real number
Picture a perfectly straight, endless ruler with no gaps anywhere. Every single point on it — whether it lands on a tidy mark like 2, a fraction like 3/4, or a wild endless decimal like pi — names one real number. The real numbers are simply all the numbers that live on this number line.
More precisely, the real numbers consist of all rational numbers together with all irrational numbers; every number on the number line is one or the other. They support addition, subtraction, multiplication, and division by nonzero values, and they come fully ordered, so for any two of them one is less than, equal to, or greater than the other.
A defining feature is that the reals have no holes: between any two reals there is always another, and the line is “complete” in the sense that limits of real numbers stay real. One honest limitation: the square root of a negative number is not a real number, because no real squared gives a negative result — handling those requires the imaginary and complex numbers.
−2, 0, 3/4, sqrt(2), and pi are all real numbers, each marking exactly one point on the number line.
Rational and irrational numbers together fill the whole line.
The set of reals is written R. The chain of containment is N ⊂ Z ⊂ Q ⊂ R: each system extends the previous one.