Did you know that..., part 3.

  1. Complex numbers are in a sense perfect while there is little doubt that perfect numbers are complex.
  2. You can position 10 defenders of a square castle so that on every side there will be 5 men.
  3. There are many things that can be multiplied: numbers, vectors, matrices, functions, equations, sets, pegs...
  4. A torus may be brushed smooth but a sphere can not.
  5. A circle may be quite useful in drawing straight lines.
  6. In the sequence of all integers, there are arbitrary long runs with no primes.
  7. With just one caveat, anything you can do with a compass and a ruler you can do with the ruler alone.
  8. There are really impossible things.
  9. You can add apples and oranges.
  10. Complex number to a complex power may be real.
  11. Irrational number to an irrational power may be rational.
  12. There are trisectable angles that are not constructible.
  13. There exist triangular numbers that are also square.
  14. No two integers are equidistant from the square root of 2
  15. Almost every integer has a digit 3 in it
  16. C0 - C0 = [-1, 1]
  17. The length of the diagonal of the unit square equals the square root of 2
  18. Every composite number is the product of some factors and also the some of the same numbers
  19. Simple quadrilaterals tessellate the plane
  20. There is a simple solution to the affirmative action problem
  21. Two simple polygons of equal area can be dissected into a finite number of congruent polygons
  22. cos(36°) = (1 + 5)/4
  23. 1/3 + 1/4 = 7/12
  24. Σ2-n = Σn·2-n
  25. Bisector of an imaginary angle may be real
  26. Any infinite set contains uncountably many nested subsets
  27. Some numbers are lucky. 13 is one
  28. How to write an equation of the union of two sets
  29. Altitudes have ears, foot, stem, and root
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