Rational points on the unit circle and the Pythagorean triples.

Trigonometry translated from it Greek roots means "the measure of triangles", most notably right triangles. The most important aspect of right triangles is the Pythagorean Theorem, usually stated as a² + b² = c².

The unit circle on the xy plane is given by the formula  x² + y² = 1. There are infinitely many points on the circle and if you pick one truly at random, it will very likely be irrational, which means neither the x or the y be written as p/q, where p and q are whole numbers. The only way it is possible for a point on the unit circle to be rational is to have two fractions of the form a/c and b/c where a² + b² = c². For example,

(3/5)² + (4/5)² = (5/5)² = 1.

In Excel, I generated 26 rational points on the unit circle, the largest denominator being 1,105. As you can see, there are a lot of gaps, but you can make out the general shape of a quarter circle.

Here's what it looks like with 662 rational points plotted. You can still see some gaps, notably near the horizontal lines at 0.8 and 0.6. If all the rational points were plotted, and there are infinitely many, you would not be able to see any gaps with the naked eye or at any magnification. In math, we say that the set of rational points is dense on the unit circle.

But here's one of the goofy things about infinity. There are also infinitely many points on the unit circle where both values are irrational AND infinitely many points where x is rational and y irrational AND infinitely many points where x is irrational and y rational. Let me write those sets out.


Set #1: { the set of points on the unit circle where both x and y are irrational}
Set #2: { the set of points on the unit circle where both x and y are rational}
Set #3: { the set of points on the unit circle where x is rational and y irrational}
Set #4: { the set of points on the unit circle where x is irrational and y rational}

All four of those sets are dense on the circle, meaning that in any segment of the circle no matter how small, there will be elements from all four of the sets.

Here's something even weirder about infinity. Sets #2, #3 and #4 are all the same size of infinity, but Set #1 is a bigger size of infinity, bigger than the other three combined.

Tomorrow, we move from goofy facts about infinity to more practical math aimed at answering the question "Is climate change or global warming real?"

The 3-4-5 triangle

The Pythagorean Theorem is often stated as


a² + b² = c²

means that the sum of the squares of the lengths of the two short sides (the ones that meet to create the right angle) is equal to the square of the length of the long side opposite the 90°, the side known as the hypotenuse.

If you pick two whole numbers at random to be the short sides, also known as the legs, the hypotenuse will be a square root of a whole number.

For example: If a = 1 and b = 3, 1² + 3² = 10, which means c² = 10. 10 is not a perfect square, so c is equal to the irrational number the square root of 10, which I will write as sqrt(10).

If we are looking at one digit numbers only, we have just one pair of legs that will add up to a perfect square. 3² + 4² = 9 + 16 = 25 = 5².

Three whole numbers that satisfy a² + b² = c² are called a Pythagorean triple. If we find such a triple, we can create infinitely more by multiplying all the sides by the same whole number.  Here are some examples.

Multiply 3-4-5 by 2: 6² + 8² = 10²
Multiply 3-4-5 by 3: 9² + 12² = 15²
Multiply 3-4-5 by 4: 12² + 16² = 20²
...

All of these triangles are similar and when discussing Pythagorean triples, the ones that are relatively prime are a special case.

The Pythagorean Theorem is named for the ancient Greek mathematician Pythagoras (570-495 BCE), but as often happens in math, this does not mean he was the first person ever to notice the pattern. There is strong evidence that the ancient Egyptians who built the pyramids understood the 3-4-5 triangle at the very least, and they lived thousands of years before Pythagoras. In many archeological digs, among the building tools are three sticks of lengths with the ratio 3:4:5. It is assumed that the builders used these when constructing walls to make sure the walls and the floors met at 90° angles.

There are other relatively prime Pythagorean triples; in fact, there are infinitely many. The next two smallest ones are 5² + 12² = 13² and 8² + 15² = 17².

Tomorrow, we will discuss how to create all the Pythagorean triples, both those that are relatively prime and those that aren't.

Two picture proofs of the Pythagorean Theorem.

Here are two squares that are the same size. In the left square, we will label the blue square a² and the red square b².  The two white rectangles both have area ab. Together, they show the famous "middle term" representation of

(a + b)² = a² + 2ab + b²

The square on the right has four white right triangles that are the equivalent of the two white rectangles sliced diagonally, where the diagonal is the hypotenuse, which we usually label c. The yellow square with the gap in it has area c².  Since the squares are the same size we get

c² + 4(½ab) = a² + 2ab + b²  Next step: get rid of the parentheses

c² + 2ab = a² + 2ab + b²  Next step: subtract 2ab from both sides

c²  = a² + b²  and we are done.

Second proof just using the yellow square and the gap inside.

The gap in the middle of the yellow square is (a - b)² = a² - 2ab + b². The four yellow triangles are 2ab, exactly the same as the four white triangles. That means this picture tells us

c² = a² - 2ab + b² + 2ab

So we combine like terms to get
 
c²  = a² + b²  and once again, Q.E.D., the Latin abbreviation for "that which has been demonstrated".

Tomorrow: number theory and Pythagorean Theorem.

The Pythagorean Theorem and its connection to trigonometry.

When teaching math, if I ask a class "What is the Pythagorean Theorem?", invariably someone will eventually pipe up the statement

a² + b² = c²

This is correct as far as it goes, but asking what a, b and c stand for is not always remembered.

Ignoring what names we assign the sides the statement of The Pythagorean Theorem is:

The sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse.

Quite often, the phrase "of the length(s)" is removed, since talking about squaring something implies the thing is a number and the number associated with a side of a triangle is the length almost always.

Using the names from the picture above, the equation would change to adj² + opp² = hyp².

So far, so good.

Let's divide all the sides by the hypotenuse.

adj/hyp = cosine
opp/hyp = sine
hyp/hyp = 1

These are still sides of a right triangle with 1 being the hypotenuse, so the equation changes to a form the is the main trigonometric identity sin² + cos² = 1. When I teach the class, I call this The Trigonometric Identity since all the other trig identities are just re-arrangements of this.

Tomorrow, I'll present a proof of the Pythagorean Theorem using basic facts from geometry, the areas of right triangles and the areas of squares.