the previous remark

that |a8| = |2|||. Use this to show that if m,n e Z can each be expressed as a sum of

four integer squares, then mn can also be expressed as a sum of four integer squares.

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

• Quaternions of the form u = ui + uj + wk are called imaginary. We can also view u as

the vector (u, v, w) in R3. Explain how the product of imaginary quaternions is related

to the dot product and the cross product of vectors.

• If u is imaginary of length 1, show that u? = -1. It follows that the polynomial

22 +1 € H[2] of degree 2 has infinitely many roots in H. Why does this not contradict

Descartes’ Factor Theorem? (Hint: H is not commutative.

• Show that every quaternion a e H can be written in polar form as a = |a|cos 0+usin ),

where u is imaginary of length 1.

• For any a, x € H with a 60 and x imaginary, show that a-‘xa is imaginary.

• Suppose that a = cos 0 + usin0 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 in R3.

Explain why a-?xa corresponds to the rotation of x around the axis u by angle 20.

. Let u e H be imaginary of length 1 and let 0 € R be real. Explain why it makes sense

to define the exponential notation du = cos 0 + usin 0.

• Consider the following set of 24 quaternions: {+1, ti, Ej, tk, (+1+i+j+k)/2). Explain

how this set is related to a regular tetrahedron. Hint: A regular tetrahedron has 12

rotational symmetries.

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Tags:

complex numbers

Exponential Form

dimensional number system

imaginary units

William Rowan Hamilton

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