Constructing Real Numbers, Method 1 (see construction below)
If AB=1, BC=1, and ABD is right in the construction below, what is the exact value of AC?
If A is the center of the circle, what does the coordinate of C' tell you about the value of AC? Use your observation to approximate the value of the quantity in #1. (Click the "Show Values" box at the top to show more exact answers.)
Drag points B and C in order to locate on the number line. What is its approximate value?
Drag points B and C to locate other irrational numbers on the number line, and along with their approximate values. What irrational numbers can you locate? (You can also adjust the precision to see how that helps.)
Are there irrational numbers whose values you cannot locate on the number line using this construction?
Scroll down to Construction Method 2 for more fun!
Constructing Reals, Method 2 (see below)
The triangle with vertices A, B, and D is similar to the triangle with vertices A, C, and D. Write a statement showing their corresponding vertices. (You can click on Show Angles to help!)
Use your statement to write a ratio between sides AB, AC, and x.
If AB=1 and AC=2, what will the exact value of x be?
If A is the center of the circle, what does the coordinate of D' tell you about the value of x? Use your observation to approximate the value of the quantity in #3. (Click on Show Values at the top to show more exact answers.)
Drag point B to locate on the number line. What is its approximate value?
Drag points B and C to locate other irrational numbers on the number line, along with their approximate values. What irrational numbers can you find? (You can again adjust the precision to see if that helps.)
Are there irrational numbers whose values you still cannot find using this construction?
(For extra fun, click on Show Construction to see how the construction of C/d was done!)
(For even more fun, prove the similarity statement you wrote in #1.)
Scroll down to Construction Method 3 for more fun!
Construction Reals, Method 3 (see below)
Click the button Construct C to construct the circumference C of the given circle with diameter d. What is happening in this construction? What do D1, D2, and D3 represent?
Approximate the value of C using the number line, and also approximate C/d.
Move point B to change the diameter of the circle. Again, Construct C. What value did you find for C/d?
To make this easier, you can click on Show Values to see a better approximation of C.
Click on the Lock checkbox and adjust the diameter. What do you notice?
Now click on the Show C/d checkbox to see that value on the number line. Again, change B and construct C. What do you notice about C/d? What is this value commonly called?
Have we now found all of the real numbers?
(For extra fun, click on Construction of C/d to see how this construction was made.)