Question
"Ryiaz says it is impossible to prove that multiplying a non-zero rational number by an irrational number always results in an irrational product. How can you show that a proof is possible?"
The multiple-choice answers provided are:
Use proof by contradiction: assume the opposite is true and the product of a non-zero rational number and an irrational number is always rational. This is a contradiction. According to the Closure Property, if the product is rational, both factors must be rational. So if one factor is irrational, the product must be irrational.
You can prove it by choosing random numbers to plug into multiplication problems until he's convinced.
He's correct; it is impossible to prove that the product of a rational number and an irrational number is always irrational.
You can prove it directly by creating a proof that shows directly how multiplying a rational and irrational number leads to an irrational answer.
The multiple-choice answers provided are:
Use proof by contradiction: assume the opposite is true and the product of a non-zero rational number and an irrational number is always rational. This is a contradiction. According to the Closure Property, if the product is rational, both factors must be rational. So if one factor is irrational, the product must be irrational.
You can prove it by choosing random numbers to plug into multiplication problems until he's convinced.
He's correct; it is impossible to prove that the product of a rational number and an irrational number is always irrational.
You can prove it directly by creating a proof that shows directly how multiplying a rational and irrational number leads to an irrational answer.
Answers
Answered by
GPT-4o mini
To demonstrate that multiplying a non-zero rational number by an irrational number always results in an irrational product, we can indeed use a proof by contradiction, which is one of the most effective ways to illustrate this property.
Let's analyze the options provided:
1. **Use proof by contradiction: assume the opposite is true and the product of a non-zero rational number and an irrational number is always rational. This is a contradiction. According to the Closure Property, if the product is rational, both factors must be rational. So if one factor is irrational, the product must be irrational.**
This option correctly outlines the approach to proving the statement via contradiction. If we assume that a rational number multiplied by an irrational number produces a rational number, we arrive at a contradiction about the definition of rational and irrational numbers.
2. **You can prove it by choosing random numbers to plug into multiplication problems until he's convinced.**
This is not a valid proof. While it may provide some examples that support the claim, it does not constitute a formal proof. A proof requires showing that the statement holds for all cases, not just specific instances.
3. **He's correct; it is impossible to prove that the product of a rational number and an irrational number is always irrational.**
This option is incorrect. It is indeed possible to prove this statement rigorously.
4. **You can prove it directly by creating a proof that shows directly how multiplying a rational and irrational number leads to an irrational answer.**
While this option points towards a valid direction for a direct proof, it lacks the clarity of methodical reasoning present in the proof by contradiction method specified in option 1.
In conclusion, the best choice for demonstrating the validity of the claim that multiplying a non-zero rational number by an irrational number always results in an irrational product is:
**Use proof by contradiction: assume the opposite is true and the product of a non-zero rational number and an irrational number is always rational. This is a contradiction. According to the Closure Property, if the product is rational, both factors must be rational. So if one factor is irrational, the product must be irrational.**
Let's analyze the options provided:
1. **Use proof by contradiction: assume the opposite is true and the product of a non-zero rational number and an irrational number is always rational. This is a contradiction. According to the Closure Property, if the product is rational, both factors must be rational. So if one factor is irrational, the product must be irrational.**
This option correctly outlines the approach to proving the statement via contradiction. If we assume that a rational number multiplied by an irrational number produces a rational number, we arrive at a contradiction about the definition of rational and irrational numbers.
2. **You can prove it by choosing random numbers to plug into multiplication problems until he's convinced.**
This is not a valid proof. While it may provide some examples that support the claim, it does not constitute a formal proof. A proof requires showing that the statement holds for all cases, not just specific instances.
3. **He's correct; it is impossible to prove that the product of a rational number and an irrational number is always irrational.**
This option is incorrect. It is indeed possible to prove this statement rigorously.
4. **You can prove it directly by creating a proof that shows directly how multiplying a rational and irrational number leads to an irrational answer.**
While this option points towards a valid direction for a direct proof, it lacks the clarity of methodical reasoning present in the proof by contradiction method specified in option 1.
In conclusion, the best choice for demonstrating the validity of the claim that multiplying a non-zero rational number by an irrational number always results in an irrational product is:
**Use proof by contradiction: assume the opposite is true and the product of a non-zero rational number and an irrational number is always rational. This is a contradiction. According to the Closure Property, if the product is rational, both factors must be rational. So if one factor is irrational, the product must be irrational.**
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