The Real Numbers: example 1.111

Given the numbers $0.58 \overset{–}{3}, 0.47, 3.605551275 . . .$ list the ⓐ rational numbers ⓑ irrational numbers.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

For the given numbers list the ⓐ rational numbers ⓑ irrational numbers: $0.29, 0.81 \overset{–}{6}, 2.515115111 … .$

How did it go?

ⓐ Look for decimals that repeat or stop.	The 3 repeats in $0.58 \overset{–}{3}$ . The decimal 0.47 stops after the 7. So $0.58 \overset{–}{3}$ and 0.47 are rational.
ⓑ Look for decimals that neither stop nor repeat.	$3.605551275 …$ has no repeating block of digits and it does not stop. So $3.605551275 …$ is irrational.

ⓐ Look for decimals that repeat or stop.	The 3 repeats in $0.58 \overset{–}{3}$ . The decimal 0.47 stops after the 7. So $0.58 \overset{–}{3}$ and 0.47 are rational.
ⓑ Look for decimals that neither stop nor repeat.	$3.605551275 …$ has no repeating block of digits and it does not stop. So $3.605551275 …$ is irrational.