Vanishing Roots

Stage: 4 and 5
Article by Alison Kiddle

When you first learned about quadratic equations, you may have talked about equations with no real roots. No real roots...

Take a look at the video showing the graph of $y = x^2 - 6x +c$:



When $c>9$ the red dots disappear.
But what if those two red dots are still around somewhere?
Where could they be?
Is there a mathematical answer to that question that makes sense?

Let's think a little more about $y=x^2-6x+c$.
If we complete the square, we can write it as $y=(x-3)^2 - 9+c$.
The roots are the solutions to $(x-3)^2-9+c=0$,
which we can solve by rearranging to give $(x-3)^2=9-c$.

That's fine if $c \leq 9$, but if $c>9$ we would have to find the square root of a negative number, and we can't do that! Or can we?...


For a little more background on how our number system expands in order to solve a wider variety of equations, click below and read on...


Imagine a world where the only numbers are positive
integers: $1,2,3,...$

We call this set the natural numbers, $\mathbb{N}$.


Convince yourself that the following expressions are also positive integers, irrespective of the values of x and y (where x and y are natural numbers):

     $x+y$     

$xy$     

$x^2$     


This property is known as closure

Now consider the expression $x-y$. We no longer have closure because there are values that are not positive integers. How can we extend our world of numbers so that we have closure again?





In addition to the natural numbers we also need zero and negative integers.
We call this set the integers, $\mathbb{Z}$


Now consider the expression $\frac{x}{y}$ (where x and y are integers, and y is not 0).
We no longer have closure because there are values that are not integers. How can we extend our world of numbers so that we have closure again?







In addition to the integers we also need fractions.
We call this set the rational numbers, $\mathbb{Q}$.



Now consider the expression $\sqrt{x}$ when $x$ is a positive rational number. We no longer have closure because there are values that cannot be expressed as rational numbers. How can we extend our world of numbers so that we have closure again?








In addition to the rationals we also need irrational numbers.
Together, the rational and irrational numbers form the set of real numbers, $\mathbb{R}$.





This article includes a proof that $\sqrt{2}$ is irrational.
 



Now consider the expression $\sqrt{x}$ when $x$ is negative. We no longer have closure because there are values that are not real numbers. How can we extend our world of numbers so that we have closure again?

Let's include a new number, $i$, such that $i^2=-1$.
We call $i$ and its multiples imaginary numbers. 






In addition to the real numbers, we need imaginary numbers and combinations of real and imaginary Numbers.
We call this set the complex numbers, $\mathbb{C}$.


It can be shown that $\mathbb{C}$ is closed under addition, subtraction, multiplication, division (by non-zero numbers), and square-rooting.


 


Rather satisfyingly, the complex numbers not only give us two roots for all quadratic equations, but three roots for all cubics, four roots for all quartics and so on. (Note that these roots may not necessarily all be distinct - we can have repeated roots, where two or more roots are the same.) This is known as the Fundamental Theorem of Algebra.

 



To explore Complex Numbers, the mathematical idea that explains where the roots go, try Opening the Door.



Notes and Background
This article gives some of the theoretical perspective about why complex numbers are important mathematically, but you may be interested to know that complex numbers are also of huge practical importance and are used in medicine, engineering, physics, technology, computing and many other fields. Watch out for more NRICH resources on complex numbers and their applications, which will be published later in 2017 as part of a joint project between the University of Oxford and the University of Cambridge.