Did you know that..., part 3.
- Complex numbers are in a sense perfect while there is little doubt that perfect numbers are complex.
- You can position 10 defenders of a square castle so that on every side there will be 5 men.
- There are many things that can be multiplied: numbers, vectors, matrices, functions, equations, sets, pegs...
- A torus may be brushed smooth but a sphere can not.
- A circle may be quite useful in drawing straight lines.
- In the sequence of all integers, there are arbitrary long runs with no primes.
- With just one caveat, anything you can do with a compass and a ruler you can do with the ruler alone.
- There are really impossible things.
- You can add apples and oranges.
- Complex number to a complex power may be real.
- Irrational number to an irrational power may be rational.
- There are trisectable angles that are not constructible.
- There exist triangular numbers that are also square.
- No two integers are equidistant from the square root of 2
- Almost every integer has a digit 3 in it
- C0 - C0 = [-1, 1]
- The length of the diagonal of the unit square equals the square root of 2
- Every composite number is the product of some factors and also the some of the same numbers
- Simple quadrilaterals tessellate the plane
- There is a simple solution to the affirmative action problem
- Two simple polygons of equal area can be dissected into a finite number of congruent polygons
- cos(36°) = (1 + √5)/4
- 1/3 + 1/4 = 7/12
- Σ2-n = Σn·2-n
- Bisector of an imaginary angle may be real
- Any infinite set contains uncountably many nested subsets
- Some numbers are lucky. 13 is one
- How to write an equation of the union of two sets
- Altitudes have ears, foot, stem, and root
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