Types of Numbers Explained: Natural, Rational, Irrational, and Imaginary

Tap a number type to see its definition.

Natural Numbers (ℕ)

The most basic positive integers used for counting things, like 1, 2, 3. (Whether 0 counts as a natural number depends on the definition used.)

Integers (ℤ)

A number system that extends the natural numbers to include 0 and the negative integers (-1, -2, ...).

Rational Numbers (ℚ)

Numbers that can be expressed as a ratio (fraction) of two integers — this includes integers themselves, along with terminating and repeating decimals.

Irrational Numbers

Numbers that cannot be expressed as a fraction, with digits after the decimal point that continue infinitely without repeating — pi (π) and the square root of 2 are classic examples.

Real Numbers (ℝ)

A concept covering every number that can be placed on a number line, combining both rational and irrational numbers.

Imaginary Numbers (i)

Numbers that become negative when squared — introduced to solve equations that can't be solved using real numbers alone (defined by i² = -1).

Complex Numbers (ℂ)

Numbers that combine a real and an imaginary part in the form (a + bi), with practical applications across fields like electrical engineering and quantum mechanics.

Prime Numbers

Natural numbers greater than 1 that are divisible only by 1 and themselves — like 2, 3, 5, and 7 — and critically important in cryptography.

The World of Numbers Has Kept Expanding

Humanity originally only needed natural numbers for counting objects, but in the process of solving equations and describing phenomena, we gradually invented broader and broader number concepts — negative numbers, fractions, irrational numbers, imaginary numbers. Each new number system emerged to solve problems the previous system couldn't handle.

Frequently Asked Questions

Are imaginary numbers actually useful for anything?

Yes — they're used in practical fields that can't be fully described using real numbers alone, such as calculating alternating-current circuits in electrical engineering or in quantum mechanics.

Is zero considered a natural number?

It depends on the definition used. In many countries' school curricula, 0 is typically not included among the natural numbers, but some countries and areas of set theory do include 0 as a natural number.