The Quaternion group

 The concept of a square root of a negative number dates back to ancient Greece, and is credited to Heron, whose name is immortalized in the method for finding the area of a triangle given the lengths of the three sides. The name imaginary number was coined by Rene Descartes in the 1600s, who did not like the concept and hoped he could kill it by mocking it. The concept did not die, and about a century later, Leonhard Euler became the first to use the lowercase letter i as the square root of -1.

 

Numbers of the form a + bi, where a and b are real numbers constitute the complex numbers C. The common way to represent the complex numbers as a picture is the complex plane, where the real numbers are represented by the x-axis and the imaginary numbers are lined up on the y-axis. The axes intersect at zero.

 

Just like the real numbers R, the complex numbers C are an abelian group under addition and C - {0} is an abelian group under multiplication. We have discussed that ({1, -1}, ✕) is a finite abelian group and a subgroup of the real numbers. Similarly, ({1, i, -1, -i}, ✕) is a finite abelian group and a subgroup of the complex numbers. Note that -i is also a square root of -1, so it has two square roots in much the same way any positive number has two square roots. For example, 3²=9 and (-3)²=9.

 

Enter William Rowan Hamilton (1805-1865), widely regarded as the greatest Irish mathematician of all time. He created a mathematical system he called the quaternions, which has not two square roots of -1, but six. The general form of a number in the quaternion system is a + bi + cj + dk, where a, b, c and d  are arbitrary real numbers and i, j and k are square roots of -1, as are -i, -j and -k.

 

The quaternion system under multiplication is non-abelian. Here are the rules for multiplying i, j and k.

 

ij = k      jk = i     ki = j

ji = -k     kj = -i    ik = -j

 

Here is the full Cayley table representation, where we can find ij in the square corresponding to row i, column j.

There are three subgroups of order 4, and since 8/2 = 4, these are all normal subgroups.

Here are the subgroups.

{1, i, -1, -i}

{1, j, -1, -j}

{1, k, -1, -k}

 

Since 8 is divisible by 8, 4, 2 and 1, we might expect there is a subgroup of order 2, and {1, -1} fills the bill.

The quaterions are a non-abelian group where all subgroups are normal. This cannot be said for any other non-abelian group we have studied.

I present group theory in terms of pure mathematics, but groups play a vital role in the study of modern physics, including the quaternions. This is a link to the Wikipedia page that tells more.

Commentary

I mentioned four mathematicians today, so let me present links to their biographies. Far be it for me to talk smack about any of these guys, but the general public is probably better acquainted with Rene Descartes than nay of the other names, because Descartes was also a philosopher. Very few mathematicians after the Greeks are famous for just their mathematical work. If we judged this quartet solely on their mathematics, Leonhard Euler (1707-1783) is the true giant. For the quantity and quality of his work, he could be compared to the composer Johann Sebastian Bach (1685-1750).

Link to the biography of William Rowan Hamilton.

 

Link to the biography of Leonhard Euler.

 

Link to the biography of Heron of Alexandria.

Link to the biography of Rene Descartes.