To determine whether the number 5.131131113 is rational or irrational, we need to understand the properties of rational and irrational numbers. A rational number is any number that can be expressed as a fraction of two integers, where the denominator is not zero. On the other hand, an irrational number is a number that cannot be expressed as a fraction and has a non-terminating and non-repeating decimal expansion.
Looking at the number 5.131131113, we can see that it has a decimal part that repeats the sequence “131113.” This indicates that the decimal expansion is non-repeating. We can also observe that the decimal part does not terminate, as it continues indefinitely. These characteristics suggest that the number is likely to be irrational.
To further confirm this, we can perform a proof by contradiction. Let’s assume that 5.131131113 is a rational number. This means it can be expressed as a fraction a/b, where a and b are integers with b not equal to zero. We can write:
5.131131113 = a/b
Multiplying both sides by 10^9 (to eliminate the decimal part), we get:
51311311.3 = a/b * 10^9
Simplifying the equation, we have:
51311311.3 * b = a * 10^9
Since both sides of the equation are integers, the left-hand side (51311311.3 * b) must also be an integer. However, multiplying a decimal by an integer generally results in a non-integer value. This contradiction suggests that our assumption that 5.131131113 is rational is incorrect.
Therefore, we can conclude that 5.131131113 is an irrational number. It cannot be expressed as a fraction of two integers and has a non-terminating and non-repeating decimal expansion.
I personally find the concept of irrational numbers fascinating. They challenge our intuitive understanding of numbers and highlight the infinite nature of mathematics. It is intriguing to think that there are numbers that cannot be fully represented or understood using fractions or whole numbers. Irrational numbers like 5.131131113 add richness and complexity to the realm of mathematics, making it a field that continually surprises and captivates me.
The number 5.131131113 is an irrational number due to its non-terminating and non-repeating decimal expansion.