Nov 7, 20195 min read

The Fundamental Theorem of Algebra

Updated: Dec 15, 2019

Many of you reading this have used algebra at some point in your lives. We know it as the core component of maths, and we use it daily, whether to do maths homework or figure out how much money is needed for groceries. Letters are used to represent unknowns, and we solve for those letters and call it algebra.

Finding out that most of maths uses letters rather than numbers must have been shocking enough when you were 8 years old – you’d rather have spent your free time drawing pretty pictures than figuring out why the letter x was showing up in a maths class. But what’s even more confusing is that the most important discovery relating to this vital tool was probably never even mentioned in your classroom. This discovery is known as the Fundamental Theorem of Algebra (FTA).

Considering how ‘fundamental’ this theorem is, why has it never been mentioned, let alone taught, in most basic algebra courses? This is mostly because it involves complex numbers, which is a difficult concept to understand.

For those who don’t know, complex numbers have two parts; a real number, such as 3, -5, 3.5, or π, and an imaginary number, which is a real number multiplied by the constant i. This constant is equal to the square root of -1, which you were told doesn’t exist for the majority of your schooling. We commonly use the letter z to represent a complex number, in the same way we use x to represent a real number.

There are two important consequences of defining a complex number like this. The first is that, just like a real number, complex numbers can be added, subtracted, multiplied and divided. The second is that complex numbers can also be represented on a complex (or Argand) plane. The complex plane is the extension to the number line you learnt when you were little. The vertical line contains all the pure imaginary numbers (i,2i,3i etc) and the horizontal line contains all real numbers (1,2,3 etc). This is definitely not a full explanation of complex numbers, and further reading on this topic is both recommended and listed at the end.

The FTA says that any polynomial with complex coefficients has at least one root. Let’s break this down. A polynomial is a function which takes in an input, z, and spits out an output, f(z). A polynomial must be in the form c(z^n) + c(z^n-1) + … + cz + c0, where c can be any complex number coefficient and n is an integer. z^2, 3z^3 + 5z^2 -z+6, and (4+i)z^4 + (3-4i)z^3 + (-9-9i)z + (4+3i) are all polynomials. A “root”, or solution, is a number which, when plugged into the formula, gives an output of 0. The FTA guarantees that, if our formula is a polynomial, there will always be at least one number which is a root.

Why is this true? Well, let’s imagine that we have an arbitrary polynomial (c(z^n) + c(z^n-1) + … + cz + c0). The coefficients and the largest power can be anything, but we want to show that, regardless of what they are, we can always find a root. How do we do this?

Well, first we realise that if we input 0, our output will always be c0. This is because if our input z is 0, any term in the output with a z in it will become 0, as multiplying by 0 produces 0. The only term in our arbitrary polynomial without a z in it is c0, so if z = 0, f(z) must equal c0. The point c0 has been represented as a specific point in the animation below, but in reality, it could be any point on the plane.

Secondly, we realise that if our input number is really far away from 0, our output will be even further away from 0. This is because, in our polynomial, the largest power is dominant. This means that when your input gets really big, the largest power has a huge effect on the output, whereas the smaller powers barely have any effect at all. This makes sense – 100^5 is 100 times bigger than 100^4, so it makes sense that it should have 100 times the effect. This means a number which is far away from 0 gets even further away in the same direction. In technical terms, we say that when the modulus (read: distance from zero) of the number is much greater than 0, the largest power is dominant. This is written as |z|>> 0, f(z) ≈ z^n, where n is the largest power. All of the numbers which are 100 units away from 0 form a circle in the complex plane, and since the outputs are roughly equal to the inputs raised to their largest power, they also form a much larger circle, with at least part of each circle in all 4 corners of the plane. This won’t be a perfect circle, and it won’t be centered exactly at the origin due to the lower powers affecting the output, but they will look very similar. The output axes in the animation have been scaled up to allow the output circle to fit on the screen.

Now that we have these two key bits of knowledge, the final step is to shrink our inputs. As we shrink the input circle closer and closer to 0, the output circle shrinks closer and closer to c0. In order for the output circle to converge onto c0, it MUST hit 0 at some point. This means there MUST be some input which causes the output to hit 0, which is what we set out to prove all along. But we can do better than this; in fact, if the biggest power is n, we can prove that there are exactly n solutions. Let’s say we know what point on the input circle gives us an output of 0. As we move our input point along the input circle, the corresponding output point on the output circle moves n times as fast. This is because of the what happens to the angles of complex numbers when we raise them to a power, based on De Moivre's Theorem. This isn’t a fully rigorous proof mathematically, and further reading on the topic is given below.

Now, as interesting as that may have been to read, watch, and understand, you may think: why do I care that some function always has a root? Why is it useful at all? The Fundamental Theorem of Algebra does have some uses to scientists and mathematicians. Finding roots of equations is very useful, as it allows us to predict things such as when planets come back to the start of their orbit, when fuel in an aircraft will run out, or when the blast radius of a nuclear warhead will reach the edge of the continent. If we can express these events using polynomials, the FTA tells us, for absolute certainty, that there is a time when all these events will happen, which allows us to plan for them. From a pure maths point of view, the FTA has a more interesting application; it intrinsically links algebra to geometry. We have just shown that the solutions for certain equations (algebra) can always be represented as a point or a set of points in space (geometry). This means that, if we’re smart enough, we may never have to do algebra again; all we may need to do is draw a bunch of pretty pictures.

Isn’t that what your eight-year-old-self wanted all along?

The Fundamental Theorem of Algebra

Further Reading

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