JR Ellis, October 1992

Department of Mathematics

University of Utah

A thesis submitted to the faculty in partial fulfillment of the requirements for the degree of Master of Science

- Abstract
- Acknowledgment
- Introduction
- Arithmetic
- Square Addition
- Square Addition Tessellation
- Trigonometric Definitions
- Square Subtraction
- Square Root Construction
- Square Multiplication
- Square Division
- Square Measure
- Square Distributive Property
- Square Multiplicative Associative Property
- Cube Root
- Summary
- References

This idea is applied to forming a complete ordered field with the elements of the set of squares. After showing constructions for combining the squares, pictures which show the group properties follow. Finally, limit theory for an infinite sequence is shown graphically as a rectangular prism is iteratively changed into a cube.

The intent of this paper is to suggest that teaching math in elementary schools can be more conceptual and creative. By using approaches, as outlined here, education can be more effective and stimulating to young developing minds.

As I played with these ideas, I developed a simple system to see the properties of the real number system by using squares in order to visualize these properties. This thesis includes a video, which animates the square root construction and proof, and demonstrates the advantages and potential of computer graphics in education. The video was produced in the Art and Film department, with the help, cooperation and resources of the Mathematics and Education departments.

It is hoped that this idea will inspire a healthier approach to education of people learning mathematics. High school students are routinely taught how to factor the difference of two squares. However, the geometric interpretation of this equation is completely ignored. When I realized that the square of the hypotenuse of a right triangle equals the sum of the square of its legs really referred to four-sided squares, the square root construction followed easily. I was excited with the awareness. I was also sad that I had plodded through algebra without being exposed to its roots in geometry. I hope the pictures in this presentation will bring the equations to life.

I would like to thank Professor Robert E. Barnhill from the University of Utah for allowing me to pursue this path as a thesis and a film; and Professor Bevan K. Youse from Emory University, for introducing me to group theory and analysis and other recent private communications. And also, I would like to thank Dr. Andrew Mitz from NIH for giving me suggestions which helped me to write this thesis.

A complete ordered field is formed by using squares as the elements of the set, with the operations of addition and multiplication. Constructions for these operations are shown, as well as division and subtraction. And there are pictures for properties like inverse, distributive, and multiplicative associativity.

Finally, the square root construction is extended by applying the procedure to an original iterative cube root construction. The limit theory for this infinite sequence uses nothing more advanced than an equation for the sum of a geometric progression. As with many of the geometric assertions in this paper, the algebraic proof is also included

The paper starts with an arithmetic application to the geometric interpretation of the difference to two squares. This shortcut can be used in normal arithmetic calculations.

Let

**17 * 23 = (20 - 3) * (20 + 3) = 20**2 - 3**2 = 400 - 9 = 391**

If the squares from one to one hundred are already memorized then
two digit multiplication will be much faster. There is another advantage
because there are fewer numbers to keep track of than with regular
multiplication. The square may be computed with an arithmetic application
of the binomial theorem for

**n = 2, (a + b)**2 = a**2 + 2ab + b**2**. (sorry... picture missing )

**17**2 = (20 - 3)**2 = 20**2 + 2(-3)(20) + (-3)**2 = 400 - 120 + 9 = 289**

The advantages of this technique maybe illustrated with another example. Here the challenge is to solve the following multiplication problem without using paper and pen:

**24 * 26 = (25 -1)(25+ 1) = 25**2 -1**2**

Now we may forget **24 **and** 26 **and
remember** 25 **and** 1**. Next** 25**2 = 625.
**And Finally, **625 - 1 = 624. **At no time were there more
than six digits to keep track of at one time. First

The standard way to multiply 24 and 26 is to first multiply **26
* 4 = 104 **and then **26 * 20 = 520**.
And in order to add **(104 + 520)**, there are more numbers to remember
at one time than the previous proposed shortcut.

- Let
**a**,**b**, and**c**represent the sides of the squares. This construction may be represented by the trigonometric definitions in an algebraic equation. - Since
**cos theta = (a/c)**, then**theta = arccos (a/c)** - And since
**sin theta= (b/c)**, then**b = c * arcsin theta** - Substituting for
**theta**,**b = c * sin [arccos (a/c)]** - So
**c**2 - a**2 = b**2 = [c * sin {arccos (a/c)}]**2**

**Point 2:** The midpoint of line segment **b1** ( **a2** is the
average of the length and width.

**Point 3:** The midpoint of line segment **a2**

**Point 4:** The intersection of a circle having its center at point **3** and radius
equal to line segment **a3,** with a circle having its cetner at point **2** and radius
equal to line segment **b2**

Line segment **a4** represents the side of the square whose area equals that of the rectangle.
When the width of the rectangle is the unit width, the length of **a4** is the square root of line
segment **ab.** To find the square root of a measure, start with a unit width rectangle having its
length equal to that measure. Then use this construction to change the rectangle into a square.
The side fo the square is the square root of the length of the rectangle.

