A: a number that can be written as a fraction but not as a decimal
B: a negative number
C: a number that can be expressed as a fraction p/q where p and q are integers and q is not equal to zero
D: a number that cannot be expressed as a fraction p/q where p and q are integers and q is not equal to zero
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The correct definition of an irrational number is:
D: a number that cannot be expressed as a fraction p/q where p and q are integers and q is not equal to zero.
Irrational numbers cannot be represented as simple fractions and their decimal expansions are non-repeating and non-terminating. Examples include numbers like \( \sqrt{2} \) and \( \pi \).
The correct description of the number \( \sqrt{2} \) is:
B: irrational.
The square root of 2 cannot be expressed as a fraction \( p/q \) where \( p \) and \( q \) are integers, making it an irrational number.
The number that is irrational among the options given is:
D: square root of 18.
The square root of 18 can be simplified to \( 3\sqrt{2} \), which is not a whole number and cannot be expressed as a fraction of integers, making it irrational.
Among the options given, the number that is most likely to be irrational is:
D: -5.690134489.
This number appears to be a decimal that does not terminate or repeat visibly. However, without specific context about its decimal representation (whether it continues indefinitely or has a repeating pattern), it could still theoretically be rational if it is a finite decimal.
In contrast:
Thus, D is the best candidate for being irrational, but please note that without additional context about the nature of the decimal, it's based on its presentation rather than a definitive classification.
The correct statement that describes the number \( \sqrt{22} \approx 4.690415 \) is:
D: The number is irrational because it can be written as a non-repeating, non-terminating decimal.
The square root of 22 is an irrational number because it cannot be expressed as a fraction of two integers. Its decimal form is non-repeating and non-terminating, which is characteristic of irrational numbers.
The correct answers for which numbers are irrational are:
B: \( \frac{3}{11} = 0.\overline{27} \) - This is a repeating decimal, and hence it is rational, not irrational.
C: \( \frac{2\pi}{4} = 1.570796… \) - This is \( \frac{\pi}{2} \), which is irrational because \( \pi \) is an irrational number and any multiple of an irrational number is also irrational.
E: \( \sqrt{8} = 2.82842712 \) - This is an irrational number. It cannot be expressed as a fraction of two integers.
A: -6.398572639 and F: -\frac{15}{16} = -0.9375 are rational numbers because they can be expressed as fractions (the latter is a terminating decimal).
D: \( \sqrt{16} = 4 \) is also a rational number since it is an integer.
So, the correct selections are:
To determine which of the given options are irrational numbers, let's analyze each one:
A: ( \sqrt{6}/3 \approx 0.81649658…
B: (-15/16 = -0.8333\overline{3}
C: (-3.857634857
D: \( \sqrt{36}/14 = 6/14 = 3/7 \) (This part seems incorrectly interpreted as a decimal conversion; however, if we consider \( \sqrt{36}\) directly, it is rational.)
E: \( \frac{8}{2\pi} = 4/\pi \approx 1.273239 \) (as a value) is irrational.
F: (-18.493847583
The only options that are irrational from the list you provided are:
So the correct selections are A and E.
The number that is a rational number among the options given is:
B: square root of 121.
This simplifies to \(11\), which is a whole number and can be expressed as a fraction \(\frac{11}{1}\).
The other options are:
So, the only rational number in the list is B.
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