Pythagorean quadruple
Let's say you have three friends, and you want to play a game with them to see who can make the longest line using sticks of different lengths. You notice that some of the sticks have the same length, so you decide to group them together.
Now, you have a group of sticks that are all the same length, called side A. You also have another group of sticks that are all the same length, called side B. Finally, you have a third group of sticks that are all the same length, called side C.
You lay out the sticks on the ground to create a square, with side A and side B forming the sides of the square and side C forming the diagonal. You notice that the length of side C is just the Pythagorean theorem in action:
Side C squared = Side A squared + Side B squared.
But here's the cool part: not all combinations of side A and side B will work to create a Pythagorean triple. In fact, some combinations will give you more than just a triple – they'll give you a quadruple!
A Pythagorean quadruple is a set of four numbers that satisfy the equation:
A^2 + B^2 + C^2 = D^2
where A, B, C, and D are all integers (whole numbers).
So, going back to our stick game, let's say you find a set of sticks with side A = 3, side B = 4, and side C = 5. If you plug those numbers into the Pythagorean theorem, you'll see that 3^2 + 4^2 = 5^2, so you have a Pythagorean triple.
But if you also find a stick with side D = 6, and you plug all four numbers into the equation A^2 + B^2 + C^2 = D^2, you'll see that 3^2 + 4^2 + 5^2 = 6^2. So, you have a Pythagorean quadruple!
Pretty neat, huh?
Now, you have a group of sticks that are all the same length, called side A. You also have another group of sticks that are all the same length, called side B. Finally, you have a third group of sticks that are all the same length, called side C.
You lay out the sticks on the ground to create a square, with side A and side B forming the sides of the square and side C forming the diagonal. You notice that the length of side C is just the Pythagorean theorem in action:
Side C squared = Side A squared + Side B squared.
But here's the cool part: not all combinations of side A and side B will work to create a Pythagorean triple. In fact, some combinations will give you more than just a triple – they'll give you a quadruple!
A Pythagorean quadruple is a set of four numbers that satisfy the equation:
A^2 + B^2 + C^2 = D^2
where A, B, C, and D are all integers (whole numbers).
So, going back to our stick game, let's say you find a set of sticks with side A = 3, side B = 4, and side C = 5. If you plug those numbers into the Pythagorean theorem, you'll see that 3^2 + 4^2 = 5^2, so you have a Pythagorean triple.
But if you also find a stick with side D = 6, and you plug all four numbers into the equation A^2 + B^2 + C^2 = D^2, you'll see that 3^2 + 4^2 + 5^2 = 6^2. So, you have a Pythagorean quadruple!
Pretty neat, huh?
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