Notes on using Complex.xls

This spreadsheet allows students to explore complex numbers in a geometrical way by looking at them as a way of multiplying points in a 2D plane together. It can be used as a first introduction to complex numbers.

Background: The idea of transformations of the plane especially rotation and enlargement. Polar coordinates would be useful but not essential.

Questions for students:

• What is the effect of multiplying by (1,0)?  
• What is the effect of multiplying by (2,0)?
• What is the effect of multiplying by (3,0)?  
• What is the effect of multiplying by (k,0)?  
• What is the effect of multiplying by (0,1)?  
• What is the effect of multiplying by (0,2)?  
• What is the effect of multiplying by (0,k)?  
• What is the effect of multiplying by (-1,0)?  
• What is the effect of multiplying by (-2,0)?  
• What is the effect of multiplying by (-k,0)?  
• What is the effect of multiplying by (0,-1)?  
• What is the effect of multiplying by (0,-k)?  
• What is the effect of repeated multiplication by (0,1)?  
• What is the effect of multiplying by (1,1)?  
• What is the effect of multiplying by (3,4)?  
• Can you generalise: what is the effect of multiplying by (x,y)?  
• If (a,b)×(a,b) = (-1,0) find the possible values of a and b  
• If we regard ordinary real numbers as points with zero y-coordinate then you have just found the square roots of -1 !  
• Find the square roots of (15,8).  
• Find a formula for (1,1)ⁿ.