What is the difference between non terminating and non recurring decimal and irrational number?
Answers
Answer 1--These are non-terminating numbers that are not repeating, meaning the decimal has no pattern and constantly changes. When a decimal is non-terminating and constantly changes, it cannot be expressed as a fraction.
Answer 2--A rational number can be expressed as a ratio of two other whole numbers. The word “ratio” is where the term “rational” comes from here. A ratio can be thought of as a fraction.
So 0.33333… is 1/3. 2.6666… is 8/3. I can express them exactly as a fraction. Even more complex repeating decimal expansions still repeat some sequence. My two examples repeat the same digit, but some repeating decimal expansions will repeat a sequence of digits.
But a number like π which is 3.14159256…. cannot be expressed accurately as a ratio. It is irrational, i.e. not a ratio. There are approximations, but those are not exact just as the decimal expansion I have here is not exact. There is no repetition of a sequence of digits in the decimal expansion of π or any other irrational number.
Decimal representation of a number and fractional representations are different ways of expressing the same number. As different ways, they have pluses and minues (whoo-hoo! math pun!). Fractional representation can be cumbersome, and some things in decimal representation are repeating.
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