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Plotting any Irrational Numbers On a Number Line

Number System II - Concepts

Converting Non Terminating Repeating Decimals into Fractions

Number Line using Successive Magnification

Plotting any Irrational Numbers On a Number Line

Rationalization

Class - 9th IMO Subjects

Concept Explanation

Plotting any Irrational Numbers On a Number Line

Plotting any Irrational Numebrs On a Number Line:

For any positiv real number x, we have

$large sqrt{left ( frac{x+1}{2}right )^{2}-left ( frac{x-1}{2} right )^{2}}=sqrt{frac{x^{2}+2x+1}{4}-frac{x^{2}-2x+1}{4}}=sqrt{frac{4x}{4}}=sqrt{x}$

Therefore, to find the positive square root of a positive real number, we may follow the following algorithm.

ALGORITHM:

STEP I Obtain the positive real number x(say)

STEP II Draw a line and mark a point A on it.

STEP III Mark a point B on the line such that AB = x units.

STEP IV From point B mark a distance of 1 unit and mark the new poiint as C.

STEP V Find the mid-point of AC and mark the point as O.

STEP VI Draw a circle with centre O and radius OC>

STEP VII Draw a line perpendicular to AC passing through B and intersecting the semi-circle at D. Length BD is equal to $large sqrt{x}.$

Justification: We have,

AB = x units and BC = 1 unit.

$large therefore$ AC = (x + 1) units

$large Rightarrow ;;OA = OC = frac{x+1}{2};units$

$large Rightarrow ;; OD= frac{x+1}{2};units$ $large [because ;OA=OC=OD]$

Now, $large OB=AB-OA=x-frac{x+1}{2}=frac{x-1}{2}$

Using Pythagoras Theorem in $large Delta OBD,$ we have

$OD^{2}=OB^{2}+BD^{2}$

$Rightarrow ;;BD^{2}=OD^{2}-OB^{2}$

$Rightarrow ;;BD^{2}=left ( frac{x+1}{2} right )^{2}-left ( frac{x-1}{2} right )^{2}$

$Rightarrow ;;BD=sqrt{frac{(x^{2}+2x+1)-(x^{2}-2x+1)}{4}}=sqrt{frac{4x}{4}}=sqrt{x}$

This shows that $sqrt{x}$ exists for all real numbers x > 0.

In order to find the position of $sqrt{x}$ on the number line, we consider BC as the number line, with B as the origin to represent zero. Since BC = 1 so, C represents 1. Now, mark points $C_{1},C_{2},C_{3},......$ such that $CC_{1}=BC$ = 1, $C_{1}C_{2}=BC=1$ and so on. Clearly, $C_{1}C_{2},C_{3},.....$ represent 2, 3, 4,... respectively.

Now, draw an arc with centre at B and radius equal to BD. Suppose this arc cuts the number line BC with B as the origin at E. Then, BE = $sqrt{x}$ . Consequently, E will represent $sqrt{x}$ .

Sample Questions

(More Questions for each concept available in Login)

Question : 1

Can all irrational numbers be plotted on a number line?
A. No	B. Yes	C. May be or may not be	D. None of these
Right Option : B
View Explanation

Question : 2

Is it possible to say that $sqrt {infty }$ is defined on number line?
A. Yes	B. No	C. May be or may not be	D. None of the above
Right Option : B
View Explanation