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Understanding Rational And Irrational Values In Mathematics

As we have all read about numbers, fractions and decimals in our previous years in school. Well, the different types of numbers such as natural number, whole number etc. 

In this particular article, we’ll be reading about rational and irrational numbers with examples in detail. Specifically, their properties and characteristics and how you can differentiate between them.

In this lesson, we’ll be taking a different approach by studying the properties one by one rather than in tabular format. So let’s get started with a little brushup of numbers and their basic types before moving to rational and irrational sets.

Various types of numbers can be categorised into sets. You may have heard of or maybe know about the sets listed below.

1. Natural Numbers: The natural number system begins with 1. N = 1, 2, 3, 4, 5,…

2. Whole Numbers: The set of whole numbers begins with 0 and continues with W = 1, 2, 3,…

3. Integers: An integer is a collection of whole numbers and negative values (natural numbers with the minus sign).

–1, 0, 1 …, I or Z =…, –1, 0, 1 ..

Now, moving on to our topic;

As we can all wonder what kind of number these will be categorised in? Well, the answer is simple, it’s a set of rational numbers.

So, what does it mean to have a rational number?

By definition, it’s any number in the form (p/q), where p and q are both real numbers and q isn’t zero.

Similarly, numbers that are not logical are referred to as irrational in the case of irrational numbers. Irrational numbers can be stated in decimal but not fractional form, suggesting that they can’t be expressed as a ratio of two integers.

After the decimal point, rational numbers have an endless amount of non-repeating digits.

In a more elaborate sense, let’s understand what these two types of numbers exactly mean.

Rational numbers:

A rational number is a number made up of integers that can be stated in p/q form. It’s indicated by (Q), which is a real number (R), and its participant integers (Z) are natural numbers (N). The integers in this situation are made up of two parts: the denominator q, which cannot be 0, and the numerator p.

Every participant’s integer is also a rational number, such as 7=7/1. The p/q statement either has a finite number of digits or displays a pattern or sequence that is repeated.

If the decimal series is repeating or terminating, it is known as a rational number. In the same way, adding and multiplying distinct integers of a rational number produces a rational number.

Well, for the case of irrational numbers:

Irrational numbers are those that can’t be easily divided into fractions of natural numbers and integers. There is no recurring or finite decimal expansion for irrational numbers. Irrational numbers include surds and peculiar numbers, for example. The irrational number Pi () is the most common. 

A surd is an imperfect square or cube that cannot be simplified anymore by deleting the square or cube root.

Now that we have revisited the definitions in detail let’s understand the classification terminology and parameters that divide a rational number from an irrational number.

Let’s look at some examples of how to distinguish between rational and irrational numbers. All integers, fractions, and repeating decimals are rational numbers, and they can be stated infractions.

The following conditions can be used to identify rational numbers:

* It’s written as a/b, where b0 is the unit of measurement.

* The a/b ratio can also be simplified and represented in decimal form.

Irrational numbers are those that are not exactly traditional. Irrational numbers can be stated in decimal but not fractional form, suggesting that they can’t be expressed as a ratio of two integers.

Rational numbers have an infinite number of non-repeating digits after the decimal point.

We’ll now look at some key features of rational and irrational numbers.

* A rational number is always the sum of two or more rational numbers. This means that under addition, the set of rational numbers is closed.

I.e. If two rational numbers are x and y, then x y and y x are likewise rational numbers.

* A rational number is always the difference between two rational numbers. This means that subtraction closes the set of rational numbers.

I.e. If two rational numbers are x and y, then x – y and y – x are both rational numbers.

* A rational number is always the product of two or more rational numbers. This means that multiplication closes the set of rational numbers.

I.e. If two rational numbers are x and y, then xy and yx are likewise rational numbers.

* When a rational number is divided by a non-zero rational number, the result is always a rational number. If the divisor is less than zero, the set of rational numbers is closed under division.

I.e. When x and y are any two rational numbers, then y 0 and x 0 are both rational numbers.

*  Addition, subtraction, multiplication and division of any two irrational numbers is not necessarily irrational.

Here are some pointers to help you comprehend the ideas of rational and irrational numbers.

The growth of the decimal

Rational numbers are either terminating or non-terminating, and they repeat themselves.

Examples:

(1/8) = 0.125 is Terminating decimal expansion

 (4/7) = 0.5714… is Non-terminating recurring decimal expansion

  • Irrational numbers are non-terminating and non-recurring.

Example:   (22/7) = 3.1415… is Non-terminating, non-recurring decimal expansion

Terminating decimal: A terminating decimal does not continue. After the decimal point, a terminating decimal will have a finite number of digits.

Non-terminating and recurring (repeating) decimal: A non-terminating and repeating decimal is when the digits repeat themselves indefinitely. Fractions can be used to represent decimals of this type.

Non-terminating and non-recurring (non-repeating) decimal: A non-terminating and non-repeating decimal is one in which no collection of digits repeats itself indefinitely. This sort of decimal cannot be expressed as a fraction.

Conclusion:

Finally, with the help of these key facts, we can clearly understand the distinction between rational and irrational numbers:

  • Perfect squares terminating decimals repeating decimals = rational numbers
  • Irrational numbers are made up of surds and non-repeating decimals.

All you have to do is take a quick look at a number. It’s a rational number if it’s a perfect square, terminating decimal, or repeating decimal. A surd or non-repeating decimal, on the other hand, indicates that the number is irrational.

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