When we use vectors for geometric and physical research, have we ever considered that vectors actually originated from “numbers”?
Before vectors were fully developed (although the concept of “quantities with both direction and magnitude” was recognized early on), complex numbers had already gained acceptance and seen preliminary applications. When we connect complex numbers with vectors, we might assume that complex numbers were linked to geometry because the operations of complex numbers on the complex plane share similarities with vectors. However, the truth is quite the opposite: vectors were gradually separated from the study of complex numbers and a system known as “quaternions.” In other words, there was a stage in history where “quaternions” and vectors were studied separately for geometry. Maxwell treated the scalar and vector parts of quaternions as distinct entities and performed extensive vector analysis. The establishment of three-dimensional vector analysis and its formal split from quaternions was completed independently in the 1880s by Gibbs and Heaviside. Vector algebra was extended to vector functions and vector calculus, leading to a debate between quaternions and vector analysis, in which vector analysis eventually prevailed. Consequently, “quaternions” gradually disappeared from textbooks. However, some special and ingenious applications of quaternions ensure that we do not forget them.
What are quaternions? First, let’s look at how complex numbers came to be. As is well known, after introducing i^2 = -1, the resulting “imaginary numbers” combined with real numbers to form the set of complex numbers. However, we can set aside this explanation and abstract complex numbers into a pure “notation”: a complex number is a number of the form a + bi, where a and b are real numbers. Operations follow the standard rules of arithmetic for real numbers, with the rule that when two i’s are multiplied, the result is recorded as -1; in all other cases, the notation i is maintained (meaning we don’t need to worry about what i actually is). The benefit of “inventing” such a number is that it is convenient for studying problems in geometry and physics. That is, we don’t need to consider what this number represents; we only need to know that it facilitates our work and allows us to obtain results relatively easily.
As complex number theory developed, people realized that real numbers are a type of “one-dimensional number,” meaning points on a number line can be in one-to-one correspondence with all real numbers, while complex numbers are “two-dimensional numbers,” requiring a plane (the complex plane) for a one-to-one correspondence. At this time, the Irish mathematician William Rowan Hamilton contemplated creating a “three-dimensional number” that would possess properties similar to complex numbers and correspond to three-dimensional space. Its form would be a + bi + cj, where a, b, c are real numbers, i is the imaginary unit from complex numbers, and j is a unit similar to but independent of 1 and i. This is equivalent to saying that complex numbers would be a subset of his envisioned “three-dimensional numbers.” Such a “creation” is simple; we can easily write down “four-dimensional” or “five-dimensional” numbers by analogy. The key question is whether such numbers can actually function. Hamilton required j^2 = -1 (noting that i \neq j), and the remaining problem was to determine ij = ?.
Hamilton proposed several requirements for “n-dimensional numbers”:
An “n-dimensional number” is of the form a_0 + a_1 i_1 + a_2 i_2 + \dots + a_{n-1} i_{n-1}, where i_1^2 = i_2^2 = \dots = i_{n-1}^2 = -1, similar to the imaginary unit in complex numbers. Its “norm” is defined as r = \sqrt{a_0^2 + a_1^2 + \dots + a_{n-1}^2}.
By setting universal results for i_p \cdot i_q (p \neq q), the norm of the product of two “n-dimensional numbers” must equal the product of their norms (this is known as the “norm law”).
The distributive law of multiplication must be satisfied.
It is easy to see that complex numbers satisfy these rules. Hamilton used these requirements to consider his envisioned “three-dimensional numbers,” which meant calculating ij = ?. First, I must remind the reader that after introducing “hypercomplex numbers” like “three-dimensional numbers,” numerical operations do not necessarily follow the rules we generally take for granted. For example, before confirming the specific properties of three-dimensional numbers, we cannot assume that the square roots of 1 are only \pm 1.
Starting with the simplest case, let’s consider ij = A. According to the “norm law,” we must have |A| = 1, so we might let A = x + yi + (\pm \sqrt{1 - x^2 - y^2})j.
