No. If the decimal expansion falls into a repeating pattern (however long) then the number is rational.
For example, 0.33... is the rational number 1/3.
or
0.04142857142857... where the pattern 142857 continues forever is the rational number 29/700.
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No. If the decimal expansion falls into a repeating pattern (however long) then the number is rational.
It is proven that between two irrational numbers there's an irrational number. There's no method, you just know you can find the number.
The square root of Pi is a transcendental number whose decimal expansion begins 1.77245385090552....(and on and on) The answer is 772004514666935
An irrational number is a real number which can not be expressed in rational form, i.e. in form of a common fraction. If written in decimal form, an irrational number will contain an infinite number of decimal positions without any periodic repetition. Common examples of irrational numbers are Pi (3.14159...), e (2.71828...) and any non perfect root as for example, the square root of 2 (1.41421...), the square root of 7 (2.64575...), and so on. Any real number which does not fall into the the irrational number subset, must be a rational number. The rational number thus are real numbers which can be expressed in rational form, this means as the division of two integers (remembering that you can not divide into o). A rational number written in decimal form either will have a finite number of decimal positions or an infinite numbers with a periodic repetition. For example, 0.5 is a rational number because it can be written as (1/2). Another example is 1.5 which can be written as (3/2). Any integer is a rational number because it can be written as the integer divided by 1 or by any other integer, for instance, 8 = (8/1) = (16/2) = (32/4) and so on. Example of infinite periodic decimals are for instance (1/3) = 0.3333...., (4/9) = 0.4444..., (168/999) = 0.168168168...
pi is a transcendental number, which is a kind of irrational number. That means that the decimal representation of pi does not end (nor does it have a recurring sequence). There is, therefore, no last digit.
The square root of 121 is 11 which is not an irrational number.