The image you've uploaded asks for the converse, inverse, and contrapositive of five conditional statements. I will help you work through each of these logical transformations in sentence form.

Let's start by recalling what each of these transformations means for a conditional statement of the form "If $P$, then $Q$":

• Converse: "If $Q$, then $P$".
• Inverse: "If not $P$, then not $Q$".
• Contrapositive: "If not $Q$, then not $P$".

Statement 1:

Original: If $\pi$ is an irrational number, then it is a number that goes on forever.

• Converse: If it is a number that goes on forever, then $\pi$ is an irrational number.
• Inverse: If $\pi$ is not an irrational number, then it is not a number that goes on forever.
• Contrapositive: If it is not a number that goes on forever, then $\pi$ is not an irrational number.

Statement 2:

Original: If $x$ is the height of a box, then it is a positive number.

• Converse: If it is a positive number, then $x$ is the height of a box.
• Inverse: If $x$ is not the height of a box, then it is not a positive number.
• Contrapositive: If it is not a positive number, then $x$ is not the height of a box.

Statement 3:

Original: If the year 2020 is a leap year, then there has to be 366 days in that year.

• Converse: If there are 366 days in that year, then the year 2020 is a leap year.
• Inverse: If the year 2020 is not a leap year, then there are not 366 days in that year.
• Contrapositive: If there are not 366 days in that year, then the year 2020 is not a leap year.

Statement 4:

Original: If $i$ is a complex number, then it is not a real number.

• Converse: If it is not a real number, then $i$ is a complex number.
• Inverse: If $i$ is not a complex number, then it is a real number.
• Contrapositive: If it is a real number, then $i$ is not a complex number.

Statement 5:

Original: If $x$ is less than 0, then it is negative.

• Converse: If it is negative, then $x$ is less than 0.
• Inverse: If $x$ is not less than 0, then it is not negative.
• Contrapositive: If it is not negative, then $x$ is not less than 0.

Would you like more clarification on any of these?

Here are 5 related questions to explore this further:

1. What is the difference between the contrapositive and the inverse?
2. Can you explain why the converse of a statement may not always be true?
3. How do you determine the truth value of an inverse statement?
4. Why is the contrapositive logically equivalent to the original statement?
5. How would you apply these logical transformations in a mathematical proof?

Tip: The contrapositive of a statement always has the same truth value as the original statement!