Mathematical Constants

For a pure imaginary number: {i = sqrt(-1)} times a real coefficient {r}, you have i*r. The cube_root(i*r) = cube_root(i)*cube_root(r), so find the cube root of r in the normal way, then we just need to find the cube root of i. For any cubic function (which has a polynomial, in which the highest term is x3) will always have 3 roots. There are 3 values, which when cubed will equal the imaginary number i:

• -i will do it: (-i)3 = (-i)2 * (-i) = -1 * (-i) = i
• The other two are complex: sqrt(3)/2 + i/2 and -sqrt(3)/2 + i/2

If you cube either of the two complex binomials by multiplying out, you will end up with 0 + i as the answer in both cases.

Note: the possible roots for any cubic are: 3 real roots, or 1 real root and 2 complex root, or 1 pure imaginary root, and 2 complex roots.

For your original question, if you want to stay in the pure imaginary domain, then you can use: Cube_root(i*r) = -i * cube_root(r) to find an answer.

Pi is defined as the ratio of the circumference of a circle to its diameter (pi = circumference / diameter). So that explains why it's in the circumference formula. For the area, a little knowledge of calculus is helpful. Here's a 'simple' explanation.

Suppose you have a polygon with many sides, which is inscribed into a circle. The area of the polygon can be found by adding up the areas of isosceles triangles formed by a polygon side and line segments extending to the center. If you have enough sides, then the base of each small triangle will approximately be equal to the corresponding arc of the circle. For sufficiently small triangles (a very large number of sides for the polygon), a 'long leg' of the isosceles triangle will be approximately equal to its altitude.

Say that we have a 100 sided polygon, with radius R and diameter = 2*R. So lets find the approximate area of the polygon. Each base (polygon side) will be approximately equal to Circumference/100, and each altitude will be approximately equal to R. So the area of one triangle is equal to Base*Altitude/2, which is approximately (Circumference/100)*R/2. Substitute Circumference = pi*Diameter = pi*2*R.

Each triangle area is approx (pi*2*R/100)*R/2 = (pi/100)*R2. Since there are 100 triangles, the polygon area is approximately pi*R2 (which is the area of the circle). Note the word approximate, because there is slight space between the polygon and the circle, but [here's the calculus part], as the number of polygon sides approaches infinity, the space between the polygon and the circle approaches zero, and the errors in approximation approach zero, so that the area of the circle and the infinite sided polygonare the same.

Dr. Daisuke Takahashi and Yasumasa Kanada did their analysis of PI

at the University of Tokyo, in 1997.

Computer Centre, University of Tokyo
Bunkyo-ku Yayoi 2-11-16
Tokyo 113 Japan

http://www.cecm.sfu.ca/~jborwein/Kanada_50b.htmlh

Yes, if the discriminant is zero, then there will be a double root, which will be real.

Also, If the discriminant is positive, there will be two distinct real solutions. But if the discriminant is negative, then you will have two complex solutions.