Asked by Rosella
Ryiaz says it is impossible to prove that multiplying a rational number by an irrational number always results in an irrational product. How can you show that a proof is possible?(1 point)
Responses
Use proof by contradiction: assume the opposite is true and the product of a 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.
Use proof by contradiction: assume the opposite is true and the product of a 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.
He’s correct; it is impossible to prove that the product of a rational number and an irrational number is always irrational.
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.
You can prove it directly by creating a proof that shows directly how multiplying a rational and irrational number leads to an irrational answer.
You can prove it by choosing random numbers to plug into multiplication problems until he’s convinced.
You can prove it by choosing random numbers to plug into multiplication problems until he’s convinced.
Responses
Use proof by contradiction: assume the opposite is true and the product of a 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.
Use proof by contradiction: assume the opposite is true and the product of a 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.
He’s correct; it is impossible to prove that the product of a rational number and an irrational number is always irrational.
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.
You can prove it directly by creating a proof that shows directly how multiplying a rational and irrational number leads to an irrational answer.
You can prove it by choosing random numbers to plug into multiplication problems until he’s convinced.
You can prove it by choosing random numbers to plug into multiplication problems until he’s convinced.
Answers
Answered by
GPT-4o mini
To show that it is possible to prove that the product of a rational number and an irrational number is always irrational, the best response would be:
**Use proof by contradiction: assume the opposite is true and the product of a 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 approach effectively demonstrates the impossibility of having a rational product when one factor is irrational, therefore supporting the assertion that the product of a rational number and an irrational number must be irrational.
**Use proof by contradiction: assume the opposite is true and the product of a 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 approach effectively demonstrates the impossibility of having a rational product when one factor is irrational, therefore supporting the assertion that the product of a rational number and an irrational number must be irrational.
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