Understanding the Representation of Irrational Numbers

Irrational numbers play a significant role in mathematics, much like their rational counterparts, but they have a different and often more complex representation. In this article, we explore the intricacies of writing and representing irrational numbers, focusing on concepts like algebraic, transcendental numbers, and the special golden ratio base.

Writing Irrational Numbers in Simplest Form

The square root is a common way to represent irrational numbers. Let's consider an example:

Write 8

First, factor the radicand into perfect squares: 4×2 Take the square root of each part: 4 × 2 Simplify the square roots of the perfect squares: 2 × 2

This answer is exact and in its simplest form. Now let's try another example: 15330615300

Factor the radicand into perfect squares: 4×9×25×49×121×169×17 Simplify the square roots of the perfect squares: 2 × 3 × 5 × 7 × 11 × 13 × 17

Approximate Representations

It's worth noting that sometimes we need to use approximate representations for irrational numbers, especially when dealing with calculations. For instance, the number pi; can be approximated as 22/7. However, this is an approximation and not the exact value.

Special Cases: e, pi;, and phi;

Some irrational numbers, like , are defined by specific sequences. The number phi;, known as the golden ratio, has an interesting representation:

(phi 0.1001000100001ldots)

Each “1” is followed by one more zero than the “1” to its left, forming a never-ending sequence.

Defining a Numeral System

At its core, any number we write is a symbolic representation of an abstract mathematical concept. Consider the number 3. When we write it down, we assume it represents a concept, not just the symbol itself. We use numeral systems, such as the decimal system, to represent numbers. However, many numbers cannot be represented finitely within these systems.

We denote certain irrational numbers using special symbols, like the square root symbol for 2, or the letter pi; for the mathematical constant pi; and e for the base of the natural logarithm.

Algebraic and Transcendental Numbers

Algebraic numbers can be represented exactly using algebraic expressions. For example:

(phi frac{1 - sqrt{5}}{2})

Numbers that can be expressed this way are algebraic numbers, which are roots of polynomials with integer coefficients. However, some numbers, like pi; and e, are transcendental and do not have such algebraic representations. These numbers are not roots of any non-zero polynomial equation with rational coefficients.

Positional Notation with Irrational Bases

Positional notation with irrational bases can also be used to write certain numbers simply. For instance, using the golden ratio base phi;, non-negative integers can be represented in a standard finite form. Here are a few examples:

[ begin{array}{l|l} 1 phi^0 1 2 phi^1 phi^{-2} 3 phi^2 phi^{-2} 4 100.01 end{array} ]

The golden ratio base is fascinating because every non-negative integer has a finite representation, which is unique and efficient.