Let x be an irrational number then what can be said about x2?
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.
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 βNβ. 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,β¦β¦β¦β¦β¦}
- 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 βWβ. The set can be represented as W={0,1,2,3,4,5,β¦β¦β¦β¦β¦β¦}
- 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 βQβ.
- 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 βPβ.
- 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 doesnβt include fractions and decimals. The set of integers is represented by βZβ. Example: Z={β¦β¦β¦..,-5.-4,-3,-2,-1,0,1,2,3,4,5,β¦β¦β¦β¦.}
- 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 βRβ.
- Complex numbers: They are a set of numbers that include imaginary numbers are complex numbers. It can be expressed as a+bi where βaβ and βbβ are real numbers. It is denoted by βCβ.
What are Irrational Numbers?
Before explaining irrational numbers, letβs 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 βaβ is the numerator and βbβ is the denominator, where a and b are integers and b β 0. Some example, the fractions 1/5 and β2222/8 are both rational numbers. All integers are included in the rational numbers and we can write any integer βzβ 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 βPβ.
The number which is not rational or we cannot write in form of fraction a/b is defined as Irrational numbers. Here β2 is an irrational number, if calculated the value of β2, it will be β2 = 1.14121356230951, and will the numbers go on into infinity and will not ever repeat, and they donβt ever terminate. It canβt 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 β3 while calculating, β3 = 1.732050807. The pattern received is non-recurring and non-terminating. So β3 here is also an irrational number.
But in the case of β9 here β9 = 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 = β2
therefore x2 = (β2)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, Ο, 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.
Ο is the only irrational here which canβt 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.