The following are two possible breakdowns. Positive, zero and negative.
Integer, fraction and mixed.
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The set of real numbers can be divided into rational numbers and irrational numbers.
Natural numbers Integers Rational numbers Real numbers Complex numbers
All rational numbers including integers can be be expressed as fractions whereas irrational numbers can't be converted into fractions
There is only one type of rational number so there are no different types which you can add.
You can multiply any pair of rational numbers as well as any irrational number and its reciprocal (or a rational multiple of its reciprocal. Thus pi * 3/7*(1/pi) is rational.