Estimating value of √37.
We know that
[tex]6^{2} = 36[/tex] and [tex]7^{2} =49[/tex], so
6 < √37 < 7
If we take the average of 6 and 7, we get
[tex]\frac{6+7}{2} = \frac{13}{2} = 6.5[/tex]
Since, [tex]6.5^{2} = 42.25[/tex]
6 < √37 < 6.5
If we take the average of 6 and 6.5 , we get
[tex]\frac{6+6.5}{2} = \frac{12.5}{2} = 6.25[/tex]
Since, [tex]6.25^{2} = 39.0625[/tex]
6 < √37 < 6.25
If we take the average of 6 and 6.25 , we get
[tex]\frac{6+6.25}{2} = \frac{12.25}{2} = 6.125[/tex]
Since, [tex]6.125^{2} = 37.515625[/tex]
6 < √37 < 6.125
If we take the average of 6 and 6.125 , we get
[tex]\frac{6+6.125}{2} = \frac{12.125}{2} = 6.0625[/tex]
Since, [tex]6.0625^{2} = 36.75390625[/tex]
6.0625 < √37 < 6.125
If we take the average of 6.0625 and 6.125 , we get
[tex]\frac{6.0625+6.125}{2} = \frac{12.125}{2} = 6.09375[/tex]
Since, [tex]6.09375^{2} = 37.1337890625[/tex]
6.0625 < √37 < 6.09375
If we take the average of 6.0625 and 6.09375 , we get
[tex]\frac{6.0625+6.09375}{2} = \frac{12.15625}{2} = 6.078125[/tex]
Since, [tex]6.078125^{2} = 36.943603515625[/tex]
6.078125 < √37 < 6.09375
If we take the average of 6.078125 and 6.09375 , we get
[tex]\frac{6.078125+6.09375}{2} = \frac{12.171875}{2} = 6.0859375[/tex]
Since, [tex]6.0859375^{2} = 37.03863525390625[/tex]
Therefore,
√37 ≈ 6.0859375.
And if we round it to the nearest tenth, we get
√37 ≈ 6.1
Locating √37 on number line.
In order to locate √37 on number line first draw a line 0 to 6 on number line.
Then draw a perpendicular line segment of 1 unit on number 6 on number line.
Join the number 0 on the number line by the top point of perpendicular line segment on number 6 we drew in above step.
Finally, draw a curve by taking radius as Hypotenuse of the right trinagle form in the diagram shown.
The curve would cut the number line exactly at √37 on number line.
