What are Irrational Numbers?

What are Irrational Numbers?

Irrational numbers in the number system are real numbers. While this is to be noted that these numbers are quite contradictory to the real numbers as they cannot be represented in a simple fractional form or cannot be expressed in the form of a ratio. Thus, the irrational numbers are represented with a backslash which symbolizes the ‘set minus’ form. Suppose a and b are irrational numbers then it can be represented like this – a\b (‘\’ this is the backslash form). While this can also be represented like – a – b, this represents the difference between a set of real numbers and the set of rational numbers.

The calculation which is done with these numbers is a bit complicated, so for the arithmetic operations first convert the values under roots then do the calculations.

Definition of Irrational Numbers

An irrational number is a real number that cannot be expressed in the form of a ratio of integers. Also, the decimal expansion of these irrational numbers is not termination or recurring. This means the irrational numbers cannot be expressed in any way other than in root values. 

How Will You Spot an Irrational Number?

You can spot irrational numbers when you see the number cannot be expressed in the form of a/b where a and b are integers. √ 2 and √ 3 are irrational numbers. Another way of spotting these numbers is by their root overs. 

Is Pi an Irrational Number?

A pi is an irrational number as it is not terminating. The approximate value of pi is said to be 22/7.

What are Rational Numbers?

We have studied ‘integers’ in our previous classes, hence now we can deal with rational numbers. (Students are required to have acknowledged the integers before they start their concept of rational numbers).

Rational Numbers can be defined as the common types of numbers which are represented in a/b format. These rational numbers can take the formation of any type of integer. Rational Numbers and fractions seem equal, but they are in contrast with each other, students must not confuse between them. To know more let us check what rational numbers are and how they are different than fractions.  

How did Rational Numbers Originate?

“Ratio” is the term that gave the origin of the word ‘Rational Number’. With this relation of ratio and rational number, we can understand that rational number consists of the concept of ratio as well.

Define Rational Numbers

As previously discussed, the rational number is in the form of a/b, where ‘a’ and ‘b’ are the perfect integers to each other. These perfect integers are not equal to 0. Now, if we represent the rational numbers in a set form, then it is to be represented by Q.

While, if we represent the rational numbers in the fractional form, then this is to be noticed that both the numerator and denominator take the form of the integers, when this is happening, then the number is said to be a rational number.

Give Some Examples of Rational Numbers

As studied, if the number can be expressed in the form of a fraction where both the numerator and denominator are the integers of each other then the number is a rational number. Examples of rational numbers can be:

  • 9/2
  • -9/6
  • 0.6 or 6/10
  • -0.4 or -4/10

What are the Types of Rational Numbers?

Rational Numbers are of varied types. Considering only the fractions as the rational numbers will be a wrong assumption. Thus, let us check different types of rational numbers, which are as follows: 

  • There are rational numbers in the form of integers like -5, 0, 9, etc.
  • There are also fractions where the numerators and the denominators are the integers.
  • There are terminating decimals which also act as rational numbers, which are in a repetitive manner.

Properties of Rational Number

There are a few fain properties of rational number, they are as under:

  1. Closure Property
  2. Commutative Property
  3. Associative Property
  4. Distributive Property
  5. Identity Property
  6. Inverse Property

All these properties state different characteristics of rational numbers. 

This was all about Rational and Irrational Numbers. Further, if you want to study more, visit Cuemath to explore a range of mathematical concepts to feed your interest in the subject.

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