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.
Irrational numbers can be added or subtracted.
Irrational numbers can be added or subtracted.
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.
The sum of an irrational and rational number can’t be predicted.
The sum of an irrational and rational number can’t be predicted.
Well, let me tell you a joke to explain this. Why did the rational number invite the irrational number to the party? Because they knew their sum would be unpredictable! So, in mathematical terms, the closure property shows that the sum of a rational and an irrational number can't be predicted, meaning it is irrational.
The sum or difference of a rational number and an irrational number is always an irrational number. Therefore, the closure property, which states that the sum or difference of two numbers in a given set is also in that set, proves that the sum of a rational and an irrational number is irrational.
The correct answer is: The sum or difference of a rational and irrational number is always an irrational number.
To understand why the sum of a rational and irrational number is irrational, we can use the Closure Property of addition. The Closure Property states that when you add or subtract two numbers from a specific set, the result will also be in the same set.
In this case, let's consider the set of rational numbers (numbers that can be expressed as a fraction of two integers). Now, suppose we have a rational number, let's call it "a", and an irrational number, let's call it "b".
According to the Closure Property, when we add these two numbers together, the result will also be a rational number, since "a" is rational and we know that the sum or difference of two rational numbers is always a rational number.
However, if we assume that the sum of a rational and irrational number is rational, we contradict the definition of an irrational number. By definition, an irrational number cannot be expressed as a fraction of two integers. Therefore, the sum of a rational and irrational number must be an irrational number.
In conclusion, the Closure Property of addition combined with the definition of an irrational number confirms that the sum of a rational and irrational number is always an irrational number.