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Algebraic numbers

A number that is a root of some polynomial with integer coefficients is called an algebraic number. What does that mean in practice? We can take any number and play a mathematical game with it. The aim of the game is to take a number x and reduce it to zero, but we may only use addition, subtraction, multiplication and powers, and we may only use integers in our manipulations. The examples will make it clearer.

Let’s start with the number 42. How could we change it to get zero? We can subtract the integer 42 from it and get zero (multiplying by zero is not allowed):

42−42=0

Since we managed to reduce 42 to zero, 42 is an algebraic number. That was fairly trivial. What about a fraction, say 73? Since we can only work with integers, we cannot subtract 73 from this fraction to get zero. We have to do something else. First we multiply the fraction by three:

3⋅73=3⋅73=7

We have got an integer, and we already know how to turn that into zero: subtract seven. So we can write

3⋅73−7=0

and say that the fraction seven thirds is also an algebraic number. Let’s take an even more complicated number, the square root of two: 2. No multiplication or addition will help us here, so we first have to get rid of the root. We square the expression:

(2)2=2

We have got an integer, which we can turn into zero by subtracting two:

(2)2−2=0

So even the square root of two, which is an irrational number, is an algebraic number. So far it looks as if every number we try is algebraic. What about Euler’s number e? That is also irrational, so it has an infinite non-recurring decimal expansion, like the square root of two. Its approximate value is 2.71. Unlike the square root of two, however, there is no way to reduce it to zero using the operations we have allowed ourselves. Whatever we add to Euler’s number, multiply it by or raise it to, we always get yet another irrational number. So Euler’s number is not an algebraic number, hooray!

Similarly, pi (3.14) is not an algebraic number.

Numbers that are not algebraic are called transcendental.

A more precise definition

So for every algebraic number there must exist a polynomial with integer coefficients that has this algebraic number as a root. A root of a polynomial is a number for which the polynomial equals zero. Going back to the examples above, for the number 42 we would get the equation

x−42=0

and since the root of the polynomial x − 42 is exactly x = 42, this means that 42 is an algebraic number. For the rational number 73 it would be the equation

3x−7=0

because the root of the polynomial 3x − 7 is exactly x=73. And finally, for the square root of two we would get

x2−2=0

because the root of the polynomial x2 − 2 is exactly the square root of two. For Euler’s number or for pi we cannot build any such equation or polynomial, because they are transcendental numbers and not algebraic numbers.

Examples of algebraic numbers

  • Every integer is also algebraic.
  • Every rational number (fraction) is algebraic.
  • Irrational numbers are more complicated. As we saw, some irrational numbers are algebraic, such as the square root of two, but some are not, such as Euler’s number.
  • Every nth root of an integer is algebraic.
  • The golden ratio is an algebraic number.