Real numbers
Real numbers are all the numbers we can write using a finite or infinite decimal expansion. We can say that we get the real numbers as the union of the set of rational and irrational numbers.
Notation and meaning
We write the real numbers with the letter R, usually with a double vertical stroke: ℝ.
If you imagine an infinite number line, the real numbers represent all the possible distances between two points that we can find on the line.
The real numbers contain all the positive integers, integers, rational numbers and irrational numbers.
Most of the functions we commonly work with in maths have the set of real numbers as their domain (for example sine and cosine, linear functions) or some connected subset of it (for example the logarithm).
Examples of real numbers: 1, −3, 0.126, −42.47, e, 0, −2π.
Properties
- The real numbers form an infinite uncountable set.
- The real numbers are closed under addition, subtraction, multiplication and division. If we take two real numbers and multiply them, we get a real number again.
- A real number is either algebraic, that is, a root of some polynomial with integer coefficients, or transcendental, that is, not a root of any such polynomial.
- Every real number has an absolute value, which is its distance from the origin of the number line.
- Every non-negative real number has a square root, which is also a real number. On the other hand, no negative real number has a real number as its square root.
