"My" seventh grader was asked to tabulate the ways to make 40 cents change without pennies, so I had couldn't resist showing him (c72) (1/(1-q^5)/(1-q^10)/(1-q^25),%%=taylor(%%,q,0,40)) 1 (d72)/T/ ---------------------------- = 5 10 25 (1 - q ) (1 - q ) (1 - q ) 5 10 15 20 25 30 35 40 1 + q + 2 q + 2 q + 3 q + 4 q + 5 q + 6 q + 7 q + . . . and then allowing for pennies by dividing by 1-q : (c73) %/(1-q) 1 (d73)/T/ ------------------------------------ = 5 10 25 (1 - q) (1 - q ) (1 - q ) (1 - q ) 2 3 4 5 6 7 8 9 10 1 + q + q + q + q + 2 q + 2 q + 2 q + 2 q + 2 q + 4 q 11 12 13 14 15 16 17 18 + 4 q + 4 q + 4 q + 4 q + 6 q + 6 q + 6 q + 6 q 19 20 21 22 23 24 25 26 + 6 q + 9 q + 9 q + 9 q + 9 q + 9 q + 13 q + 13 q 27 28 29 30 31 32 33 + 13 q + 13 q + 13 q + 18 q + 18 q + 18 q + 18 q 34 35 36 37 38 39 40 + 18 q + 24 q + 24 q + 24 q + 24 q + 24 q + 31 q + . . . But dividing by 1-q is the "undifference" operator, replacing the nth coefficient by the running sum of the coefficients from the 0th thru nth. Obviously! To allow for pennies, make change for anything, then pennies quantum sufficit. Then I illustrated the situational nature of the question by revoking the pennies, but including the 2, 3, and 20 cent pieces from 19th century coinage: (c74) d72/(1-q^2)/(1-q^3)/(1-q^20); 1 (d74)/T/ -------------------------------------------------------- = 2 3 5 10 20 25 (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) (1 - q ) 2 3 4 5 6 7 8 9 10 11 1 + q + q + q + 2 q + 2 q + 2 q + 3 q + 3 q + 5 q + 4 q 12 13 14 15 16 17 18 19 + 6 q + 6 q + 7 q + 9 q + 9 q + 10 q + 12 q + 12 q 20 21 22 23 24 25 26 + 17 q + 15 q + 19 q + 20 q + 22 q + 27 q + 27 q 27 28 29 30 31 32 33 + 30 q + 34 q + 35 q + 44 q + 42 q + 49 q + 52 q 34 35 36 37 38 39 40 + 56 q + 65 q + 66 q + 72 q + 79 q + 82 q + 97 q + . . . However, there were also half-cents. Dividing by 1-sqrt(q) gives ... + 1046 q^40 + ... . Restoring pennies gives an amazing 10135 ways to make 40 cents. Quick: How many ways if we just exclude half-cents? Answer: 1046 again! It doesn't matter if we fill out a lesser amount with pennies or half cents. Two new mathematical truths offered as a contribution to education to be used only by the subscribers to Math-Fun free of cost by paying any royalties whatever on the same, provided it is accepted and adopted by the official action of the membership: It has been found that the relationship of Fahrenheit to Centigrade is not C = (F - 32) 5/9, as previously supposed, but rather C = (F - 32) sin (3 pi/16), which is approximately C = (F - 32) .555570 and more elegantly C = (F - 32) sqrt(2-sqrt(2-sqrt(2)))/2. Moreover, the ratio of feet per second to miles per hour, previously believed to be 22/15 (= 1.4666...), is in reality, pi/(pi-1) (= 1.4669...) when one remembers to allow for the curvature of the Earth. The remainder of this contribution is quotes from that 7th grade Prentice Hall "astronomy" book, with my comments and questions in [brackets]. p17. You are completely uprepared for the spectacle that awaits you when you put on your glasses. To your left you notice a mass of brilliant stars that shine mainly with X-rays. [Gee, then how come the Hertzsprung-Russell diagram (p34?) only goes up to blue on the temperature scale?] p18, fig 1-8 Each galaxy in this cluster of galaxies might hold several trillion or more stars. [10^12? Are there hundredfold bigger galaxies than Andromeda out there?] p32 fig1-24 The brightness of these stars in theTrifid Nebula, as seen from Earth, is called absolute brighness. What term is used for the actual brighness of a particular star? [HUH??] p33: The Hertzsprung-Russell diagram is the single most important diagram astronomers use today. On the Hertzsprung-Russell diagram, the surface temperature of stars is plotted along the horizontal axis. The absolute magnitude, of actual brightness, of stars is plotted along the vertical axis. [It's a scatter plot with about twenty stars, all named, and grouped into either supergiants, giants, main sequence, or dwarfs. But there is absolutely no clue that this is anything but a poop sheet on these particular twenty stars. Why or how is this diagram is used? Later, in the section] Measuring star distance [they "explain" parallax and redshift, but don't