The set of real numbers includes which of the following?

Correct — C. The real numbers are made of two groups that do not overlap.

Rational: can be written as p⁄q with integers p and q, q ≠ 0, e.g. 3, −2, 3⁄4, 0.25
Irrational: cannot be written that way, e.g. √2, √7, π

Every real number is rational or irrational, so the reals are rationals together with irrationals → option (c)

Why the others are wrong

• (a)
  Only Rational Numbers — Only Rational Numbers leaves out √2 and π. Both sit on the number line, so they are real, but no fraction p⁄q equals either of them.
• (b)
  Only Irrational Numbers — Only Irrational Numbers leaves out every integer and every fraction. 5 and 3⁄4 are real numbers, and both are rational.
• (d)
  Only Whole Numbers — Whole numbers (0, 1, 2, 3 …) are a small part of the reals. They miss negatives such as −2, fractions such as 3⁄4, and every irrational number.

The number system nests: natural ⊂ whole ⊂ integers ⊂ rationals ⊂ reals. Each set sits inside the next.

Irrational numbers are not a further step in that chain. They are the reals left over once the rationals are taken out, so rationals and irrationals split the reals between them with nothing in common.

A decimal test tells them apart: a rational number's decimal ends or repeats, an irrational number's decimal goes on without repeating.

Key facts

• A rational number can be written as p⁄q with integers p and q, q ≠ 0.
• √2, √7 and π are irrational.
• Every real number is rational or irrational, never both.
• Whole numbers are 0, 1, 2, 3 … and all of them are rational.

Common traps

• Treating irrational numbers as outside the real numbers: √2 lies on the number line between 1.41 and 1.42.
• Calling a recurring decimal such as 0.333… irrational: it equals 1⁄3.

No directly related past PYQ was found.