### Roots and Coefficients

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?

### Target Six

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

### 8 Methods for Three by One

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?

# Two and Four Dimensional Numbers

##### Age 16 to 18 Challenge Level:

If you can show for two systems that whatever operations you carry out in one are always exactly mimicked in the other, then you can work in whichever system is the more convenient to use and all the results carry over to the other system. We say that the systems are isomorphic.

Using a set of matrices exhibits all the algebraic structure of complex numbers including a matrix with real entries that corresponds to $\sqrt -1$. Having established the model it is more convenient to use the $x+i y$ notation rather than use the matrices.

Using a set of linear combinations of matrices exhibits all the algebraic structure of quaternions including three different matrices corresponding to the three different square roots of -1. Again, having established the model, it is more convenient to use the $a + {\bf i}x + {\bf j}y + {\bf k}z$ notation rather than to use the matrices.