Difference between revisions of "Non-terminating recurring decimal expansions"
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== Non-Terminating Decimal Definition == | == Non-Terminating Decimal Definition == | ||
Non-terminating decimals are defined as those decimal numbers that do not have an endpoint in their decimal digits and keep continuing forever. This happens when a dividend is divided by a divisor but the remainder is never <math>0</math> and hence the process keeps repeating and the non-terminating decimal is obtained in the quotient where the decimal digits keep occurring and never come to an end. A non-terminating decimal has an infinite number of decimal places and it is named as non-terminating because the decimal will never terminate. For example, <math>1.333333....</math>,<math>4.65675747775 | Non-terminating decimals are defined as those decimal numbers that do not have an endpoint in their decimal digits and keep continuing forever. This happens when a dividend is divided by a divisor but the remainder is never <math>0</math> and hence the process keeps repeating and the non-terminating decimal is obtained in the quotient where the decimal digits keep occurring and never come to an end. A non-terminating decimal has an infinite number of decimal places and it is named as non-terminating because the decimal will never terminate. For example, <math>1.333333....</math>,<math>4.65675747775....</math> etc. | ||
== Types Non-Terminating Decimal Expansion == | == Types Non-Terminating Decimal Expansion == | ||
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=== Non-Terminating Recurring Decimal Expansion === | === Non-Terminating Recurring Decimal Expansion === | ||
Non-terminating recurring decimal is also known by the name non terminating repeating decimal. In this expansion, the decimal places will continue forever | Non-terminating recurring decimal is also known by the name non terminating repeating decimal. In this expansion, the decimal places will continue forever with the repetition of the decimal values in a specific pattern. | ||
For example, <math>\frac{2}{9}=0.2222222 | For example, <math>\frac{2}{9}=0.2222222....</math> is a non-terminating recurring decimal expansion. The repetition of the decimal can also be indicated by showing a bar on top of the numbers that are repeating i.e., <math>0.2222222....</math> can also be represented as <math>0.\bar{2}</math>. Similarly, <math>\frac{1}{7}=0.142857142857....</math> which can also be written as <math>0.\bar{1}\bar{4}\bar{2}\bar{8}\bar{5}\bar{7}</math> is also a non-terminating repeating decimal expansion as the block of decimals 142857 is repeating after every 6 digits. Non-terminating recurring decimals can always be converted to a rational number. | ||
=== Non-Terminating Non-Recurring Decimal Expansion === | === Non-Terminating Non-Recurring Decimal Expansion === | ||
Non-terminating non-recurring decimal is also known by the name non terminating non-repeating decimal as the values after decimal do not repeat or terminate. For example, <math>1.4142135...</math> , <math>2.35638745...</math> Unlike non-terminating recurring decimal, the decimal places do not form any pattern. A non-terminating non-recurring decimal cannot be converted to a rational number. Hence, non terminating non-recurring decimals are also known as irrational numbers. Note that pi (<math>\pi</math>) is an irrational number as its expansion is non-terminating non-recurring i.e., <math>3.1415926535 897...</math> | Non-terminating non-recurring decimal is also known by the name non terminating non-repeating decimal as the values after decimal do not repeat or terminate. For example, <math>1.4142135....</math> , <math>2.35638745....</math> Unlike non-terminating recurring decimal, the decimal places do not form any pattern. A non-terminating non-recurring decimal cannot be converted to a rational number. Hence, non terminating non-recurring decimals are also known as irrational numbers. Note that pi (<math>\pi</math>) is an irrational number as its expansion is non-terminating non-recurring i.e., <math>3.1415926535 897....</math> |
Latest revision as of 10:29, 4 May 2024
Non-terminating decimals are decimals that have never-ending decimal digits and continue forever.
Non-Terminating Decimal Definition
Non-terminating decimals are defined as those decimal numbers that do not have an endpoint in their decimal digits and keep continuing forever. This happens when a dividend is divided by a divisor but the remainder is never and hence the process keeps repeating and the non-terminating decimal is obtained in the quotient where the decimal digits keep occurring and never come to an end. A non-terminating decimal has an infinite number of decimal places and it is named as non-terminating because the decimal will never terminate. For example, , etc.
Types Non-Terminating Decimal Expansion
A non-terminating decimal expansion has an infinite number of places and its expansion continues forever. We have two types of non-terminating decimal expansions and they are as follows:
- Non terminating recurring decimal expansion
- Non terminating non-recurring decimal expansion
Non-Terminating Recurring Decimal Expansion
Non-terminating recurring decimal is also known by the name non terminating repeating decimal. In this expansion, the decimal places will continue forever with the repetition of the decimal values in a specific pattern.
For example, is a non-terminating recurring decimal expansion. The repetition of the decimal can also be indicated by showing a bar on top of the numbers that are repeating i.e., can also be represented as . Similarly, which can also be written as is also a non-terminating repeating decimal expansion as the block of decimals 142857 is repeating after every 6 digits. Non-terminating recurring decimals can always be converted to a rational number.
Non-Terminating Non-Recurring Decimal Expansion
Non-terminating non-recurring decimal is also known by the name non terminating non-repeating decimal as the values after decimal do not repeat or terminate. For example, , Unlike non-terminating recurring decimal, the decimal places do not form any pattern. A non-terminating non-recurring decimal cannot be converted to a rational number. Hence, non terminating non-recurring decimals are also known as irrational numbers. Note that pi () is an irrational number as its expansion is non-terminating non-recurring i.e.,