Question

Part A: Find a rational number that is between 7.7 and 7.9. Explain why it is rational. (2 points) I kinda understand this part, that it would be 7.8 because you can turn it into a fraction but the others you cannot but I don't know if that's correct.
Part B: Find an irrational number that is between 7.7 and 7.9. Explain why it is irrational. Include the decimal approximation of the irrational number to the nearest hundredth. (3 points)

Answer 1

Answer:

Part A: 7.8. It equals 78/10.

Part B: √61. It is about 7.8102

Step-by-step explanation:

Part A:

Yes, you are correct. I'm going to expand on your answer, however.

So, recall what rational numbers are. They are any number that cannot be written as a fraction between two integers. Another way of saying this is that if the number doesn't repeat and doesn't terminate, then it's irrational.

So, yes, 7.8 is a rational number between 7.7 and 7.9.

In fact, there are infinite rational numbers between 7.7 and 7.9. 7.8 is rational. So is 7.88. And 7.888. And 7.8888. Their fractions are, respectively: 78/10, 788/100, 7888/1000, and 78888/10000. They can all be written as a fraction (between two integers), and so, they are rational.

Part B:

So, unlike rational numbers, irrational numbers do not terminate and they do not repeat. An example is √2. Its approximation is 1.41424...  This doesn't repeat nor terminate.

To find an irrational number between 7.7 and 7.9, the strategy is to square the two numbers. 7.7^2 is 59.29 and 7.9^2 is 62.41. Now, we just need to find a number within this range that when you take the square root of it, it is irrational.

After thinking, 61 is a valid candidate. The square root of 61 is 7.81024... It doesn't repeat and doesn't terminate. It is irrational.

In fact, the square root of any prime number is irrational.

And 7.81024... is between 7.7 and 7.9. Therefore, an answer we could give is √61.

And it's approximately 7.8102.

Edit: Typo