Pi is used in work with circles or ellipses. For example, how far a car will move in one rotation of its tyre. That may seem a silly question but that is how the odometer works.
Irrational square roots will occur in many measurements. The diagonal of a square or the principal diagonal of a cube will always be irrational. That will also be the case for most rectangles and cuboids.
The number e has many uses in advanced mathematics.
In art and architecture, the Golden Ratio is an irrational number. The Fibonacci sequence, which crops up in nature in many situations, is closely related to the Golden Ratio.
These are some examples.
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Irrational numbers can not be expressed as fractions whereas rational numbers can be expressed as fractions
Two of the most important numbers in advanced mathematics are pi and e and both are irrational.
yes * * * * * No. Rational and irrational numbers are two DISJOINT subsets of the real numbers. That is, no rational number is irrational and no irrational is rational.
Not necessarily. The sum of two irrational numbers can be rational or irrational.
The history of irrational numbers is quite simple in that any number that can't be expressed as a fraction is an irrational number as for example the value of pi as used in the square area of a circle.