Terminating decimals are decimals that have a finite number of digits after the decimal point, meaning they end or terminate. For example, 0.25, 0.50, and 0.75 are all terminating decimals because they have a specific end point.
On the other hand, repeating decimals are decimals in which a pattern of digits repeats indefinitely after the decimal point. This pattern is typically denoted by placing a bar above the repeating portion of the digits. For example, 0.333... and 0.1212... are repeating decimals.
The end behavior of a decimal can determine whether it is a rational or irrational number. A rational number is a number that can be expressed as the ratio of two integers, while an irrational number cannot.
Terminating decimals, given their finite nature, can always be expressed as a fraction with an integer numerator and denominator. For example, 0.25 can be expressed as 25/100, and 0.5 can be expressed as 1/2. Therefore, all terminating decimals are rational numbers.
Repeating decimals also represent rational numbers. The repeating portion can be represented as a fraction by setting up an equation using algebra. For example, 0.333... can be denoted as x, so multiplying both sides by 10 gives 10x = 3.333..., and subtracting the original equation gives 10x - x = 3.333... - 0.333..., which simplifies to 9x = 3. Solving for x gives x = 1/3, so 0.333... is equivalent to 1/3. Therefore, repeating decimals are also rational numbers.
In contrast, irrational numbers do not have repeating or terminating decimals. Their decimal representation goes on indefinitely without any pattern or repetition. Examples of irrational numbers include pi (3.14159...) and the square root of 2 (1.41421...).
What is the difference between terminating and repeating decimals? How does the end behavior of a decimal relate to rational number or irrational number?
3 answers
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Terminating decimals have a definite end, while repeating decimals have a pattern after the decimal point that repeats indefinitely.
The end behavior of a decimal determines whether it is a rational or irrational number. Terminating decimals always represent rational numbers because they can be expressed as fractions. Repeating decimals also represent rational numbers because the repeating portion can be converted into a fraction. Irrational numbers have decimals that go on forever without repeating.
The end behavior of a decimal determines whether it is a rational or irrational number. Terminating decimals always represent rational numbers because they can be expressed as fractions. Repeating decimals also represent rational numbers because the repeating portion can be converted into a fraction. Irrational numbers have decimals that go on forever without repeating.