Rational number

A rational number is a specific type of value in mathematics. It is any number that can be expressed as a quotient of two integers. This means you can write it as a fraction. The top number is called the numerator. The bottom number is called the denominator. The denominator must be a non-zero integer.

Every whole number, or integer, is also a rational number. You can represent an integer like 5 as the fraction 5/1. This makes the set of rational numbers, often called "the rationals," a very broad group.

Rational numbers have specific decimal patterns. If you write a rational number as a decimal, it will do one of two things. First, the decimal might terminate, meaning it stops after a finite number of digits. For example, 1/4 becomes 0.25. Second, the decimal might repeat. It will eventually start a sequence of digits that repeats over and over. This rule works in base 10, but it also works in other bases like binary or hexadecimal. Numbers that do not follow these patterns are called irrational numbers. Examples of irrational numbers include the square root of 2 and the golden ratio.

In the study of arithmetic, rational numbers follow strict rules. You can add, subtract, multiply, and divide them. When you perform these operations on rational numbers, the result is always another rational number. Because of this, mathematicians call the rationals a "field." This is a special structure where these operations work consistently.

Every rational number also has an additive inverse, which is its opposite. For example, the opposite of 1/2 is -1/2. Every non-zero rational number has a multiplicative inverse, also known as a reciprocal. The reciprocal of 3/4 is 4/3.

There is a special way to write every rational number called canonical form. This is also known as an irreducible fraction. In this form, the numerator and denominator are coprime. This means they share no common factors other than 1. You can find this form by dividing both numbers by their greatest common divisor. For example, 6/8 can be simplified to 3/4. This simplified version is the unique canonical representative of that value.

The history of these numbers is quite surprising. The word "rational" was used to describe numbers as early as 1570. However, the word "ratio" did not appear with its modern meaning until about 1660. This means the term "rational" actually predates the word "ratio." The concept of "irrational" numbers has deep roots in ancient Greece. Some Greek thinkers found certain lengths to be illogical. They avoided thinking of these lengths as numbers. They called them irrational because they were "not to be spoken about."

Rational numbers have a unique property called density. This means that between any two rational numbers, there is always another rational number. You can keep finding new ones forever. Because of this, the rationals are a dense subset of the real numbers.

Despite being everywhere, the set of rational numbers is "countable." This means you can create a list that includes every single rational number. You can do this by sorting them by the sum of their numerator and denominator.

Even though they are countable, the set of real numbers is "uncountable." This implies that almost all real numbers are actually irrational.

Mathematically, the field of rational numbers is a "prime field." This means it is the smallest field that contains the integers. Any other field containing the integers must also contain the rationals. In mathematical analysis, the real numbers can actually be built using the rational numbers. This is done through processes called completion, using things like Cauchy sequences or Dedekind cuts. The rational numbers are a fundamental building block for the entire system of real numbers.