Math

QuestionWhich decimals are rational numbers: terminate, repeat, or both?

Studdy Solution

STEP 1

Assumptions1. A rational number is a number that can be expressed as a fraction where both the numerator and the denominator are integers and the denominator is not zero. . A decimal is a way of representing a number in terms of fractions of ten.

STEP 2

First, let's consider the first option "The decimal repeats and does not terminate."A repeating decimal is a decimal fraction that eventually becomes periodic (repeating its digits in the same order) and does not end. For example, the fraction1/ =0.333... is a repeating decimal.This is a characteristic of rational numbers. When a fraction is expressed as a decimal, the decimal either terminates (ends) or repeats. So, the first option is correct.

STEP 3

Now, let's consider the second option "The decimal terminates and does not repeat."
A terminating decimal is a decimal number that has a finite number of digits. For example, the fraction1/2 =0.5 is a terminating decimal.This is also a characteristic of rational numbers. When a fraction is expressed as a decimal, the decimal either terminates (ends) or repeats. So, the second option is also correct.

STEP 4

Next, let's consider the third option "The decimal neither terminates nor repeats."
A decimal that neither terminates nor repeats is an irrational number. For example, the number Pi (approximately3.14159...) is a decimal that neither terminates nor repeats.This is not a characteristic of rational numbers. Therefore, the third option is incorrect.

STEP 5

Finally, let's consider the fourth option "The decimal terminates or repeats."
As we have already established, a rational number, when expressed as a decimal, either terminates or repeats. Therefore, the fourth option is correct.
So, the correct answer is "The decimal terminates or repeats."

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