Why we keep inventing new numbers
Numbers did not arrive all at once. They grew up one need at a time. You start by counting sheep: 1, 2, 3, … — the natural numbers. Then someone asks how many sheep are left after you sell them all, and you need zero; the natural numbers plus zero are the whole numbers. Owe someone two sheep and you need negatives, which gives the integers …, −2, −1, 0, 1, 2, …. Split one sheep three ways and you need fractions — the rational numbers. Each new kind of number is invented to answer a question the older numbers could not.
A rational number is any number you can write as a ratio a/b of two integers with b not zero — that includes 7 (= 7/1), −3/4, and 0.25 (= 1/4). For a long time people believed every number was rational. Then geometry produced sqrt(2), the length of the diagonal of a unit square, and it turned out no fraction equals it. Numbers like that are the irrational numbers.
Boxes inside boxes
The crucial picture is that these sets nest. Every natural number is a whole number; every whole number is an integer; every integer is a rational number; and every rational number, together with all the irrationals, makes up the real numbers — the entire number line with no gaps. So 5 is a natural number, an integer, a rational, and a real, all at once. The label you use just depends on which box you are pointing at.
naturals ⊂ wholes ⊂ integers ⊂ rationals ⊂ reals 1, 2, 3 + 0 + (−1,−2) + 3/4, −7/2 + sqrt(2), π Classify each number (smallest box it fits): 6 → natural 0 → whole −4 → integer 2/5 → rational sqrt(9)=3 → natural (careful: it simplifies!) sqrt(2) → irrational π → irrational