Mathematical Matryoshka: Counting Numbers, Integers, Rational Numbers, and Real Numbers

When you first start dealing with numbers, you learn about the four main sets, or groups, of numbers, which, like Russian matryoshka dolls, nest inside one another:

  • Counting numbers (also called natural numbers): The set of numbers beginning 1, 2, 3, 4, . . . and going on infinitely
  • Integers: The set of counting numbers, zero, and negative counting numbers
  • Rational numbers: The set of integers and fractions
  • Real numbers: The set of rational and irrational numbers (which can’t be written as simple fractions)

The sets of counting numbers, integers, rational, and real numbers are nested, one inside another, similar to the way that a city is inside a state, which is inside a country, which is inside a continent. The set of counting numbers is inside the set of integers, which is inside the set of rational numbers, which is inside the set of real numbers.

Counting, or natural, numbers

The set of counting numbers is the set of numbers you first count with, starting with 1. Because they seem to arise naturally from observing the world, they’re also called the natural numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, . . .

When you add two counting numbers, the answer is always another counting number. Similarly, when you multiply two counting numbers, the answer is always a counting number. Another way of saying this is that the set of counting numbers is closed under both addition and multiplication.


The set of integers arises when you try to subtract a larger counting number from a smaller one. For example, 4 – 6 = –2. The set of integers includes the following:

  • The counting numbers
  • Zero
  • The negative counting numbers

Here’s a partial list of the integers:
. . . –4, –3, –2, –1, 0, 1, 2, 3, 4, . . .

Like the counting numbers, the integers are closed under addition and multiplication. Similarly, when you subtract one integer from another, the answer is always an integer. That is, the integers are also closed under subtraction.

Rational numbers

The set of rational numbers includes all integers and all fractions. Like the integers, the rational numbers are closed under addition, subtraction, and multiplication. Furthermore, when you divide one rational number by another, the answer is always a rational number. Another way to say this is that the rational numbers are closed under division.

Real numbers

Even if you filled in all the rational numbers on the number line, you’d still have points left unlabeled. These points are the irrational numbers.

An irrational number is neither a whole number nor a fraction. In fact, an irrational number can only be approximated as a non-repeating decimal. In other words, no matter how many decimal places you write down, you can always write down more; furthermore, the digits in this decimal never become repetitive or fall into any pattern.

The most famous irrational number is π, 3.14159265358979323846264338327950288419716939937510 . . .

Together, the rational and irrational numbers make up the real numbers, which comprise every point on the number line.