What are irrational, rational, real, whole, and integers numbers
An irrational number is a number that can't be represented as a fraction. The classic example of an irrational number is the square root of 2.
A rational number is a number that can be represented as a fraction (so long as the denominator of the fraction isn't zero). 1/3 would be rational, but then so would 3/1=3.
Real numbers are, for the most part, any number that isn't imaginary. (Imaginary numbers include the imaginary unit [lower case I]*, which equals the square root of -1.) Real numbers are basically any number that can be represented by a decimal, even if it's a nonrepeating decimal, so all the other types of numbers you mentioned are also real numbers. The square root of 2, -3.2, 7/11, and 1 are all real numbers.
"Whole number" isn't that definite a technical term in mathematics. In this case you should probably use the term "natural number". A natural number is any of the positive counting numbers, like 1, 2, 3, and 50,006,393,431. Sometimes this includes 0, but that's the exception.
An integer is like a natural number, only it includes all their negative values, as well as 0. So -2, -1, 0, 1, and 2 would be integers.
*It looks like FunAdvice automatically capitalizes a lone lower-case I. It's that letter with the dot over it. :)
Irrational: top bigger than bottom (5/3)
Rational: bottom bigger than top (3/5)
Real: a number between -infinity and +infinity (67554365647.76546)
Integer: same as a whole number (6)
Whole number = a non fraction or non decimal number.
I'll let somebody else tackle the others.