Rational and irrational numbers are two distinct types of numbers that exist in mathematics. The main difference between rational and irrational numbers is that a rational number can be expressed as the ratio of two integers, while an irrational number cannot be expressed as the ratio of two integers.
A fraction is an example of a rational number, which is composed of a numerator and denominator that represent the ratio of two numbers. On the other hand, square root, some decimals, and pi are examples of irrational numbers. Irrational numbers cannot be written as the ratio of two numbers and they have infinite non-repeating digits after the decimal point.
In summary, one main difference between rational and irrational numbers is that a rational number can be expressed as the ratio of two integers while an irrational number cannot be expressed as such.
Rational vs. Irrational Numbers
| Rational numbers | Irrational numbers |
|---|---|
| Rational numbers can be expressed as a ratio of two integers (a fraction i.e a/b). So, it includes all integers, fractions, and decimals that can be written as a ratio of two integers. | Irrational numbers cannot be expressed as a ratio of two integers. So, it is any real number that cannot be written as a ratio of two integers. |
| The decimal expansion of rational numbers is either terminating or recurring. | The decimal expansion of irrational numbers is non-terminating and non-recurring. |
| It includes perfect squares such 4, 9, 16, 25 and so on. | It includes surds such √2, √3, √5, √7 and so on. |
| A fraction is an example of a rational number having a numerator and denominator where denominator is not zero. More Rational numbers: 3/2 = 1.5, 3.6767 | Pi is an example of irrational number. More Irrational numbers examples: √5, √11 |
The Different Types of Numbers
Numbers are everywhere around us and play an important role in our lives. They can be used to quantify, measure, and explain many things. There are different types of numbers that we use in mathematics, each with its specific purpose and function:
An integer is a whole number or a natural number that does not contain any decimal point or fraction.
Whole numbers are the same as integers but they may also include zero.
A fraction is a part of an integer which is expressed as two separate numbers separated by a slash (e.g., 1/2).
Decimals are fractions with digits after the decimal point (e.g., 0.5). Irrational numbers are real numbers that cannot be expressed as fractions and do not terminate or repeat in their decimal form (e.g., pi).
Rational numbers are real numbers that can be expressed as fractions and terminate or repeat in their decimal form (e.g., 3/4). Both rational and irrational numbers can be classified as either terminating or non-terminating, repeating or non-repeating decimals depending on their properties of expression.
Famous Irrational Numbers
The most famous irrational numbers are π (pi) and √2 (the square root of 2). While rational numbers can be written using fractions, irrational numbers can only be expressed using decimals that never end or repeat in a predictable pattern. In fact, pi continues infinitely without any repeating pattern whatsoever.
Other famous irrational numbers include e, the base of natural logarithms, and the golden ratio. Irrational numbers are used in mathematics to solve problems that involve ratios, such as calculating the circumference of a circle or finding the square root of a number.
Although irrational numbers are fascinating to study and use, they will never perfectly match up with rational numbers no matter how many decimal places you calculate them to.
Examples of Rational and Irrational Numbers
Rational and irrational numbers are two distinct types of numbers. A rational number is any number that can be expressed as a fraction, with the numerator and denominator being integers. Common examples of rational numbers include integers such as -5, 0, or 10, and fractions like 1/2 or 4/5.
In contrast, an irrational number cannot be represented as a fraction with integers in both the numerator and denominator. A few famous examples of irrational numbers include π (pi), which is approximately equal to 3.14159…; e (Euler’s Number), which is approximately equal to 2.71828…; and recurring decimals such as 0.333333… An irrational number never ends or repeats itself when written as a decimal.
Natural, Whole, and Integer Numbers
Natural, whole, and integer numbers are all important in mathematics. Natural numbers are the most basic type of number, often referred to as counting numbers because they start from 1 and go on forever; they have no fractional or decimal parts. Whole numbers also include natural numbers, but with the addition of 0.
Integers are whole numbers that also include negative numbers. All three kinds of numbers have their own unique characteristics and functions in mathematics. Natural numbers are used for counting objects, while whole numbers can be used for measuring distances or adding up different values.
Integers can be used for subtracting a smaller value from a larger one or comparing two values to determine which is greater or lesser than the other. In any case, understanding the distinctions between these three types of number can help make working with them much easier.
Key Differences Between Rational and Irrational Numbers
The key difference between rational and irrational numbers is that rational numbers can be expressed as a ratio of two integers (a fraction) whereas irrational numbers cannot.
A rational number includes all integers, fractions, and decimals that can be written as a ratio of two integers. An irrational number is any real number that cannot be written as a ratio of two integers; this includes some decimals, but not all.
The decimal form of an irrational number never ends or repeats. In contrast, the decimal form of a rational number either terminates or else it will repeat itself after some point in the decimal form.
When writing a fraction, the denominator should never equal zero since any fraction with a zero denominator is undefined. The sum of two rational numbers is also a rational number and the sum of two irrational numbers is also an irrational numbers. Here is the comparison between rational and irrational numbers:
Why pi is not a rational number?
Pi, often written as the Greek symbol π, is an irrational number that is used in mathematics to represent a constant. It is the ratio of a circle’s circumference to its diameter and is equal to approximately 3.14159. The number pi cannot be expressed as a fraction, making it an irrational number, meaning that it cannot be expressed in terms of integers.
Pi is also considered an irrational number because its decimal expansion continues infinitely without any repeating pattern or set of numbers. This infinite nature means that pi never ends and can never be fully expressed or represented by rational numbers.
In addition, pi has been proven to be irrational and transcendental, meaning that it is not the root of any polynomial equation with rational coefficients. For these reasons, pi cannot be classified as a rational number and will always remain an irrational one.
