Wikipedia:Reference desk/Archives/Mathematics/2006 August 12
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Meaning of a billion
[edit]Dears, I wish to be educated on how you came to give the definition of one billion to be a number with ten digits instead of thirteen as below. 1,000,000,000.To me this is one thousand million 1,000,000,000,000. To me this is one billion. Going by definition of english .Billion...the bi means two,thus billion means a\"a million million" You can not say 500,000,000 +500,000,000 =one billion. instead is one thousand million. sebastian mgimba Tanzania
- Please see our article on Long and short scales which discusses the different meanings of the word billion. Road Wizard 08:08, 12 August 2006 (UTC)
- In the UK 500,000,000 plus 500,000,000 is definitely called a billion. The article named by Road Wizard explains that some countries use 1,000,000,000 and some 1,000,000,000,000. - Adrian Pingstone 08:27, 12 August 2006 (UTC)
deception in compound interest
[edit]60x^37-63x^36+3=0 find x?
- This doesn't appear to have anything to do with compound interest. StuRat 20:50, 12 August 2006 (UTC)
- Nonetheless, x = 1 is obviously a solution, and Newton's method is possibly your best bet for finding the others numerically. -- Meni Rosenfeld (talk) 21:16, 12 August 2006 (UTC)
- ((Homework answers will not be provided here. See guidelines at top of page.—KSmrqT)) Fredrik Johansson 21:26, 12 August 2006 (UTC)
- Nonetheless, x = 1 is obviously a solution, and Newton's method is possibly your best bet for finding the others numerically. -- Meni Rosenfeld (talk) 21:16, 12 August 2006 (UTC)
- Deception, indeed! But I found x! It was fiendishly hidden just next to the 60 and the 63 in that equation. By the way, 3 real roots, 17 pairs of complex roots. The 3 real ones should be quite easy to find... digfarenough (talk) 00:11, 14 August 2006 (UTC)
- First remove a factor of 3, leaving 20x37−21x36+1 = 0. Because the sum of the coefficients is 0 we know that 1 is a root; so, as suggested, we may divide out a factor of x−1 leaving 20x36−∑k=0..35 xk.
- However, a more interesting idea is to use Sturm's theorem. The first derivative of X0 = 20x37−21x36+1 is 740x36−756x35, from which we can remove a factor of 4 leaving X1 = 185x36−189x35. The reduced negative of the remainder of X0 by X1 is X2 = 3969x35−6845. The next Sturm sequence polynomial is X3 = −185x+189; and finally we obtain a constant polynomial, X4 = −3.46×1082…, approximately. Now we can count unique real roots.
X0 X1 X2 X3 X4 σ −∞ − + − + − 4 −2 − + − + − 4 −1 − + − + − 4 0 + 0 − + − 3 +1 0 − − + − 2 +2 + + + − − 1 +∞ + + + − − 1
- We already know that 1 is a root; now we can see that there are only two other real roots, one of which lies between −1 and 0, and the other of which lies between 1 and 2. (None of the roots has magnitude greater than 1+21⁄20 = 2.05, by the Cauchy bound alone.) --KSmrqT 06:10, 14 August 2006 (UTC)