Question:- 1. state whether the following statements are true or false. Jutify your answer.
(a). Every irrational number is a real number.
(b.)Every point on the number line is of the form √m,where m is a natural number.
(c.) Every real number is an irrational number.
(a)Consider the rational numbers and real numbers separately.
* The irrational numbers are that numbers which cannot be written in the form of p/q where p and q are integers and q≠ 0
(E.g.. √5, 6𝛑 , 0.022022022……)
* We already know that real number is a collection of rational numbers and irrational numbers .So,we got that ,every irrational number is a real number
(b)represent a number line. on number line ,we can represent negative as well as poitive numbers .
* positive numbers can be written in the form of √2,√2.2,√2.3,…….
* But we cannot write a negative number after taking square root of any number .
E.g. √-5=5i is a complex number (which you will be able to understand in higher class )
So,we got that every point on the number line is not in the form of √m where m is a natural number
(c) consider the irrational numbers and the real number separately .
* Irrational numbers are the numbers that cannot be written in the form of p/q where p and q are integers and q≠ 0.
* A real number is the collection of rational numbers and irrational numbers (e.g.√3, 3𝛑, 0.011011011)
So,we conclude that every irrational number is a real number.
Therefore .every real number is not an irrational number.
Quetion:- 2.Are the quare roots of all positive integers irrational ?If not , give an example of the square root of a number that is a rational number .
Solution:-As we know that sqiare root of every positive integer will not yield an integer (e.g.√3,√5,√7,…….)which are called irrational numbers.
But √9=3 which is an integer.
Therefore,we conclude that square root of every positive integer is not an irrational number.
Question:- 3.Show how √5 can be represented on the number line.
Solution:-Firt of all ,we have to draw a line segment AB of 2 unit on the number line.After then draw a prependicular line segment BC of 1 unit. After then join the points C and A to create a line segment AC.
Now, According to pythagorus theorem,
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