How does the Closure Property prove that the sum of a rational and irrational number is irrational?
The Closure Property is a fundamental property of real numbers, meaning that if you add two real numbers, the sum will also be a real number.
To prove that the sum of a rational and irrational number is irrational, we can use a proof by contradiction.
Let's assume that the sum of a rational number (p/q, where p and q are integers) and an irrational number (r) is rational.
So, p/q + r = a/b, where a and b are integers and b is not equal to zero.
Rearranging this equation, we get r = (a/b) - (p/q).
Since a/b and p/q are rational numbers, their difference (a/b) - (p/q) is also a rational number.
However, this contradicts the assumption that r is an irrational number.
Therefore, our assumption that the sum of a rational and an irrational number is rational must be false.
Hence, we can conclude that the sum of a rational and an irrational number is always irrational.
can i ask another question
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How does the Closure Property prove that the sum of a rational and irrational number is irrational?(1 point)
Responses
The sum or difference of two rational numbers is a rational number.
The sum or difference of two rational numbers is a rational number.
The sum or difference of a rational and irrational number is a rational number.
The sum or difference of a rational and irrational number is a rational number.
Irrational numbers can be added or subtracted.
Irrational numbers can be added or subtracted.
The sum of an irrational and rational number can’t be predicted.
The sum of an irrational and rational number can’t be predicted.
The correct response is:
The sum or difference of a rational and irrational number is a rational number.