To determine which of the numbers listed are considered irrational, we need to know the definitions of rational and irrational numbers:
- Rational numbers are numbers that can be expressed as the quotient of two integers (i.e., a fraction a/b where a and b are integers, and b ≠ 0). Rational numbers can either terminate (like 0.5) or repeat (like 0.333...).
- Irrational numbers are numbers that cannot be expressed as a simple fraction, meaning they cannot be written as a ratio of two integers. They have non-repeating, non-terminating decimal expansions.
Now, let's analyze each of the numbers you've listed:
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2.77777…: This number is a repeating decimal (2.7 with the 7 repeating), which can be expressed as a fraction. Therefore, it is a rational number.
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-7/3: This is a fraction where both the numerator and denominator are integers. It can be written as -2.333..., which is also a repeating decimal. Hence, this is a rational number.
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2.7182818459…: This is a non-repeating, non-terminating decimal. It is an approximation of the mathematical constant \( e \), which is known to be irrational. Therefore, this is an irrational number.
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3: This is an integer and can also be expressed as the fraction 3/1. Therefore, it is a rational number.
In summary:
- Irrational number: 2.7182818459…
- Rational numbers: 2.77777…, -7/3, 3