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Irrational numbers are real numbers that can't be expressed as fractions of integers. This piece delves into examples like π, e, the golden ratio φ, and the square root of two. It explores their decimal representation, which is infinite and non-repeating, unlike rational numbers. The prevalence of irrational numbers in the real number system is also discussed, highlighting their uncountable nature and their significance in mathematics.

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## Definition of Irrational Numbers

### Fundamental category of real numbers

Irrational numbers cannot be written as a simple fraction of two integers

### "Gaps" between rational numbers on the number line

Irrational numbers fill in the "gaps" between rational numbers on the number line

### Incommensurable line segments

In geometry, incommensurable line segments have an irrational ratio and cannot be expressed using a common unit of measure

## Famous Examples of Irrational Numbers

### Pi (π)

Pi is the ratio of a circle's circumference to its diameter and is crucial in trigonometry and geometry

### Euler's number (e)

Euler's number is the base of the natural logarithm and has applications in calculus, complex analysis, and financial mathematics

### Golden ratio (φ)

The golden ratio is often found in art, architecture, and nature due to its aesthetically pleasing properties

## Decimal Representation of Irrational Numbers

### Infinite and non-repeating decimal expansion

Irrational numbers have an infinite and non-repeating decimal expansion, while rational numbers either terminate or have a repeating pattern

### Hallmark of irrational numbers

The non-repetitive and non-terminating nature is a hallmark of irrational numbers

### Proven through mathematical proofs

The characteristics of irrational numbers are rigorously established through mathematical proofs

## Expressions and Properties of Irrational Numbers

### Represented by non-terminating continued fractions

Irrational numbers can also be represented by non-terminating continued fractions, which may be periodic in some cases

### Diversity of mathematical expressions

There are numerous mathematical expressions for irrational numbers, showcasing the diversity of mathematical notation

### Unique properties and role within the set of real numbers

Studying irrational numbers involves exploring their unique properties and understanding their role within the set of real numbers

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