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### Number and algebra

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# Two and Four Dimensional Numbers

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### Roots and Coefficients

### Target Six

### 8 Methods for Three by One

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Age 16 to 18

Challenge Level

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This problem starts by using matrices as a model for complex numbers and showing that the structure

$$ \left( \begin{array}{cc} 1& 0\\ 0& 1\end{array} \right)x+ \left( \begin{array}{cc} 0& -1\\ 1& 0\end{array} \right)y $$ behaves in the same way as $x+{\text i}y$.

The idea is then extended to introduce three different two by two matrices which all square to give $$ \left( \begin{array}{cc} -1& 0\\ 0& -1\end{array} \right) $$.At the end of the problem, there are some links and videos for further information about quaternions.

There are more matrix problems in this feature.

If xyz = 1 and x+y+z =1/x + 1/y + 1/z show that at least one of these numbers must be 1. Now for the complexity! When are the other numbers real and when are they complex?

Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.

This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different? Which do you like best?