Simanaitis Says

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OUR TOUR OF numbers here at SimanaitisSays has included “You Can Count on Me—Or Maybe Not,” “Exploring Rationality,” and “Euler’s Elegance.” 

Along the way, we’ve encountered three expanding sets of numbers: There’s N, the Natural Numbers, …, -3, -2, -1, 0, 1, 2, 3, …. There’s Q, the Rational Numbers, those of the form m/n where m and n are natural numbers. And there’s R, the Real Number line centered at 0 and stretching infinitely far in either direction.

The Real Line, R, containing both rational and irrational numbers. Image from Wikipedia.

In two dimensions, we’ve encountered C, the Complex Plane, with numbers of the form a + bi where imaginary number i has the defining property that i2 = -1. The horizontal axis is the real axis; the vertical one is the imaginary one. 

The Complex Plane, C. Image from

Just as the real line extends off infinitely in either direction, the complex plane reminds me of Steven Leacock’s horseman “flinging himself upon his horse and riding madly off in all directions.” And just as the real numbers have many linear applications, the complex plane has many applications, including those in physics, electrical engineering, flow dynamics, and aerodynamics.

The Next Expansion. Quaternions H are of the form a + bi + cj + dk, where i, j, and k interact in special ways, as shown in the table below. In particular, i2 = -1,  j2 = -1, and k2 = -1, each behaving like the imaginary number i in C.

The basic Quaternion Multiplication Table. Image from Wikipedia.

A Geometric Interpretation. Just as C has a two-dimensional interpretation, lurking behind this curious multiplication is a geometric interpretation of H in four dimensions. 

Graphical representation of products of quaternion units as 90° rotations in the planes of 4-dimensional space spanned by two of {1, i, j, k}. The left factor can be viewed as being rotated by the right factor to arrive at the product. Visually  j = −( i).
  • In blue:
    • 1 ⋅ i = i (1/i plane)i ⋅ j = k (i/k plane)
    In red:
    • 1 ⋅ j = j (1/j plane)j ⋅ i = −k (j/k plane)
  • Image by Machen from Wikipedia.

    A Non-Commutative Multiplication. Note, ij = – ji is but one example of quaternion multiplication being generally non-commutative. (In most multiplications we know, e.g., 3 x 2 = 2 x 3 = 6 either way. Not necessarily so in H.)

    The Discovery of Quaternions. Sir William Rowan Hamilton thought of himself as a pure mathematician, not a physicist. Nonetheless, he made important contributions in a reformulation of Newtonian mechanics, now known as Hamiltonian mechanics. Wikipedia notes “This work has proven central to the modern study of classical field theories such as electromagnetism and to the development of quantum mechanics.”

    Sir William Rowan Hamilton, 1805–1865, Irish mathematician, Andrews Professor of Astronomy at Trinity College, Dublin, and Royal Astronomer of Ireland.

    Hamilton’s discovery of quaternions arose in a walk along Dublin’s Royal Canal with his wife on October 16, 1843. He had been musing about a way to extend complex numbers’ two-dimensional representation to higher dimensions. Then, alongside Brougham Bridge (now called Broom Bridge), he thought of a four-dimensional interpretation wherein i 2 = j2 = k2 = ijk = -1. 

    Hamilton was so delighted with this discovery that he carved this equation on Brougham Bridge. 

    Hamilton’s Quaternion Equation plaque on today’s Broom Bridge, Dublin. Below, the bridge.

    Real-World Applications. In The Irish Times, October 4, 2018, Peter Lynch describes “The Many Modern Uses of Quaternions.” Lynch notes that quaternions “languished in obscurity for a century or more but have re-emerged recently in several contexts. Today, quaternions have applications in astronautics, robotics, computer visualisation, animation and special effects in movies, navigation and many other areas.”

    In 2005, dignitaries traveled Dublin’s Royal Canal to unveil a new plaque. Image from The Irish Times, October 4, 2018.

    A Personal Note. In undergraduate school back in the 1960s, I was learning computer programing, including Machine Language, Assembly Language, Dartmouth Professor John Kemeny’s BASIC, and FORTRAN. (“What’s ‘Machine Language,’ Grandpa?” “Look at two fingers,” Grandpa began….)

    At one point, I needed a programming project that was straightforward enough for my chosen allotment of time, yet original enough to satisfy my instructor. I had been interested in the history of mathematics and decided my project would be computerized quaternion arithmetic. 

    Thanks, Professor Hamilton, for the inspiration. ds 

    © Dennis Simanaitis,, 2021  

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