You use an approximate decimal representation or part of an infinite series.
For example,
pi = 4*[1 - 1/3 + 1/5 - 1/7 + ... ]
The above series is infinite but you can stop whenever you want to get an approximation. There are many such series.
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How would you do that ?? You'd have to change the value of the number. As long as the irrational number keeps its original value, it's an irrational number.
The value of the sum depends on the values of the rational number and the irrational number.
You can either express it as a number in a radical - possibly in the hope or expectation that it will cancel out. Or you approximate it using one of several methods. Since it is irrational, you can never write down its value in the form of a ratio, terminating or recurring decimal.
Usually, but not necessarily. A logarithm that is not an integer-value is irrational. For example log10100 = 2 which is a rational number. log1012 = 1.0791812460476... which is an irrational number.
Pi is a mathematical constant and does not depend on a particular circle. It is an irrational (transcendental) number whose approximate value is 3.14159 although 3.14 is also used.