4 Important Sets of Numbers

The number line grows in both the positive and negative directions and fills in with a lot of numbers in between. Here is a quick tour of how numbers fit together as a set of nested systems, one inside the other.

A set of numbers is really just a group of numbers. You can use the number line to deal with four important sets of numbers:

  • 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

The sets of counting numbers, integers, rational, and real numbers are nested, one inside another. This nesting of one set inside another is similar to the way that a city (for example, Boston) is inside a state (Massachusetts), which is inside a country (the United States), which is inside a continent (North America).

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 on the counting 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 . . .

The counting numbers are infinite, which means they go on forever.

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.

Introducing integers

The set of integers arises when you try to subtract a larger 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.

Staying rational

Here's the set of rational numbers:

  • Integers

  • Counting numbers

  • Zero

  • Negative counting numbers

  • 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.

Getting real

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

An irrational number is a number that's neither a whole number nor a fraction. In fact, an irrational number can only be approximated as a nonrepeating 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:


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

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