The number world is fascinating. It forms the basis of several important concepts of mathematics and other sciences. Rational and irrational numbers are important topics that every student must know as it helps build a solid mathematical foundation.
Read this to understand the characteristics of identifying rational vs irrational numbers.
What are Rational numbers?
Rational numbers originated from the word ratio. Thus, anything that can be expressed in a ratio of rational format is a rational number.
Rational numbers are a type of real number. They are represented in the form p by q, where q is not equal to zero.
Examples of rational numbers are widespread-
- Any number in fractional format ½, ¾, ⅚, etc.
- Positive and negative Integers with a denominator
A simple definition of rational number can be any number and can be represented in the form p by q. It is easy to say that all fractions can fall under the blanket category of rational numbers.
Both numerator and denominator in rational numbers are integers, and the denominator is never zero. The decimal obtained after dividing the fraction can be either terminated or repeated.
How to know if it’s a rational number?
To check if a number is a rational number, you must be checking if it can be represented by p/q and q is not equal to zero and if the ratio of p by q can be represented in a decimal form. It is essential to know that rational numbers can include positive, negative, and 0 and should be expressed as fractions.
Different types of rational numbers
There are several types of rational numbers. All real numbers are rational numbers. It also includes integers and rational numbers. Thus, every whole number which can be expressed as a fraction is called a rational number.
Important characteristics of rational numbers
Some important characteristics of a rational number are:
- If you multiply, add, or subtract any two rational numbers, the result is always a rational number.
- The rational number always remains the same if you divide and multiply more than in a numerator and denominator with the same factor.
- All the rational numbers are closed when you add, subtract, or multiply them.
Examples of rational numbers
If a number is easily expressible in a fraction format when both the numerator and denominator are integers, the number is called a rational number.
Many types of rational numbers exist. It includes integers, positive, negative, zero, and fractions. Rational numbers can be easily divided and also represented like any other fraction.
What are Irrational numbers?
Irrational numbers are real numbers and cannot be represented in any ratio form. It means real numbers, which are not rational numbers, are irrational. These were first discovered by a Greek philosopher (Pythagoras) but weren’t given much attention to.
Years later, this concept was recognized, and the existence of a rational number was established. These irrational numbers are the set of real numbers, and these numbers cannot be described in a ratio format, a fraction format, or in the form of p/q where p and q are integers.
It is important to know that the denominator q is not equal to zero. The decimal expansion of any irrational number cannot be repeated or terminated. Thus, it is fine to say that irrational numbers are the exact opposite of rational numbers.
How to know if it’s an irrational number
Irrational numbers have particular characteristics and properties. It helps in differentiating them from real numbers. Some essential characteristics of these numbers are-
- They have a non-terminating and non-recurring decimal.
- Irrational numbers are also real numbers.
- When an irrational and rational number undergoes addition, the sum is always an irrational number.
- When you multiply any irrational number on a non-zero rational number, the product is an irrational number.
- Irrational numbers may or may not have the least common multiple.
- It is also important to know that when you add, subtract, multiply or divide two irrational numbers; the answer cannot be a rational number.
Thus, to identify an irrational number, it is important to know that it is a real number that cannot be expressed in the form of p by q, where p and q are both integers and q is not equal to zero.
Some examples of irrational numbers
- Pi is an example of an irrational number because the decimal value is never-ending. The value of pi is taken as 3.14.
- The Euclidean number is also an example of an irrational number.
Importance of rational and irrational numbers
Rational and irrational numbers are very important to solve different mathematical problems in the real world. If there were no such numbers, then theoretical calculations would be difficult.
These form an important part of the fundamental field of mathematics and sciences. They also have a property called completeness, and both rational and irrational numbers cannot exist without each other.
Difference between rational and irrational numbers
|Rational number||Irrational numbers|
|Rational numbers can be written in the form of p by q or a ratio of two numbers||Irrational numbers cannot be written as a ratio of p by q or as a ratio of two numbers|
|Rational numbers are always recurring and are finite||These numbers are non-terminating and non-repeating|
|The numerator and denominator are both whole numbers, and the denominator can never be zero||These numbers are always written in functional forms|
|Rational numbers are perfect squares||Irrational numbers give surds|
The number world is fascinating, and students must understand the concept of rational and irrational numbers clearly. Thus, rational numbers can be expressed as a fraction when the denominator is not zero, and these numbers cannot be expressed as a fraction of two integers such that the denominator is never zero.
Having a clear knowledge and learning the different properties make it easy to work with these numbers. Rational and irrational numbers thus are an important part of the number system world.