mention H-R:] To determine the distance to a star more than 100 light years away, astronomers use the brightness of the star. They plug the star's apparent magnitude and its absolute magnitude into a complicated formula. The formula provides a close approximation of the distance to that star. [Well, if the formula is complicated enough, there's no telling what it might reveal. About redshift, they say] As you have read, light from a star moving away from the Earth has a red shift in its spectrum. Astronomers measure the amount of red shift in a star's spectrum and use complex formulas ... [The book earlier mentions that farther objects look dimmer and redder, but apparently this is not useful in "Measuring star distance".] [The parallax illustration has the star at about 1 AU, i.e., about a 90 degree swing. No mention of how wildly out of scale this drawing is, with the nearest star's actual parallax about 1/3 arc second, and all others less. But that would require numbers. And not just ordinary numbers. Big numbers (shudder).] [Even photographs can be misleadingly out of scale. A (necessarily) overexposed photo of Sirius shows Sirius B nearly submerged in the glare of Sirius A. I'll bet even the teacher would agree that B must be barely outside A's corona, when in fact, the photo only establishes which is brighter, not which is larger. The book blithely depicts brighter stars with bigger dots, and nowhere explains how small the images of stars should really be.] p42 [End of section problems include:] List and describe the four main layers of the sun. [Diagram p39 "Layers of the sun", shows corona, chromosphere, photosphere "about 550 km thick", and core, drawn with < 1/3 solar radius. This leaves 96% of the sun's volume unnmamed! Then they ask for the names of four kinds of solar storm. The book lists flares, prominences, and sunspots, but damned if I can find a fourth. The teacher probably couldn't either, since she omitted this set from the homework.] p25 According to the [big bang] theory, all the matter and energy in the universe was [a lucky mistake] once concentrated into a single place. This place, of course, was extremely hot and dense. Then some 15 to 20 billion years ago, an explosion--the big bang-- shot the concentrated matter and energy in all directions. The fastest moving matter traveled farthest away. Energy, too, began moving away from the area of the big bang. [I.e., there was already a big, empty universe, and a bang went off in the middle.] p27 Quasars If the universe is expanding, then objects near the edge of the universe are the oldest objects in the universe. Put another way, these objects took longer to reach their current position than objects closer to the center of the universe did. [HUH??] p46 In a massive star, gravity continues to pull together the carbon atoms in the core. When the core is squeezed so tightly that the heat given off reaches 600,000,000 deg C, tbe carbon atoms begin to fuse together to form new and heavier elements such as oxygen and nitrogen. The star has begun to become a factory for the production of heavy elements. The core of the massive star is so hot that fusion continues until the heavy element iron forms. But not even the tremendous heat of a massive star can cause iron atoms to fuse together. [Wouldn't tremendous heat *fission* heavier elements? What sort of heat would fission Pb, e.g.? Maybe a big ball of lead would burn via fission?] [According to, e.g., http://csep10.phys.utk.edu/guidry/violence/supernovae-info.html, iron production is so weakly exothermic as to only transpire in the final week before supernova, if at all. Doesn't most iron come from the supernova itself?] Supernovas By the time most of the nuclear fusion in a massive star stops [why?], the central core [is there a peripheral core?] is mainly iron.[?] Although the process is not well understood, the iron atoms begin to absorb energy. Soon this energy is released, as the star breaks apart in a tremendous explosion called a supernova. [Oh, is THAT what happens! No wonder Geritol cures tired blood.] --rwg PS, July 2 is the 36th anniversary of X-ray astronomy, inadvertently initiated by US spy satellites. Absent gamma ray bursts, what's the next best evidence for black holes?