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# Two and Four Dimensional Numbers

##### Age 16 to 18Challenge Level To do this problem you only need to know how to add and multiply two by two matrices. The problem gives models for complex numbers and 4-dimensional numbers called quaternions. Although you don't need the information to do this problem you may like to read the NRICH article What are Complex Numbers? and the Plus article: Curious Quaternions .

(1) Let $C^*$ be the set of $2 \times2$ matrices of the form $$\left( \begin{array}{cc} x& -y\\ y& x\end{array} \right)$$ where $x$ and $y$ are real numbers and addition and multiplication are defined according to the usual rules for adding and multiplying matrices.

(a) Add and multiply the matrices $$\pmatrix {x & -y \cr y & x} {\rm \ and }\ \pmatrix {u & -v \cr v & u}.$$ (b) What are the identities and inverses for addition and multiplication?

(c) Consider also the subset $R^*$ for which $y=0$. Investigate the arithmetic of $R^*$ (addition, subtraction, multiplication, division, identities, inverses, distributive law) and compare it with the arithmetic of real numbers.

(d) Compare the arithemetic of $C^*$ with that of complex numbers.

(e) The matrix $\pmatrix {-1 & 0 \cr 0 & -1}$ is equivalent to the real number -1. What is the matrix equivalent to $i = \sqrt {-1}$ in the set of complex numbers?

(2) Complex numbers are two-dimensional numbers $x + iy$ where $x$ and $y$ are real numbers and $i = \sqrt {-1}$. Quaternions are four-dimensional numbers of the form $a+{\bf i}x+{\bf j}y+{\bf k}z$ where $a, x, y, z$ are real numbers and ${\bf i, j, k}$ are all different square roots of $-1$.

Can such a number system exist? To answer this, let's assume that in part (1) you have established that if real numbers and 2 by 2 matrices exist then so does the complex number $i = \sqrt {-1}$. The model for the system of quaternions is the set of linear combinations of 2 by 2 matrices: $$aI +{\bf i}x + {\bf j}y + {\bf k}z\ = a\pmatrix {1 & 0\cr 0 & 1}+ x\pmatrix {i & 0\cr 0 & -i}+y\pmatrix {0 & 1\cr -1 & 0}+ z\pmatrix {0 & i\cr i & 0}.$$ where $a, x, y, z$ are real numbers.

(a) Work out the matrix products: ${\bf i^2, \ j^2}$ and ${\bf k^2}$ showing that these matrices give models of three different square roots of -1.

(b) Work out the matrix products: ${\bf i j}$, ${\bf j i}$, ${\bf j k}$, ${\bf k j}$, ${\bf k i}$ and ${\bf i k}$ showing that the three matrices ${\bf i,\ j,\ k}$ model unit vectors along the three axes in 3-dimensional space ${\bf R^3}$ with matrix products isomorphic to vector products in ${\bf R^3}$.

You have now shown that this set of linear combinations of matrices models the quaternions.

(3) Investigate the sequence: ${\bf i},\$ ${\bf i j},\$ ${\bf i j}$ ${\bf k},\$ ${\bf i j}$ ${\bf k i},\$ ${\bf i j}$ ${\bf k i}$ ${\bf j,\ ...}$

... and the story continues in the Plus article: Ubiquitous Octonions