More Types of Numbers

Transcendental Numbers:

The next major update after irrational numbers in terms of how we classfiy things on the numberline came in 1844. Joseph Liouville proved the existence of transcendental numbers. Transcendental numbers are special irrational numbers. Their primary quality is that they cannot be a solution to a simple equation.

Imagine the number line. All the way from negative infinity to positive infinity. When we write an equation or a polynomial  $x + 2 = 5$, it has the solution 3, which we can mark on the number line. There are infinite many equations that we can make and each of their solution is added to the number line. Liouville proved that if we take all such equations and used them to populate the number line, there will always be holes left.

Some numbers simply cannot be created using the mathematical operations. These numbers are also infinitely many and they have quite a few celebrities in them. The two most famous transcendental numbers are $\pi$ and $e$. The case is simple, you are given the five operations, addition, subtraction, multiplication, division and any root like square root. Given these operations, you can take any one natural number and create all the integers and rational numbers, and you can even create many irrational numbers. For instance, look at this. Let us go from 17 to $\sqrt{3}$.

Creating one number from another

• $17 + 17 = 34$

• $\dfrac{34}{17} = 2$

• $\dfrac{2}{2} = 1$

• $1+2 = 3$

• $3 = \sqrt{3} \times \sqrt{3}$

We can perform these operations to create infinitely many numbers, even some irrational numbers. But Liouville proved that this is not true for all irrational numbers. $\pi$ is such a number. No matter how you perform your operations onto which numbers, a discrete number or even other irrational numbers like $\sqrt{2}$ cannot be turned into $\pi$. This proof is out of this course in scope, but I will include it in a later update.

Again, transcendental numbers are such numbers that cannot be the roots of any polynomial with rational co-efficients. Also, any resulting numbers that come out of a transcendental number through any operation involving other rational numbers is also a transcendental number.$\pi$, $\pi^2$, $\pi + 1$, $3\pi$ are all transcendental.

Algebraic Numbers

By contrast, a real number that is not transcendental is algebraic. $0 \text{, } 1 \text{, } \dfrac{3}{17} \text{ and } \sqrt{2}$ are all algebraic numbers.

Types of Decimals

Terminating Decimals

A terminating decimal is a decimal number that ends after a finite number of digits. These are always rational numbers. For example, $0.5 = \dfrac{1}{2}$ and $0.125 = \dfrac{1}{8}$. In general, a fraction will have a terminating decimal if its denominator (in lowest terms) only contains the prime factors $2$ and $5$.

Non Terminating Decimals

Non-terminating decimals are decimals that continue infinitely without ending. They fall into two categories: recurring and non-recurring. For example, $\dfrac{1}{3} = 0.333\ldots$ is non-terminating.

Recurring Decimals

A recurring (or repeating) decimal is a non-terminating decimal where a digit or group of digits repeats forever. These are also rational numbers. For instance, $\dfrac{2}{11} = 0.\overline{18}$ and $\dfrac{1}{3} = 0.\overline{3}$. Every rational number either terminates or recurs.

Non-Recurring Non Terminating Decimals

These are decimals that neither terminate nor repeat, continuing forever without any pattern. All irrational numbers fall into this category. For example,$\sqrt{2}$, $\pi$, and $e$ all have non-recurring, non-terminating decimal expansions. Such numbers cannot be written as a ratio of two integers. We can only approximate their values.

• $\sqrt{2} = 1.41421356237 \ldots$

• $\pi = 3.141592653 \ldots$

• $e = 2.7182818284 \ldots$