| Locate √10 on the number line
Answers
I’m supposing that square root of 10, written √10 , also referred to as radical 10. We can find its exact location on the number line with some help from Pythagoras. Mark the origin and the point 3 on the positive x-axis. Now erect a perpendicular segment of length 1 at the location x = 3. (This is a basic geometric construction that only requires a compass and straight edge, but if you aren’t familiar with those, just use a ruler and a fixed right angle.) Then connect the origin to the end of that vertical segment. That connection is the hypotenuse, h, of a right triangle whose legs are 3 and 1. Pythagoras’ formula guarantees that 32+12=h2. So h2 = 10, and therefore h = √10. Place the compass needle at the origin, and the pencil point at the end of the hypotenuse, and swing an arc down through the x-axis: You’ve just located √10. And it’s the exact location, not the decimal approximation that your calculator shows.
Now that you’ve located √10 , you can locate √11 by erecting a 1-unit perpendicular there, and repeating the process.