Is 3.05313131313131313131313131... irrational or rational? Thanks.
it repeats, rational.
Okay thank you.
To determine whether the number 3.05313131313131313131313131... is rational or irrational, we need to understand the properties of rational and irrational numbers.
A rational number can be expressed as a fraction of two integers, where the denominator is not zero. For example, 1/2, 3/4, and -5/7 are all rational numbers.
On the other hand, an irrational number cannot be expressed as a fraction of two integers. Some well-known examples of irrational numbers are π (pi) and √2 (square root of 2).
Now, let's analyze the given number:
3.05313131313131313131313131...
We can observe that there is a repeating pattern of "13" after the decimal point. As the pattern repeats endlessly, we can express this number as:
3.053 + (0.01313131313131313131313131...)
Now, let's focus on the fractional part, which is 0.0131313131313...
To determine if this is rational or irrational, we can express it as a fraction. Let x = 0.0131313131313...
Multiplying x by 100, we get:
100x = 1.31313131313...
Subtracting x from 100x, we have:
100x - x = 1.31313131313... - 0.0131313131313...
Simplifying, we get:
99x = 1.3
Dividing both sides by 99, we find:
x = 1.3/99
Hence, we can express the fractional part 0.0131313131313... as the rational number 1.3/99.
Since we have expressed the entire decimal representation as the sum of the rational number 3.053 and the rational fraction 1.3/99, we can conclude that the original number 3.05313131313131313131313131... is a rational number.
Therefore, the number 3.05313131313131313131313131... is rational.