# Algebraic Number

Most numbers we use every day are Algebraic Numbers

But some are **not**, such as π and **e**

## Algebraic Number

** Put simply**, when we have a polynomial equation like (for example)

### 2x^{2} + 4x − 7 = 0

whose coefficients (the numbers 2, 4 and −7) are rational numbers (whole numbers or simple fractions) ...

... then **x** is **Algebraic**.

We can imagine all kinds of polynomials:

**x − 1 = 0**has x =**1**,**x + 1 = 0**has x =**−****1**,**2x − 1 = 0**has x =**½**,**x**has x =^{2 }− 2 = 0**√2**,- and so on

In each case **x** is algebraic.

In fact all integers, all rational numbers, some irrational numbers (such as √2) are Algebraic.

### Testing Game

We can make a game of it!

Start with our number, and we have to get it to **zero **using only:

- whole numbers
- add, subtract, multiply (but not divide)
- whole number exponents

Here are some examples:

^{2}− 2 = 0

^{2}− 18 = 0

All those numbers are algebraic!

Try some yourself and see how you go.

Now try π (pi) and see if you have any success.

## More Formally

To be algebraic, a number must be a root of a non-zero polynomial equation with rational coefficients.

**x**is algebraic in this example:

### 2x^{3} − 5x + 39 = 0

Because all conditions are met:

- 2x
^{3}− 5x + 39 is a non-zero polynomial (a polynomial which is not just "0") **x**is a root (i.e.**x**gives the result of**zero**for the function 2x^{3}− 5x + 39)- the coefficients (the numbers 2, −5 and 39) are rational numbers

So we know **x** is algebraic

But let's see its value anyway:

### Example: 2x^{3} − 5x + 39 = 0

We need to find the value of **x** where **2x ^{3} − 5x + 39 is equal to 0**

Well **x = −3** works, because 2(−3)^{3} − 5(−3) + 39 = −54+15+39 = 0

Let's try another polynomial (remember: the coefficients must be rational).

### Example: 2x^{3} − ¼ = 0

The coefficients are 2 and −¼, both rational numbers. So **x is an Algebraic Number**

We can also discover that **x = 0.5**, because 2(0.5)^{3} − ¼ = 0

In fact:

All integers and rational numbers are algebraic,

but an irrational number **may or may not** be algebraic

## Not Algebraic? Then Transcendental!

When a number is not algebraic, it is called transcendental.

Back in 1844 Joseph Liouville created a number:

0.110001000000000000000001000000...

(it has a 1 in every factorial numbered position such as 1, 2, 6, 24, etc)

And he showed it is **not algebraic** ... he had broken mathematics!

Just kidding. But it did *transcend* algebraic numbers and was

the first known transcendental number

Then in 1873 Charles Hermite proved that * e* (Euler's number) is transcendental, and in 1882 Ferdinand von Lindemann proved that π (pi) is transcendental.

It is actually hard to prove that a number is transcendental.

## More

Let's investigate a few more numbers

### Example: the unit imaginary number **i**

**i**

Well, we know that **i**^{2} = −1, so * i* is the solution to:

**x ^{2} + 1 = 0**

*i*is an Algebraic NumberNote: using the "testing game": **i**^{2} + 1 = 0

### Example: x^{2} + 2x + 10 = 0

The solutions to this quadratic equation are complex numbers :

- x = −1 + 3
**i** - x = −1 − 3
**i**

(Try putting them into the equation, and remember that *i*^{2} = −1)

They are both **Algebraic Numbers**

## Which is More Common?

It might seem that transcendental numbers are rare, but:

**almost all** real and complex numbers are **transcendental.**

Why? Well, imagine some random real number where **each digit is randomly chosen**, and you get something like 7.17493614485672... (on for infinity). It is almost certain to be transcendental.

But in every day life we use carefully chosen numbers like 6 or 3.5 or 0.001, so most numbers we deal with (except π and * e*) are algebraic, but any truly randomly chosen real or complex number is almost certain to be transcendental.

## Properties

All algebraic numbers are computable and so they are definable.

The set of algebraic numbers is **countable**. Put simply, the list of whole numbers is "countable", and you can arrange the algebraic numbers in a 1-to-1 manner with whole numbers, so they are also countable.