| Number | Rational or Irrational? | |---|---| | \(\sqrt{3}\) | Irrational | | 78.12566 | Rational | | \(\sqrt{25}\) | Rational | | \(\sqrt{5}\) | Rational | | \(-\sqrt{16}\) | Irrational | | \(\sqrt{18}\) | Irrational | | \(-\frac{5}{3}\) | Rational | | 4.17 | Rational | | \(2\pi\) | Irrational | | 3.89 | Rational | | \(\sqrt{111}\) | Irrational | | \(4\pi\) | Irrational | | \(9\pi\) | Irrational | | 72 | Rational | | \(\frac{2\pi}{3}\) | Irrational |
Answer (1)
To determine whether a number is rational or irrational, we need to understand the definitions:
Rational Numbers : These are numbers that can be written as a fraction b a where a and b are integers and b = 0 . Rational numbers include integers, fractions, and terminating or repeating decimals.
Irrational Numbers : These are numbers that cannot be expressed as a simple fraction. They include non-repeating, non-terminating decimals. Examples include π , e , and square roots of non-perfect squares.
Let's analyze each number provided:
3 : This is a square root of a non-perfect square, hence it is irrational.
78.12566 : This is a decimal that terminates, so it is rational.
25 : 25 = 5 , which is an integer, making it rational.
5 : Like 3 , it is a square root of a non-perfect square, so it is irrational.
− 16 : 16 = 4 , and − 16 = − 4 , which is an integer, so it is rational.
18 : This is a square root of a non-perfect square, thus it is irrational.
− 3 5 : This is a fraction of two integers, hence rational.
4.17 : This is a terminating decimal, so it is rational.
2 π : Since π is irrational, multiplying it by 2 remains irrational.
3.89 : This is a terminating decimal, therefore it is rational.
111 : As this is a square root of a non-perfect square, it is irrational.
4 π : Multiplying an irrational number π by 4 keeps it irrational.
9 π : Similarly, 9 π is irrational.
72 : This is an integer, making it rational.
3 2 π : Since π is irrational, any multiple or fraction of it remains irrational.
In summary, understanding the difference between rational and irrational numbers helps in identifying them. Rational numbers can be precisely expressed as fractions or have repeating/terminating decimals, whereas irrational numbers cannot be expressed in such forms.
