As a mathematical expert, I am well-versed in the intricacies of numbers, especially when it comes to the fascinating realm of irrational numbers. Let's delve into the concept of irrational numbers and their relationship with fractions.

Irrational numbers are real numbers that cannot be expressed as a ratio of two integers, meaning they cannot be written as a fraction in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b \neq 0 \). The defining characteristic of an irrational number is its non-repeating, non-terminating decimal expansion. When we talk about non-repeating, we mean that the digits after the decimal point do not form a repeating pattern indefinitely. This is in contrast to rational numbers, which can be either terminating decimals or repeating decimals.

For instance, consider the number \( \pi \) (pi), which is the ratio of a circle's circumference to its diameter. Pi is an irrational number, and its decimal expansion goes on forever without repeating. Another well-known example is the square root of any non-perfect square, such as \( \sqrt{2} \). The decimal expansion of \( \sqrt{2} \) also does not repeat and continues indefinitely.

Now, let's address the misconception that irrational numbers can be written as fractions. It's important to clarify that this is not possible. The very definition of an irrational number precludes it from being expressed as a simple fraction. If an irrational number could be written as a fraction, it would be rational by definition, as rational numbers are precisely those that can be expressed as a ratio of two integers.

The statement that an irrational number "cannot be written as a simple fraction because there is not a finite number of numbers when written as a decimal" is accurate. However, it's crucial to understand that this is not due to the number of digits in the decimal expansion, but rather the nature of the expansion itself. The decimal expansion of an irrational number is infinite and non-repeating, which means no finite sequence of digits can capture its entirety.

It's also worth noting that while we cannot express irrational numbers as fractions, we can approximate them to any desired degree of accuracy using rational numbers. For practical purposes, we often use rational approximations of irrational numbers, such as \( \pi \approx 3.14159 \) or \( \sqrt{2} \approx 1.41421 \). These approximations are useful for calculations where exact values are not necessary or where the precision required does not demand the full complexity of the irrational number.

In conclusion, irrational numbers are a fundamental part of the real number system, and their non-repeating, non-terminating decimal expansions set them apart from rational numbers. While we cannot express irrational numbers as fractions, we can approximate them using rational numbers to a high degree of accuracy for most practical applications.

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An irrational number cannot be expressed as a ratio between two numbers and it cannot be written as a simple fraction because there is not a finite number of numbers when written as a decimal. Instead, the numbers in the decimal would go on forever, without repeating.read more >>