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Graphically Understanding Complex Roots

If you have studied complex numbers then you’ll be familiar with the idea that many polynomials have complex roots. For example x2 + 1 = 0 has the solution x = i and -i.  We know that the solution to x2 – 1 = 0 ( x = 1 and -1) gives the two x values at which the graph crosses the x axis, but what does a solution of x = i or -i represent graphically?

There’s a great post on Maths Fun Facts which looks at basic idea behind this and I’ll expand on this in a little more detail.  This particular graphical method only works with quadratics:

Step 1

You have a quadratic graph with complex roots, say y = (x – 1)2 + 4. Written in this form we can see the minimum point of the graph is at (1,4) so it doesn’t cross the x axis.

Step 2

Reflect this graph downwards at the point of its vertex. We do this by transforming y = (x – 1)2 + 4 into y =  -(x – 1)2 + 4

Step 3

We find the roots of this new equation using the quadratic formula or by rearranging – leaving the plus or minus sign in.
-(x – 1)2 + 4 = 0
(x-1) = ± 2
x = 1 ± 2

Plot a circle with centre (1,0) and radius of 2.  This will touch both roots.

Step 4

We can now represent the complex roots of the initial equation by rotating the 2 real roots we’ve just found 90 degrees anti-clockwise, with the centre of rotation the centre of the circle.

The points B and C on the diagram are a representation of the complex roots (if we view the graph as representing the complex plane).  The complex roots of the initial equation are therefore given by x = 1 ± 2i.

General case

It’s relatively straightforward to show algebraically what is happening:

If we take the 2 general equations:

1) y = (x-a)2 + b
2) y = -(x-a)2 + b  (this is the reflection at the vertex of equation 1 )

(b > 0 ).  Then the first equation will always have complex roots.  The roots of both equations will be given by:

1) a ±i√b

2) a ±√b

So we can think of (2) as representing a circle of radius √b, centred at (a,0).  Therefore multiplying √b by i has the effect of rotating the point (√b, 0 ) 90 degrees anti-clockwise around the point (a,0).  Therefore the complex roots will be graphically represented by those points at the top and bottom of this circle.  (a, √b) and (a, -√b)

Graphically finding complex roots of a cubic

There is also a way of graphically calculating the complex roots of a cubic with 1 real and 2 complex roots.  This method is outlined with an algebraic explanation here

Step 1

We plot a cubic with 1 real and 2 complex roots, in this case y = x3 – 9x2 + 25x – 17.

Step 2

We find the line which goes through the real root (1,0) and which is also a tangent to the function.

Step 3

If the x co-ordinate of the tangent intersection with the cubic is a and the gradient of the tangent is m, then the complex roots are a ± (√m)i.  In this case the tangent x intersection is at 4 and the gradient of the tangent is 1, therefore the complex roots are 4 ± 1i.

[2nd last sentence edited – thanks for the comments below!]

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