√25 = 5 is an integer, 0.272727... = 27/99 = 3/11 is a repeating decimal, and -4/9 is already a fraction of integers. 7 is not a perfect square, so √7 cannot be written as a fraction of integers and is irrational.
Question 2 True / False
The number 22/7 is exactly equal to π.
TTrue
FFalse
Answer: False
22/7 is a fraction of integers, so it is rational, while π is irrational. 22/7 = 3.142857... is a useful approximation, but π = 3.14159..., so the two differ starting in the third decimal place.
Question 3 Short Answer
Between which two consecutive tenths does √2 lie? Show how you know.
Think about your answer, then reveal below.
Model answer: Between 1.4 and 1.5, because 1.4² = 1.96 < 2 and 1.5² = 2.25 > 2.
Squaring brackets the root. Since 1.96 < 2 < 2.25, the number whose square is 2 lies between 1.4 and 1.5. Repeating the process with hundredths (1.41² = 1.9881, 1.42² = 2.0164) narrows it further, and the digits never begin to repeat.
Question 4 Short Answer
Write the repeating decimal 0.444... as a fraction.
Think about your answer, then reveal below.
Model answer: 4/9
Let x = 0.444.... Then 10x = 4.444.... Subtracting gives 9x = 4, so x = 4/9. Any repeating decimal can be converted this way, which is why repeating decimals are rational.