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Two and Four Dimensional Numbers

Age 16 to 18
Challenge Level

This problem is about establishing the existence of complex numbers and of quaternions by providing two models.

Real numbers fill a line (1 dimension), complex numbers fill a plane (2 dimensions) and quaternions fill 4-dimensional space. If we assume that real numbers exist (that is, if we know about the arithmetic of real numbers) can we be sure that complex numbers and quaternions exist?

In part (1) we study a model where the arithmetic and algebra is isomorphic to the arithmetic and algebra of complex numbers.

In part (2) we extend the model to one where the arithmetic and algebra is isomorphic to that of quaternions.

To add and take scalar multiples of quaternions just treat them like 4 dimensional vectors, for example: $$(a_1 +b_1{\bf i} + c_1{\bf j} + d_1{\bf k}) + (a_2 +b_2{\bf i} + c_2{\bf j} + d_2{\bf k})= (a_1+a_2) + (b_1+b_2){\bf i} + (c_1+c_2){\bf j} + (d_1+d_2){\bf k}).$$ Multiplication is defined by the rules of ordinary algebra where $${\bf i}^2={\bf j}^2={\bf k}^2=-1,\quad {\bf i j} = {\bf k} = {\bf -j i}, \quad {\bf j k} = {\bf i} = {\bf -k j},\quad {\bf k i} = {\bf j} = {\bf -i k} .$$ We can look at quaternions in three different but equivalent ways.

(1) Quaternions are points $(a,b,c,d)$ in 4-dimensional space ${\bf R}^4$.

(2) Quaternions are ordered pairs of complex numbers $(z,w)= (a+i b, c+i d)$. As such they are elements of ${\bf R}^2\times {\bf R}^2$.

(3) Quaternions are ordered pairs consisting of a real number $a$ and a vector $x i+y j+z k$ in 3-space that is they are elements of ${\bf R}\times {\bf R}^3$.

In the problems and articles on this site we mainly use the third representation.

While real numbers and complex numbers form fields, the arithmetic and algebra of quaternions is the same as a field in all respects except that multiplication is not commutative. For this reason the structure for the algebra of quaternions is called a skew field.

To read about number systems, where quaternions fit in, why there are no three dimensional numbers and numbers in higher dimensions, see the NRICH article What Are Numbers?

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