Rational number — Safekipedia Discoverer

In mathematics, a rational number is a number that can be written as a fraction of two integers. This means it can be shown as p over q, where p is the numerator and q is a non-zero denominator. For example, 3 over 7 is a rational number, and every whole number, like -5, is also rational because it can be written as -5 over 1.

Rational numbers are very important in math because they follow special rules when you add, subtract, multiply, or divide them. These numbers are also called the rationals, and they are usually shown with the symbol Q. When you write these numbers as decimals, they either stop after a few digits or repeat the same pattern of digits forever. For instance, 3/4 is 0.75, which stops, and 1/3 is 0.333…, which repeats.

Numbers that are not rational are called irrational numbers. Examples include the square root of 2, π, and the number e. While there are many rational numbers, most real numbers are actually irrational. Rational numbers help us understand and build the bigger world of real numbers in math.

Terminology

In math, the word "rational" is often short for "rational number." Sometimes, it means that the numbers used to describe something are rational. For example, a rational point is a point where both coordinates are rational numbers.

The word "rational" actually came before the word "ratio." People started using "rational" to talk about these numbers in 1570, while "ratio" with its modern meaning came later, around 1660. This history goes back to ancient Greeks, who thought of numbers that couldn't be written as fractions as "irrational," meaning they were hard to talk about or "illogical."

Arithmetic

Every rational number can be written in a special way as a fraction ⁠ a b {/w/0}, {\displaystyle {\tfrac {a}{b}}}, ⁠ where a and b are coprime integers and b is greater than 0. This special form is called the canonical form of the rational number. For example, the fraction 2⁄4 can be simplified to 1⁄2 by dividing both numbers by their largest common factor, which is 2 in this case.

Continued fraction representation

Main article: Simple continued fraction

A finite continued fraction is a special way to write numbers using a pattern of addition and division. It looks like this:

a₀ + 1/a₁ + 1/a₂ + 1/⋯ + 1/aₙ

Here, the aₙ values are whole numbers. Every rational number can be shown as a finite continued fraction. We can find these numbers by using a method called the Euclidean algorithm on the fraction’s top and bottom numbers.

Other representations

Rational numbers can be shown in many different ways. For example, the number ⁠

          8
          3
        
      


{\displaystyle {\tfrac {8}{3}}}

⁠ can also be written as a mixed numeral like ⁠

          2
          2
          3
        
      


{\displaystyle 2{\\tfrac {2}{3}}}

⁠, a repeating decimal such as 2.6¯ or 2.(6), or even as a continued fraction like ⁠

          2
          +
          
            1
            
              1
              
                1
                
                  2
                  
                
              
            
          
        
      


{\displaystyle 2+{\\tfrac {1}{1+{\\tfrac {1}{2}}}}}

npm

          2
          ;
          1
          ,
          2
        
      


{\displaystyle \[2;1,2\]}

⁠. These are just different ways to show the same rational number.

Formal construction

The rational numbers can be built using pairs of whole numbers. Imagine two numbers, like 3 and 7. We can write this pair as (3, 7). Many different pairs can represent the same rational number. For example, (3, 7) is the same as (6, 14) because both simplify to the same value.

We can add and multiply these pairs using special rules. This helps us understand how rational numbers behave and relate to each other. Every rational number fits into one special group, called an equivalence class, which groups together all pairs that represent the same value.

This way of building rational numbers shows that they can be organized and counted, even though there are infinitely many of them.

Real numbers and topological properties

The rational numbers are very closely packed among all real numbers. Between any two real numbers, you can always find a rational number, which means the rationals are dense in the real numbers.

Rational numbers have special properties in mathematics. They form a type of space that helps us study their behavior, but they are not complete on their own — the real numbers fill in the gaps.

p-adic numbers

Main article: p-adic number

Besides the usual way to measure distances between numbers, there is another special way to do this using a prime number. This creates a new kind of number system called the p-adic numbers. This system helps mathematicians study numbers in unique ways.