Continuing to consider i(i + j), we have: i(i + j) = -1 + ij = (-1 + x) + yi + (\pm \sqrt{1 - x^2 - y^2})j. According to the norm law, we have (-1 + x)^2 + y^2 + (1 - x^2 - y^2) = (-1 + x)^2 + 1 - x^2 = 2. Solving this gives x = 0.
Following this logic, we have found some clues and can continue the derivation. Similarly, applying the norm law to (1 + i)(i + j) = -1 + i + j + ij = (-1 + x) + (y + 1)i + (1 \pm \sqrt{1 - x^2 - y^2})j. We already found x = 0, so substituting this gives -1 + (y + 1)i + (1 \pm \sqrt{1 - y^2})j. Then |(1 + i)(i + j)|^2 = 4 = (-1)^2 + (y + 1)^2 + (1 \pm \sqrt{1 - y^2})^2. This yields y = \pm 1/\sqrt{2}. It seems we should celebrate?
To be cautious, let’s check another number: (1 + i)j = j + ij = x + yi + (1 \pm \sqrt{1 - x^2 - y^2})j. Substituting x = 0 gives (1 + i)j = yi + (1 \pm \sqrt{1 - y^2})j. Then 2 = y^2 + (1 \pm \sqrt{1 - y^2})^2, which yields y = \pm 1. This contradicts the previous result!
We find that this contradiction cannot be resolved. Ultimately, we must conclude: “three-dimensional numbers” do not exist!
After going through the above derivation, you might feel some disappointment: if three-dimensional numbers don’t exist, then there’s no point in considering four-dimensional numbers. However, this is not the case. On October 16, 1843, the mathematician Hamilton, mentioned at the beginning, discovered what is now called the “quaternion,” a four-dimensional number. This is a number of the form a + bi + cj + dk. You might think that by setting d = 0, you could get a three-dimensional number. Not so fast; first, look at the rules for four-dimensional numbers:
i^2 = j^2 = k^2 = -1
ij = k, jk = i, ki = j
ji = -k, kj = -i, ik = -j
Although one could set d = 0 to make k disappear, the product of i and j would still cause k to appear; thus, one must consider “quaternions.” These rules can be derived one by one using the same method we used for three-dimensional numbers. Furthermore, it is easy to see that quaternion multiplication does not satisfy the commutative law (AB is not necessarily equal to BA), which can be proven by calculating squares. Interested readers can derive the specific details themselves.
The day after discovering “quaternions,” Hamilton wrote a letter to his friend John T. Graves to report his findings. Based on Hamilton’s discovery, Graves extended this to what he called “Octaves” or “Octonions,” which is another type of number containing Hamilton’s quaternions. These octonions, like quaternions, allow for addition, subtraction, multiplication, and division. Hamilton was very pleased with Graves’ extension. He studied these octonions in detail and found that octonion multiplication does not satisfy the associative law; generally, for three octonions A, B, C, A \times (B \times C) \neq (A \times B) \times C. The British mathematician Cayley also independently discovered octonions. (Because Cayley was more famous, octonions were later referred to as Cayley numbers).
On January 4, 1844, John Graves wrote to Hamilton reporting the basic multiplication formulas for octonions:
i^2 = j^2 = k^2 = l^2 = m^2 = n^2 = o^2 = -1
i = jk = lm = on = -kj = -ml = -no
j = ki = ln = mo = -ik = -nl = -om
k = ij = lo = nm = -ji = -ol = -mn
l = mi = nj = ok = -im = -jn = -ko
m = il = oj = kn = -li = -jo = -nk
n = jl = io = mk = -lj = -oi = -km
o = ni = jm = kl = -in = -mj = -lk
Regarding the practical applications of quaternions, one of the most critical is 3D rotation. To learn more, you can refer to the attachment of this article. This article aims to provide a simple description of the development of quaternions and the expansion of numbers. If there is an opportunity in the future, I will certainly discuss the applications of quaternions with readers in detail.
Regarding quaternions, refer to: http://en.wikipedia.org/wiki/Quaternion
Regarding Hamilton, refer to: http://www.math123.cn/sxxs/588.htm
Attachment: Quaternions.pdf
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