Definition

The definition of a quaternion finds its shelter in a class which is called Quaternion:

class Quaternion
 
end

The first method of a quaternion will be its instantiation:

       def initialize(a,b)
                @a,@b = a,b
        end

       def a
                @a
        end
 
        def b
                @b
        end

Display

In order that it be easy to display a quaternion q with puts(q) it is necessary to redefine a method to_s for it (a case of polymorphism). There are several choices but this one works OK:

       def to_s
                '('+a.real.to_s+')+('+a.imag.to_s+')i+('+b.real.to_s+')j+('+b.imag.to_s+')k'
        end

To read it loud it is better to read from right to left. For example, a.real denotes the real part of a and q.a.real denotes the real part of the a part of q.

Modulus

The absolute value of a quaternion is a (positive) real number.

       def abs
                Math.hypot(@a.abs,@b.abs)
        end

Conjugate

The conjugate of a quaternion is another quaternion, having the same modulus.

       def conj
                Quaternion.new(@a.conj,-@b)
        end

Addition

To add two quaternions, just add their as together, and their bs together:

       def +(q)
                Quaternion.new(@a+q.a,@b+q.b)
        end

Subtraction

The use of the - symbol is an other case of polymorphism, which allows to write rather simply the subtraction.

       def -(q)
                Quaternion.new(@a-q.a,@b-q.b)
        end

Multiplication

Multiplication of the quaternions is more complex (!):

       def *(q)
                Quaternion.new(@a*q.a-@b*q.b.conj,@a*q.b+@b*q.a.conj)
        end

This multiplication is not commutative, as can be checked by the following examples:

p=Quaternion.new(Complex(2,1),Complex(3,4))
q=Quaternion.new(Complex(2,5),Complex(-3,-5))
puts(p*q)
puts(q*p)

Division

The division can be defined as this:

       def /(q)
                d=q.abs**2
                Quaternion.new((@a*q.a.conj+@b*q.b.conj)/d,(-@a*q.b+@b*q.a)/d)
        end

As they have the same modulus, the quotient of a quaternion by its conjugate has modulus one:

p=Quaternion.new(Complex(2,1),Complex(3,4))
 
puts((p/p.conj).abs)

This last example digs that ${\displaystyle \left(-{\frac {22}{30}}\right)^{2}+\left({\frac {4}{30}}\right)^{2}+\left({\frac {12}{30}}\right)^{2}+\left({\frac {16}{30}}\right)^{2}=1}$, or ${\displaystyle 22^{2}+4^{2}+12^{2}+16^{2}=484+16+144+256=900=30^{2}}$, which is a decomposition of ${\displaystyle 30^{2}}$ as a sum of 4 squares.

Definition

All the following methods will be enclosed in a class called Octonion:

class Octonion
 
        def initialize(a,b)
                @a,@b = a,b
        end
 
        def a
                @a
        end
 
        def b
                @b
        end

At this point, there is not much difference from the quaternion object. Only, for an octonion, a and b will be quaternions, not complex numbers. Ruby will know it when a and b will be instantiated.

Display

The to_s method of an octonion (converting it to a string object so that it can be displayed) is very similar to the quaternion equivalent, only there are 8 real numbers to display now:

       def to_s
                '('+a.a.real.to_s+')+('+a.a.imag.to_s+')i+('+a.b.real.to_s+')j+('+a.b.imag.to_s+')k+('+b.a.real.to_s+')l+('+b.a.imag.to_s+')li+('+b.b.real.to_s+')lj+('+b.b.imag.to_s+')lk'
        end

The first of these numbers is the real part of the a part of the first quaternion, which is the octonions's a! Accessing to this real part of the a part of the octonion's a part, requires to go through a binary tree which depth is 3.

Modulus

Same than for the quaternions:

	def abs
		Math.hypot(@a.abs,@b.abs)
	end

Conjugate

	def conj
		Octonion.new(@a.conj,Quaternion.new(0,0)-@b)
	end

Addition

Like for the quaternions, one has just to add the as and the bs separately (only now the a and b part are quaternions):

	def +(o)
		Octonion.new(@a+o.a,@b+o.b)
	end

Subtraction

	def -(o)
		Octonion.new(@a-o.a,@b-o.b)
	end

Multiplication

	def *(o)
		Octonion.new(@a*o.a-o.b*@b.conj,@a.conj*o.b+o.a*@b)
	end

This multiplication is still not commutative, but it is even not associative either!

m=Octonion.new(p,q)
n=Octonion.new(q,p)
o=Octonion.new(p,p)
puts((m*n)*o)
puts(m*(n*o))

Division

	def /(o)
		d=1/o.abs**2
		Octonion.new((@a*o.a.conj+o.b*@b.conj)*Quaternion.new(d,0),(Quaternion.new(0,0)-@a.conj*o.b+o.a.conj*@b)*Quaternion.new(d,0))
	end

Here again, the division of an octonion by its conjugate has modulus 1:

puts(m/m.conj)
puts((m/m.conj).abs)