Cayley–Dickson Construction of Complex, Quaternion and Octonion Numbers

  • Complex number \displaystyle{a + i b} can be thought of as a pair of real numbers.
  • Quaternion can be thought of as a pair of complex numbers
  • Octonion can be thought of as a pair of quaternions

When we construct the complex, quaternion and octonion numbers as pairs mentioned above then their addition and multiplication rules will look the same. This is known as Cayley-Dickson construction.


In the formulas below you need to remember the conjugate rule for the real numbers. Real number is its own conjugate. Let \displaystyle{a} be a real number,  \displaystyle{a^*=a}

If \displaystyle{(a,b)} represents either a complex number or a quaternion or an octonion the conjugate is defined as

\displaystyle{(a,b)^* = (a,-b)}

Addition rule

Let \displaystyle{(a,b)} and \displaystyle{(c,d)} represent either 2 complex numbers or 2 quaternions or 2 octonions. The addition rule is straight forward: component-wise.

\displaystyle{(a,b) + (c,d)=(a+c, b+c)}

Multiplication rule for the pairs of real numbers

Let the pair \displaystyle{(a,b)} represent a complex number, and the pair \displaystyle{(c,d)} represent another complex number. The \displaystyle{a}, \displaystyle{b}\displaystyle{c}, \displaystyle{d} are real numbers.

\displaystyle{(a,b)(c,d)=(ac-db^*, a^* d+cb)}

Remember that \displaystyle{a} and \displaystyle{b} are real numbers, therefore \displaystyle{b^*=b} and \displaystyle{a^*=a}

Multiplication rule for the pairs of complex numbers

Let the pair \displaystyle{(a,b)} represent a quaternion, and the pair \displaystyle{(c,d)} represent another quaternion. The \displaystyle{a}, \displaystyle{b}\displaystyle{c}, \displaystyle{d} are complex numbers.

\displaystyle{(a,b)(c,d)=(ac-db^*, a^* d+cb)}

Multiplication rule for the pairs of quaternions

Let the pair \displaystyle{(a,b)} represent an octonion, and the pair \displaystyle{(c,d)} represent another octonion. The \displaystyle{a}, \displaystyle{b}\displaystyle{c}, \displaystyle{d} are quaternions.

\displaystyle{(a,b)(c,d)=(ac-db^*, a^* d+cb)}

Multiplication formulas are the same

Have you noticed that the forms of the multiplication rule are the same in all three cases. The \displaystyle{a}, \displaystyle{b}, \displaystyle{c}, \displaystyle{d} are real numbers in the first case, complex numbers in the second case and quaternions in the third case but the form of the multiplication formula remains the same.

Multiplicative inverse

Real numbers, complex numbers, quaternions and octonions have multiplicative inverses.

For the complex numbers in pair notation we can check if it is true


where \displaystyle{k} is a real number, the square of the norm of \displaystyle{(a,b)}. You can check this for quaternions and octonions as well.

Whenever \displaystyle{(a,b)} is non-zero the multiplicative inverse is


About Suresh Emre

I have worked as a physicist at the Fermi National Accelerator Laboratory and the Superconducting Super Collider Laboratory. I am a volunteer for the Renaissance Universal movement. My main goal is to inspire the reader to engage in Self-discovery and expansion of consciousness.
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