Geometrically, the construction is easy to
explain. Rectangle **THE** on the top left is equal in area to square
**Z **on the bottom right. In the first step of the construction,
rectangle **THE** is divided into 3 parts. Point **1** marks square
**E, **and** **Point** 2** divides the remainder into 2 equal
rectangles:** T** and **H**.
Points **A**, **B**, **1**, **2**, **3**, and **4** correspond
to the square root construction from the previous page.
In the second step, the bottom picture shows rectangle **THE** transformed
by separating rectangle** T **and attaching it to square** E**
as shown. Rectangle **THE** is now an irregular hexagon, and also
the difference of two squares. (**AC** and **A2** are equivalent
since **E** is a square, and **H** and **T** are equivalent
rectangles.)** **Finally Square **Z** is resolved through the
subtraction construction already shown.

- Draw the two circles
**(A, B)**and**(B, A)**, then mark point**V**as the left intersection point. - Draw circle
**(V, A)**and mark point**W**at the intersection with circle**(B, A).**.**AV**= BA - Draw circle
**(W, V)**and mark point**C**at the intersection with circle**(B, A).****AC**is the diameter of circle**(B, A)**and therefore twice the length of**AB**. - Draw circle
**(C, A)**and mark points**S**and**T**at the intersections with circle**(A, B).****CA**is perpendicular to**ST**. - Draw circles
**(S, A)**and**(T, A)**, and then mark their intersection at point**M**.**ST**is perpendicular to**AM**therefore**M**lies on**AC**. and triange**CVA**is similar to triangle**VMA**. So**AV : CA = AM : AV**therefore**AM = 1/2 AB**.

Here we are given the point **P** and the circle **K**
with center **O**. Construct the circle through **P** with center
**O**, and from **P** mark off the radius three times (points
**Q**, **R**, **S**) on the circumference, and also mark off
the distance **ST = r**.
Then **PT = sqare root { 4p**2 -r**2 }**.
Draw the circles about **P **and** S **with radius
**PT**, thus obtaining the point **U**. Draw the circle with
center **U **and radius **PR = p [ square root { 3 } ]** , thus obtaining
the points **X **and **Y** (in all, eight circles are necessary).

**Given: OP = OQ = OR = OS =
PQ = QR = RS = p, UX = PR, UP = PT, OX = TS = r,
angle UOP** is a right angle, **X** lies on the
circle **(O, r)**.

so line segment

To measure a line, count off unit lengths as shown. Then proceed to
obtain the next significant digit from the remainder. Construct a
line ruler of ten arbitrary units, for base ten. The units may be
different than the unity square. Draw a circle using the length of
the ruler as radius and the center of the circle as the endpoint of
the last whole unit that fit into the length of the unit rectangle.
Find Point **T **where the circle intersects line **RS**.
Draw lines parallel to **RS** from each calibration point on the
ruler to obtain the next significant digit of the measure. Repeat
this procedure for the next significant digit. In this case, it would
be obscure to find that 6.25 is the measure of square **A**, because
the construction would be taking place in an area which is too small
to be practical.

This construction may be applied to measure the line on the side of the square that was obtained in the square root construction.

The construction for the cube root works by changing a unit rectangular
prism into a cube. The starting rectangular prism has unit squares
on two opposite faces, while the other four are unit rectangles, having
length equal to **a**. This construction is iterative, producing
a sequence that converges to a cube with a side equal to the cube
root of **a**.

At each stage of the sequence, transform and mold the prism by constructing a square from the unit rectangle. This transformation will retain the area, while producing, once again, a rectangular prism where two opposite faces are squares. Eventually, it approaches a cube.

The proof which is not shown uses limit theory of series and sequences to show that transformation of the rectangular prism in this manner does indeed converge to a cube. Limit theory may be used since the set of squares with the operations of multiplication and addition is isomorphic to the set of real numbers.

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