In the previous post we have discussed about How to tackle Irrational Number
and In today's session we are going to discuss about Rational and Irrational Numbers, Our number line consists of various types of numbers which play different types of roles in different areas. These numbers are real numbers, natural numbers, complex, whole, prime, rational and irrational numbers.
But here we are going to discuss about only rational and irrational numbers.
Rational and irrational numbers are totally opposite of each other. Because rational numbers are those which can be expressed in terms of a fraction or quotient like a/b, where a and b are integers and b is not equals to zero. On the other side irrational numbers cannot be represented in the form of a simple fraction like a/b.
In rational numbers, the decimal expansion of the number either terminates after some finite sequence of digits or repeats same finite sequence of digits over and again.
Whereas, in irrational numbers, the decimal expansion of the number continues forever without even repeating same finite sequence of digits again and again.
Rational numbers are dense in nature, by saying that we mean that in between any two integers on the real number line; there exists many rational numbers in between them.
Examples of rational numbers are: 1.333333….., 2.5, 4/5, etc.
And examples of irrational numbers are: e (Euler’s number), π (pi), √2 (square root of 2), φ (golden ratio), etc.
If we look towards the Cantor’s proof: it says that on the real number line almost all the numbers are irrational in nature, because as we know that our number line is a mixture of rational and irrational numbers, but real numbers are uncountable or infinite and rational numbers can be counted, so finally we can say that the remaining irrational numbers cannot be counted or are uncountable. Hence, the Cantor’s proof is verified.
In order to get help in the topics: rational and irrational numbers, how to find the area of a parallelogram and cbse previous years question papers, you can visit various Online Portals.
and In today's session we are going to discuss about Rational and Irrational Numbers, Our number line consists of various types of numbers which play different types of roles in different areas. These numbers are real numbers, natural numbers, complex, whole, prime, rational and irrational numbers.
But here we are going to discuss about only rational and irrational numbers.
Rational and irrational numbers are totally opposite of each other. Because rational numbers are those which can be expressed in terms of a fraction or quotient like a/b, where a and b are integers and b is not equals to zero. On the other side irrational numbers cannot be represented in the form of a simple fraction like a/b.
In rational numbers, the decimal expansion of the number either terminates after some finite sequence of digits or repeats same finite sequence of digits over and again.
Whereas, in irrational numbers, the decimal expansion of the number continues forever without even repeating same finite sequence of digits again and again.
Rational numbers are dense in nature, by saying that we mean that in between any two integers on the real number line; there exists many rational numbers in between them.
Examples of rational numbers are: 1.333333….., 2.5, 4/5, etc.
And examples of irrational numbers are: e (Euler’s number), π (pi), √2 (square root of 2), φ (golden ratio), etc.
If we look towards the Cantor’s proof: it says that on the real number line almost all the numbers are irrational in nature, because as we know that our number line is a mixture of rational and irrational numbers, but real numbers are uncountable or infinite and rational numbers can be counted, so finally we can say that the remaining irrational numbers cannot be counted or are uncountable. Hence, the Cantor’s proof is verified.
In order to get help in the topics: rational and irrational numbers, how to find the area of a parallelogram and cbse previous years question papers, you can visit various Online Portals.
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