**Octonions**build an eight-dimensional number system— the largest of the four sets of numbers of normed devision algebra [1,2]. Baez and Huerla (both with first name John) describe how they were discovered by John Graves (another John!), who called them

**octaves**[3]. Graves communicated his findings to the Irish mathematician William Rowan Hamilton. Although Hamilton was not interested in these numbers, he reported them at the Irish Royal Society. Without any written publication, Arthur Cayley, one of the

*Invariant Twins*[4], rediscovered the octonions and beat Graves to publication. Thus, octonions are also known as

**Cayley numbers**—and not as

**Graves numbers**.

Octonions are the elements in a

**Cayley algebra**[5]. Like the multiplication of quaternions, the

**multiplication**of octonions is

**noncommutative**: order matters. Multiplication of octonions is

**not**even

**associative**. But they have two “very good properties” [2]: every nonzero octonion has a

**multiplicative inverse**, and two nonzero octonions never multiply together to give zero. John Graves amateur interest (he was a lawyer) in algebra and his imagination of numbers with supernatural properties remains striking, especially, when considering their growing importance in explaining the universe and modeling the matter and forces therein.

*: multidimensional systems, vector space, supersymmetry, spinors*

**Keywords****References and further reading**

[1] John C. Baez:

**The Octonions**[math.ucr.edu/home/baez/octonions/].

[2] Timothy Gowers (Editor):

**The Princeton Companion to Mathematics**. Princeton University Press, Princeton, New Jersey, 2008; pp. 275 to 279.

[3] John C. Baez and John Huerta:

**The Strangest Numbers in String Theory**.

*Sci. Am*. May

**2011**,

*304*(5), pp. 60-65 [www.scientificamerican.com/article.cfm?id=octonions-web-exclusive].

[4] Chapter 20 “

**Invariant Twins**” in

**Men of Mathematics**by E. T. Bell. Simon & Schuster, New York,

**1937**.

[5] Wolfram MathWorld:

**Octonion**[mathworld.wolfram.com/Octonion.html].