why are fractions not always rational?
Fractions are not always rational because a rational number is defined as any number that can be expressed as the quotient or ratio of two integers, where the denominator is not zero. However, some fractions cannot be expressed in this form.
For example, the fraction 1/3 cannot be expressed as a ratio of two integers. When representing 1/3 as a decimal, it would be an infinite repeating decimal of 0.33333... This decimal representation cannot be written as a ratio of two integers.
Additionally, fractions with non-repeating or non-terminating decimals, such as Pi (π), are considered irrational. These numbers cannot be expressed as a simple fraction because they go on infinitely without a repeating pattern.
Fractions are not always rational because rational numbers are defined as numbers that can be expressed as a ratio of two integers (where the denominator is not zero). This means that the decimal representation of a rational number either terminates (ends) or repeats indefinitely.
However, there are fractions that cannot be expressed as a ratio of two integers. These are called irrational numbers. Irrational numbers have decimal representations that neither terminate nor repeat. Examples of irrational numbers include √2 (the square root of 2) and π (pi). The decimal representation of these numbers goes on forever without any pattern.
So, fractions that have decimal representations that do not terminate or repeat are considered irrational and are not classified as rational numbers.