Question

Represent ${\displaystyle \sqrt{6},}$ ${\displaystyle \sqrt{7},}$ ${\displaystyle \sqrt{8}}$ on the number line.

Answer 1

We are asked to represent ${\displaystyle \sqrt{6},}$ ${\displaystyle \sqrt{7}}$ and ${\displaystyle \sqrt{8}}$on the number line

We will follow certain algorithm to represent these numbers on real line

We will consider point A as reference point to measure the distance

(1) First of all draw a line AX and YY’ perpendicular to AX

(2) Consider AO = 2 unit and OB = 1 unit, so

${\displaystyle AB=\sqrt{2^2+1^2}=\sqrt{5}}$

(3) Take A as center and AB as radius, draw an arc which cuts line AX at A₁

(4) Draw a perpendicular line A₁B₁ to AX such that A₁B₁ = 1 unit and 

(5) Take A as center and AB₁ as radius and draw an arc which cuts the line AX at A₂.

Here 

AB₁ = AA₂

${\displaystyle =\sqrt{{\text{AA}}_1^2+A_1{B}_1^2}}$

${\displaystyle =\sqrt{{\left(\sqrt{5}\right)}^2+1}}$

${\displaystyle =\sqrt{6}}$ unit

So AA₂ = ${\displaystyle \sqrt{6}}$ unit

So A₂ is the representation for ${\displaystyle \sqrt{6}}$

(1) Draw line A₂B₂ perpendicular to AX

(2) Take A center and AB₂ as radius and draw an arc which cuts the horizontal line at A₃ such that 

AB₂ = AA₃

${\displaystyle =\sqrt{{\text{AA}}_2^2+A_2{B}_2^2}}$

${\displaystyle =\sqrt{{\left(\sqrt{6}\right)}^2+1}}$

${\displaystyle =\sqrt{7}}$ unit

So point A₃ is the representation of ${\displaystyle \sqrt{7}}$

(3) Again draw the perpendicular line A₃B₃ to AX

(4) Take A as center and AB₃ as radius and draw an arc which cuts the horizontal line at A₄ 

Here;

AB₃ = AA₄

${\displaystyle =\sqrt{{\text{AA}}_3^2+A_3{B}_3^2}}$

${\displaystyle =\sqrt{{\left(\sqrt{7}\right)}^2+1^2}}$

${\displaystyle =\sqrt{8}}$ unit

A₄ is basically the representation of ${\displaystyle \sqrt{8}}$