Pythagorean Triples – Part 1 Slideshow 38, Mathematics Mr. Richard Sasaki, Room 307

Objectives Understand the definition of a Pythagorean triple Be able to find Pythagorean triples Learn relations between Pythagorean triples and their rules and regulations through a project

Definition & Meaning What is a Pythagorean Triple? Three positive integers that satisfy 𝑎 2 + 𝑏 2 = 𝑐 2 , written in the form (𝑎, 𝑏, 𝑐). An example of a Pythagorean Triple is . (3, 4, 5) This is generally well known and refers to a “3−4−5 triangle”. If we learn (3, 4, 5) and if one number is missing, we can easily identify it as common knowledge. 3 5 4 Of course, the larger number must represent the hypotenuse. 3 ? (3, 5, 4) is not a Pythagorean Triple. 5 (4, 3, 5), however would be fine.

Factors & Multiples If (3, 4, 5) is a Pythagorean triple, then (6, 8, 10) is also a triple. But (1.5, 2, 2.5) is not because 𝑎, 𝑐∉ℤ. Conversely, if (4𝑥, 4𝑦, 4𝑧) is a triple, (2𝑥, 2𝑦, 2𝑧) and (𝑥, 𝑦, 𝑧) are also triples if 𝑥, 𝑦, 𝑧∈ℤ. A Pythagorean triple that cannot be simplified is called a primitive Pythagorean triple. Next, we will look at how to produce triples. You do not need to learn the given proof, it is simply for you to see where the following formulae come from. Note: Parts of the proof are removed as you will discover such elements of triples in your project.

Proof (Euclid’s Formulae for Triples) Consider a triple (𝑎, 𝑏, 𝑐) that satisfies 𝑎 2 + 𝑏 2 = 𝑐 2 where 𝑎, 𝑏, 𝑐∈ℤ. By manipulation, 𝑎 2 + 𝑏 2 = 𝑐 2 ⇒ 𝑐 2 − 𝑎 2 = 𝑏 2 ⇒ 𝑐−𝑎 𝑐+𝑎 = 𝑏 2 . Hence, 𝑐+𝑎 𝑏 = 𝑏 𝑐−𝑎 . As 𝑎, 𝑏, 𝑐∈ℤ, 𝑐+𝑎 𝑏 ∈ℚ. ∴ 𝑐+𝑎 𝑏 = 𝑚 𝑛 for some 𝑚, 𝑛∈ℤ. Also as 𝑐+𝑎 𝑏 = 𝑐−𝑎 𝑏 −1 , 𝑐−𝑎 𝑏 = 𝑛 𝑚 . The next step is algebra manipulation! Use the worksheet to try and manipulate the expressions in purple to make 𝑐 𝑏 = 𝑚 2 + 𝑛 2 2𝑚𝑛 and 𝑎 𝑏 = 𝑚 2 − 𝑛 2 2𝑚𝑛 .

Proof (Euclid’s Formulae for Triples) ① 𝑐+𝑎 𝑏 = 𝑚 𝑛 , ② 𝑐−𝑎 𝑏 = 𝑛 𝑚 Write ① as 𝑎= 𝑏𝑚 𝑛 −𝑐 and ② as 𝑐=𝑎+ 𝑏𝑛 𝑚 . Let’s substitute ① into ② and ② into ①. ② 𝑐−𝑎 𝑏 = 𝑛 𝑚 ① 𝑐+𝑎 𝑏 = 𝑚 𝑛 𝑐− 𝑏𝑚 𝑛 +𝑐 𝑏 = 𝑛 𝑚 𝑎+ 𝑏𝑛 𝑚 +𝑎 𝑏 = 𝑚 𝑛 2𝑐− 𝑏𝑚 𝑛 = 𝑏𝑛 𝑚 2𝑎+ 𝑏𝑛 𝑚 = 𝑏𝑚 𝑛 2𝑐= 𝑏𝑛 𝑚 + 𝑏𝑚 𝑛 2𝑎= 𝑏𝑚 𝑛 − 𝑏𝑛 𝑚

Proof (Euclid’s Formulae for Triples) 2𝑐= 𝑏 𝑛 2 𝑚𝑛 + 𝑏 𝑚 2 𝑚𝑛 2𝑎= 𝑏 𝑚 2 𝑚𝑛 − 𝑏 𝑛 2 𝑚𝑛 2𝑐 𝑏 = 𝑚 2 + 𝑛 2 𝑚𝑛 2𝑎 𝑏 = 𝑚 2 − 𝑛 2 𝑚𝑛 𝑐 𝑏 = 𝑚 2 + 𝑛 2 2𝑚𝑛 𝑎 𝑏 = 𝑚 2 − 𝑛 2 2𝑚𝑛 Assuming the numerators and denominators are in their simplest form… 𝒂= 𝒎 𝟐 − 𝒏 𝟐 , 𝒃=𝟐𝒎𝒏, 𝒄= 𝒎 𝟐 + 𝒏 𝟐 These are Euclid’s formulae and can be used to calculate Pythagorean Triples.

Answers (45, 108, 117) (55, 48, 73) (3, 4, 5) (7, 24, 25) Not Primitive Primitive Primitive Primitive (36, 160, 164) (16, 30, 34) (140, 48, 148) (75, 308, 317) Not primitive Not primitive Not primitive Primitive (32, 24, 40) (33, 56, 65) The results would just be negative. Not primitive Primitive

Project So now you know about how to find triples! We will start a project and all complete the same project in pairs or individually. As usual, each of you will at the end need to declare who did what. You may use a calculator if you wish to and may use a computer in your own time but not in class. This is not a research project, it is for discovery. The project must be written on paper. Multiple sheets are expected. Everything will be scanned so please use paper clips, not staples.

Structure & Expectations The project must contain the following, in order: 1. Title – Must be clear & Relate to Pythagorean Triples & student names / class 2. Introduction – A paragraph about what you will do and hope to do in your project 3. Hypothesis – What you expect to see, any relationships or patterns between 𝑎, 𝑏, 𝑐, 𝑚 and 𝑛

Structure & Expectations 4. Testing – Calculating results, minimum should be valid results (positive triples) for ∀ 𝑚, 𝑛≤10, 𝑚, 𝑛∈ ℤ + . Tables may be appropriate. 5. Results – Patterns that you can see, relationships, when results are primitive / not primitive, facts about primitive triples & not primitive triples, odd / even numbers 6. Conclusion – How was your hypothesis? Anything you wish you did / didn’t do / changed?

Structure & Expectations Scruffiness, crossings out and messy rubbings will be treated harshly. If you need to, make a first draft. A dark pencil or one of those cool pens that rub out is recommended. Be willing to add colour, avoid empty spaces and use a ruler when necessary! Use one-sided blank A4 white paper and guidelines as a guide only. This will be due on our first lesson back after the winter holiday. Only work with a friend if you are able to work together during the holiday!