Number System: 9 Math

NCERT Exercise1.2 - Irrational Number: 9th math

A number `s` is called IRRATIONAL, if it cannot be written in the form of `p/q` where p and q are integers and `q!=0`.

Example: `sqrt2,\ sqrt3,\ sqrt(15), pi` 0.101101110 . . . . . . . . etc.

Real Number

All Rational Numbers and Irrational Numbers together are called Real Number. Real Number is defined by letter R.

Therefore, a Real Number is either rational or irrational.

The Real Number Line

It can be said that every Real Number is represented by a unique point on the Number Line. Also, every point on the number line represents a unique Real Number.

Thus, the Number Line is called The Real Number Line.

In the 1870s two German mathematicians G. Cantor and R. Dedekind, showed that: Corresponding to every real number, there is a point on the real number line, and corresponding to every point on the number line, there exists a unique real number.

Solution of NCERT Exercise 1.2 (mathematics class nine)

Question (1) State whether the following statement are true or false. Justify your answers.

(i) Every irrational number is a real number.

Answer: True

Explanation: Since all rational and irrational numbers together with comprises Real Number, thus every irrational number is a real number.

(ii) Every point on the number line is of the form `sqrt\m`, where m is a natural number.

Answer: False

Explanation: A number line have negative and positive both types of numbers. And root of a natural number never be a negative number. Thus, the given statement is false.

(iii) Every real number is an irrational number

Answer: False

Explanation: All rational and irrational numbers collectively are known as Real Number. Thus every irrational number is a real number, but every real number cannot be an irrational number.

Question (2) Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.

Answer: No. Square roots of all positive integers are not irrational.

Example: `sqrt4` = 2

Here, 4 is a positive integer and its square root, 2 is a rational number. Because 2 can be written in the form of `p/q` i.e. `2/1`

Question (3) Show how `sqrt5` can be represented on the number line.

Solution:

OQ = OE `=sqrt5`

Steps to draw `sqrt5` on a number line:

(a) Take OA = 1 unit on a number line.

(b) Draw `AB_|_OA` where OA = AB = I unit

Now, in `triangle OAB, /_A = 90^0`

∴ According to Pythagoras theorem,

OB² = OA² + AB²

= 1 ² + 1²

= 1 + 1 = 2

`:. OB = sqrt2`

(c) Draw a perpendicular CB on OB where CB = OA = 1 unit

Now, in `triangle OBC, /_B = 90^0`

∴ According to Pythagoras theorem,

OC² = OB² + BC²

`=(sqrt2)^2 + 1^2`

= 2 + 1 = 3

`:. OC = sqrt3`

(d) Draw a perpendicular DC on OD where CD = OA = I unit

Now, in `triangle OCD, /_C = 90^0`

∴ According to Pythagoras theorem,

OD² = OC² + DC²

`= (sqrt3)^2 + 1^2`

=3 + 1 = 4

`:. OD^2 = sqrt4`

⇒ OD = 2

(e) Draw a perpendicular ED on OD, where ED = OA = I unit

Now, in `triangle ODE, /_D = 90^0`

∴ According to Pythagoras theorem,

OE² = OD² + ED²

= 2² + 1 ²

= 4 + 1 = 5

`:. OE = sqrt5`

Now, cut a line OA = OE `=sqrt5` on number line

Thus, `OQ = sqrt5`

Question (4) Classroom activity (Constructing the 'square root spiral'): Take a large sheet of paper and construct the 'square root spiral' in the following fashion. Start with a point O and draw a line segment OP₁ of unit length. Draw a line segment P₁P₂ perpendicular to OP₁ of unit length (see Fig). Now draw a line segment P₂P₃, perpendicular to OP₂. Then draw a line segment P₃P₄ perpendicular OP₃. Continuing in this manner, you can get the line segment P_n–1P_n by drawing a line segment of unit length perpendicular to OP_n–1. In this manner, you will have created the points P₂, P₃, . . . ., P_n, and joined them to create a beautiful spiral depicting `sqrt2, sqrt3, sqrt4`, . . .

Solution:

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