Natural number

Natural numbers are the numbers we use to make sense of the world. They include 0, 1, 2, 3, and continue forever in a fixed order.

You might know them as counting numbers or whole numbers. We use them to answer questions about how many things exist in a group. This idea is called cardinality. When we use numbers to show the size of a collection, they are called cardinal numbers. These numbers help us understand the scale of everything around us.

There are two main ways to use these numbers. First, we use them to show the size of a group. You can find the size by pairing objects from two different groups. If every apple has exactly one orange to match it, the groups have the same cardinality.

If one group has objects left over, that group is larger. Second, we use them to show position in a list. These are called ordinal numbers, like saying something is the third item in a row. Every number has a successor, which is just the very next number in the sequence.

Mathematicians have spent a long time defining these numbers very carefully. One famous way is through the Peano axioms, named for Giuseppe Peano. These axioms are a list of rules that must be true for natural numbers. One rule says that every number has a unique successor. Another rule says that 0 is a natural number but is not the successor of any other number.

There is also a way to define them using set theory. A mathematician named John von Neumann created a famous method using sets. In his version, each number is built out of an empty set.

We use ten special symbols to write these numbers down. These symbols are called numerals, and they include 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Most people use the decimal system to write them. This system uses these numerals and specific rules for their positions. Sometimes, people use numbers as labels rather than for counting. For example, a jersey number on a sports team is a nominal number. These labels look like natural numbers, but they do not have the same math properties.

Natural numbers are the foundation for almost all other math. You can use them to perform basic operations like addition and multiplication. Addition is like putting groups together to make a larger total. Multiplication is like having many equal groups of the same size. These numbers are also found inside larger systems like integers and real numbers. Even the study of complex patterns and shapes relies on them. Without natural numbers, we could not easily count, measure, or organize our world.