Select all the true statements.

Question

You can compare irrational numbers using rational approximation.
Square roots can be compared and ordered by comparing and ordering the numbers underneath the radical symbol.
You cannot compare the value of rational and irrational numbers.
The closer together the numbers being compared, the more decimal places you need to use.
All irrational numbers have some repeating pattern which can be used to compare them to rational numbers.

Answer 1

Let's evaluate each statement:

1. You can compare irrational numbers using rational approximation.
   True. You can approximate irrational numbers with rational numbers, allowing for comparison between them.

2. Square roots can be compared and ordered by comparing and ordering the numbers underneath the radical symbol.
   True. For example, √4 is compared to √9 by comparing 4 and 9, since √4 = 2 and √9 = 3.

3. You cannot compare the value of rational and irrational numbers.
   False. You can compare rational and irrational numbers even though they belong to different sets. For example, √2 (irrational) is greater than 1 (rational).

4. The closer together the numbers being compared, the more decimal places you need to use.
   True. When two numbers are very close, having more decimal places helps to see the difference more accurately for comparison.

5. All irrational numbers have some repeating pattern which can be used to compare them to rational numbers.
   False. Irrational numbers do not have repeating patterns in their decimal expansions; they are non-terminating and non-repeating.

Based on the evaluations, the true statements are:

1. You can compare irrational numbers using rational approximation.
2. Square roots can be compared and ordered by comparing and ordering the numbers underneath the radical symbol.
3. The closer together the numbers being compared, the more decimal places you need to use.