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Let x be an irrational number then what can be said about x2?

  • Last Updated : 30 Sep, 2021

Numerals are the mathematical figures used in financial, professional as well as a social field in the social world. The digits and place value in the number and the base of the number system determine the value of a number. Numbers are used in various mathematical operations as summation, subtraction, multiplication, division, percentage, etc which are used in our daily businesses and trading activities.  

Numbers are the mathematical figures or values applicable for counting, measuring, and other arithmetic calculations. Some examples of numbers are integers, whole numbers, natural numbers, rational and irrational numbers, etc.

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The number system is a standardized method of expressing numbers into different forms being figures as well as words. It includes different types of numbers for example prime numbers, odd numbers, even numbers, rational numbers, whole numbers, etc. These numbers can be expressed in the form on the basis of the number system used.  



The number system includes different types of numbers for example prime numbers, odd numbers, even numbers, rational numbers, whole numbers, etc. These numbers can be expressed in the form of figures as well as words accordingly.

For example, the numbers like 40 and 65 expressed in the form of figures can also be written as forty and sixty-five.

The elementary system to express numbers is called a number system. It is the standardized method for the representation of numerals in which numbers are represented in arithmetic and algebraic structure.

Types Of Numbers

There are different types of numbers categorized into sets by the number system. The types are described below:

  • Natural numbers: Natural numbers counts from 1 to infinity. They are the positive counting numbers that are represented by ‘N’. It is the numbers we generally use for counting. The set of natural numbers can be represented as N = {1,2,3,4,5,6,7,……………}
  • Whole numbers: Whole numbers count from zero to infinity. Whole numbers do not include fractions or decimals. The set of whole numbers is represented by ‘W’. The set can be represented as W={0,1,2,3,4,5,………………}
  • Rational numbers: Rational numbers are the numbers that can be expressed as the ratio of two integers. It includes all the integers and can be expressed in terms of fractions or decimals and is represented by ‘Q’.
  • Irrational numbers: Irrational numbers are numbers that cannot be expressed in fractions or ratios of integers. It can be written in decimals and have endless non-repeating digits after the decimal point. They are represented by ‘P’.
  • Integers: Integers are the set of numbers including all the positive counting numbers, zero as well as all negative counting numbers which count from negative infinity to positive infinity. The set doesn’t include fractions and decimals. The set of integers is represented by ‘Z’. Example: Z={………..,-5.-4,-3,-2,-1,0,1,2,3,4,5,………….}
  • Decimal numbers: Any numeral value that consists of a decimal point is a decimal number. It can be expressed as 2.5,0.567, etc.
  • Real number: The set of numbers that do not include any imaginary value and are constituent of all the positive integers, negative integers, fractions, and decimal values are real numbers. It is generally denoted by ‘R’.
  • Complex numbers: They are a set of numbers that include imaginary numbers are complex numbers. It can be expressed as a+bi where “a” and “b” are real numbers. It is denoted by ‘C’.

What are Irrational Numbers?

Before explaining irrational numbers, let’s have a brief about rational numbers. The numbers which can be expressed as a ratio between two integers are defined as rational numbers. It is the form of a/b, here “a” is the numerator and “b” is the denominator, where a and b are integers and b ≠ 0. Some example, the fractions 1/5 and −2222/8 are both rational numbers. All integers are included in the rational numbers and we can write any integer “z” as the ratio of z/1.

Irrational numbers are numbers that cannot be expressed in fractions or ratios of integers. It can be written in decimals and have endless non-repeating digits after the decimal point. They are represented by ‘P’.

The number which is not rational or we cannot write in form of fraction a/b is defined as Irrational numbers. Here √2 is an irrational number, if calculated the value of √2, it will be √2 = 1.14121356230951, and will the numbers go on into infinity and will not ever repeat, and they don’t ever terminate. It can’t be written in a/b form where b is not equal to zero. The resultant value is actually non-terminating and there is no pattern in the digits after the decimal. These types of numbers are called irrational numbers.

Consider √3 while calculating, √3 = 1.732050807. The pattern received is non-recurring and non-terminating. So √3 here is also an irrational number. 



But in the case of √9 here √9 = 3 this is a rational number. Square root of perfect square will always be a rational number. 

The square root of any number which is not a perfect square will always be an irrational number. Irrational numbers can have a decimal expansion that never ends and does not repeat. 

Let x be an irrational number then what can be said about x2?

Solution: 

As per the Question: x is an irrational number 

So lets assume x = √2 

therefore  x2 = (√2)2

                  = 2 which is Rational number 

Hence the x2 will be Rational number if x is an irrational number 

Similar Questions

Question 1: Which of these are Irrational numbers?

1.5, π, 1/3, 0.857857

Solution:

The numbers that cannot be expressed as fraction are irrational numbers. So here 1.5 can be written as 3/2 and 1/3 itself a fraction, 0.857857 can be written as 8578/1000 .so these are rational numbers. 

π is the only irrational here which can’t be expressed as fraction. 

Question 2: Identify is 7.5 a rational or irrational number?

Solution:

The number 7.5 is a rational number. Since rational numbers can also be expressed as decimals with repeating digits after the decimal point. Here we can write 7.5 as 75/10 and further write it as 15/2 = 7.5 so its a rational number.

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