### 628, Tau, Pi, and Level 6

The circumference of a circle with a radius of one is approximately 6.28. That’s an important enough number that it has been given the symbol “τ ” which is pronounced “tau”. τ looks a little like half of the number π, but τ = 2π.

Some people think we should get rid of π and only use τ. Other people feel that π has been used for centuries, and there is no compelling reason to change now.

π is perfect for finding the area of a circle: Area = πr². Here’s the area of a circle using tau: Area = r²τ/2.

τ is very good for finding the circumference of a circle: Circumference = τr, but that looks strange compared to 2πr. In fact, it can be difficult to tell if τr is one character or two.

The Tau Manifesto shows angle measurements in degrees, π radians and τ radians. You might want to look at some videos, too. Some people think the τ radians are simpler because the radians correspond exactly to the fractional pieces the circumference of a circle or, get this, to the fractional pieces of a pie. (τ does that, not π.) Other people think that π radians are just as good because we’re used to them, and they correspond exactly to the area of any wedge in a unit circle or the area of any slice of pie. (Which would you rather eat the circumference or the area of a pie?)

Until I wrote this post and read the link shared in the comments, I hadn’t heard anybody say that π is better for some situations while τ is better for others. (Actually it appears that π is better except in formulas that use 2π.) Diameters and radii have co-existed peacefully for centuries. I don’t understand why π and τ can’t do the same. Here’s a great video that shows both sides of the argument.

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22² + 12² = 628.

628 is the hypotenuse of the Pythagorean triple 340-528-628. The greatest common factor of those three numbers is the same as the greatest common factor of 22² and 12².

7² + 11² + 13² + 17² = 628. Thank you Stetson.edu for that fun fact about the squares of those four consecutive prime numbers.

Print the puzzles or type the solution on this excel file: 12 Factors 2015-09-21

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- 628 is a composite number.
- Prime factorization: 628 = 2 x 2 x 157, which can be written 628 = (2^2) x 157
- The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2 = 6. Therefore 628 has exactly 6 factors.
- Factors of 628: 1, 2, 4, 157, 314, 628
- Factor pairs: 628 = 1 x 628, 2 x 314, or 4 x 157
- Taking the factor pair with the largest square number factor, we get √628 = (√4)(√157) = 2√157 ≈ 25.059928.

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