Yes, 0.12112111211112 is a rational number. A rational number is a number that can be expressed as the quotient or ratio of two integers, where the denominator is not equal to 0.
We can convert the decimal representation of the number 0.12112111211112 to a fraction in two ways.
Method 1: Using Place Values
Firstly, we need to recognize the pattern in the decimal representation. We can see that the repeating pattern is 1211121112. Let us represent this pattern by a variable ‘x’.
x = 1211121112
Now, we need to find the value of x divided by the appropriate power of 10 so that it becomes a fraction. Since there are 10 digits in the repeating pattern, we need to divide x by 10^10 as shown below:
x/10^10 = 1211121112/10^10
Simplifying this fraction, we get:
x/10^10 = 0.1211211121
Now, we can see that 0.1211211121 is a fraction with a numerator and denominator which are both integers. Therefore, 0.1211211121 is a rational number.
Method 2: Using Algebra
We can also use algebra to convert 0.12112111211112 into a fraction. Let’s represent the decimal number by the variable ‘a’.
a = 0.12112111211112
Multiplying both sides by 10^14, we get:
10^14 a = 12112111211112
Let’s subtract the left-hand side from the right-hand side:
(10^14 a) – a = 12112111211112 – 0.12112111211112
Simplifying, we get:
10^14 a – a = 12112111211111.87887888788888
Simplifying further, we get:
(10^14 – 1) a = 12112111211111.87887888788888
Dividing both sides by (10^14 – 1), we get:
a = 12112111211111.87887888788888 / (10^14 – 1)
This fraction can be simplified by dividing both the numerator and denominator by their greatest common factor. However, we can see that both the numerator and denominator are very large numbers. Therefore, it is easier to use the first method to convert the decimal to a fraction.
Whether we use algebra or place values, we can see that 0.12112111211112 can be expressed as a fraction with integer numerator and denominator. Therefore, it is a rational number.
