It is impossible to have a surd that is not irrational.
Surds are defined to be an irrational number (square root of a number).
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How one understands surds depends on the person. If you would like to know what surds are, or have help understanding surds A surd is an unresolved radical, meaning that it is a root with the radical sign still on it. It is easier (and more accurate) to express it this way than writing it out for many numbers if the root is irrational. The concept of irrational numbers (which is what surds are, usually) can be confusing. In short, they are numbers that are not rational, that is, they cannot be written as a fraction. When using surds in math, you treat them just as you would a written out number.
Yes.
No. The square root of negative one is an example of an imaginary (not real) number. Pi is irrational, but real.
The irrational numbers are real numbers. An irrational number is one that cannot be written as a fraction (ie, they have an infinite, non-repeating sequence of decimal places), such as pi. Most square roots are also irrational, like the square root of 2 for example.
Assuming you want an example of one, √10 is an irrational number between 3 and 4. If you want all of them, you're out of luck - there is an infinite number of them.