Is 1.875 rational or irrational? Explain Identify the number as rational or irrational square root of 112
The correct answer and explanation is:
Correct Answers:
- 1.875 is a rational number.
- The square root of 112 is an irrational number.
Explanation:
A rational number is any number that can be written as a fraction or a ratio of two integers, where the denominator is not zero. Rational numbers include integers, fractions, and decimals that terminate or repeat.
The number 1.875 is rational because it is a terminating decimal. It can be written as a fraction:
1.875 = 1875/1000 = 15/8 (after simplifying).
Since both 15 and 8 are integers, and 8 is not zero, this confirms 1.875 is a rational number.
On the other hand, an irrational number is a number that cannot be written as a simple fraction. Its decimal representation is non-terminating and non-repeating. Common examples include π (pi) and non-perfect square roots like √2 or √7.
The square root of 112 is irrational. To determine this, look at whether 112 is a perfect square. The perfect squares closest to 112 are 100 (which is 10²) and 121 (which is 11²). Since 112 is not a perfect square and cannot be simplified into the square root of a perfect square, √112 is irrational.
For confirmation, we can try to simplify √112:
- √112 = √(16 × 7) = √16 × √7 = 4√7
Now √7 is a well-known irrational number, and multiplying it by 4 still results in an irrational number. Therefore, √112 = 4√7 is irrational.
In summary, 1.875 is rational because it can be written as a fraction (15/8), and √112 is irrational because it cannot be expressed as a simple fraction and contains the square root of a non-perfect square.