Using vocabulary terminating non terminating repeating non repeating explain why the value 3.14is rational while the value for pie is irrational
PI is a non-terminating quanitity, it has infinite number of digits.
3.14 has no other digits. It terminates at the hundreds digit.
To determine whether a value is rational or irrational, we need to understand the concepts of terminating, non-terminating, repeating, and non-repeating decimals.
A terminating decimal is a decimal number that ends after a certain number of decimal places. For example, the value 0.75 is a terminating decimal because it ends at the 2nd decimal place.
On the other hand, a non-terminating decimal is a decimal number that continues forever without repeating or terminating. An example of a non-terminating decimal is the value 0.333..., where the digit 3 repeats infinitely.
Now, let's discuss the value 3.14. This is a rational number because it can be expressed as a fraction of two integers. We can represent 3.14 as the fraction 314/100. In this fraction, both the numerator (314) and the denominator (100) are integers, making it rational. Furthermore, 314/100 can be simplified to 157/50, showing that it can be expressed without a decimal.
On the other hand, the value for pi (π) is an irrational number. Pi is a non-terminating, non-repeating decimal. Its decimal representation starts with 3.14159... and continues indefinitely without ever repeating or terminating. It cannot be expressed as a fraction or ratio of two integers. Therefore, pi is considered an irrational number.
In summary, the value 3.14 is rational because it can be expressed as a fraction, while the value for pi is irrational because it is a non-terminating, non-repeating decimal that cannot be expressed as a fraction.