Can you explain the answer of this question below:A number is irrational if and only if its decimal representation is:A:TerminatingB:non- terminating and repeatingC:non- terminatingD:non- terminating and non-repeatingThe answer is d.

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The number is irrational if it's decimal representation is non terminating and non repeating.

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A number is irrational if its decimal representation is non terminating and non repeating.......eg 1.23478917995815484748576544....

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Definition of an Irrational Number:
An irrational number is a real number that cannot be expressed as a simple fraction or ratio of two integers. It cannot be written as a terminating or repeating decimal.

Understanding the Answer:
The correct answer is option 'D' - a number is irrational if and only if its decimal representation is non-terminating and non-repeating. Let's understand this in detail:

Terminating Decimals:
A terminating decimal is a decimal that ends after a finite number of digits. For example, 0.25 is a terminating decimal because it ends after two digits. Terminating decimals can always be written as fractions or ratios of integers. For instance, 0.25 can be written as 25/100 or 1/4. Therefore, terminating decimals are rational numbers, not irrational.

Repeating Decimals:
A repeating decimal is a decimal that has a repeating pattern of digits. For example, 0.333... is a repeating decimal because the digit 3 repeats indefinitely. Repeating decimals can also be written as fractions. For instance, 0.333... is equivalent to 1/3. Therefore, repeating decimals are rational numbers, not irrational.

Non-terminating and Non-repeating Decimals:
An irrational number is characterized by having a decimal representation that neither terminates nor repeats. These decimals continue indefinitely without any pattern. Some examples of irrational numbers are √2, π (pi), and e (Euler's number). It is impossible to write these numbers as fractions or ratios of integers. Their decimal representation goes on forever without any repeating pattern.

Conclusion:
To summarize, a number is considered irrational if and only if its decimal representation is non-terminating and non-repeating. This means that the digits after the decimal point continue indefinitely without any pattern. Irrational numbers cannot be expressed as fractions or ratios of integers, and they are an essential part of the real number system.

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Because irrational number can't be terminate and repeat

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Non terminating non repeating because non terminating and non repeating numbers are counted in irrational

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A non-terminating, non-repeating decimal is a decimal number that continues endlessly, with no group of digits repeating endlessly. Decimals of this type cannot be represented as fractions, and as a result, are irrational numbers. Pi is a non-terminating, non-repeating decimal.

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