How good is your "number sense"? How many of the following can you answer without using a calculator or looking up a conversion factor?
Are there more inches in a mile, or Sundays in 1000 years?
Are there more seconds in a week, or feet in 100 miles?
Are there more millimeters in a mile, or seconds in a month?
Which is larger, multiplying all the numbers from 1 to 10, or multiplying just the even numbers from 1 to 16?
Which is longer, 666 days or 95 weeks?
Which is longer, 666 inches or 55 feet?
Which is longer, 666 hours or 28 days?
Are there more ounces in a ton or inches in a kilometer?
Which is hotter, $0^{\circ}C$ or $0^{\circ}F$?
Which is larger, $e^\pi$ or $\pi^e$?
Click below for the answers.
Inches in a mile. (63,360. There can be up to 52,178 Sundays in 1000 years.)
Seconds in a week. (604,800, compared to 528,000 feet in 100 miles.)
Seconds in a month. (Even if the month only has 28 days, that's 2,419,200 seconds, compared to only 1,609,340 millimeters in a mile.)
Just the even numbers from 1 to 16. (Multiplying all the numbers from 1 to 10 gives you 3,628,800. Multiplying the even numbers from 1 to 16 give you 10,321,920.)
666 days. (95 weeks is only 665 days.)
666 inches. (55 feet is 660 inches.)
28 days (which is 672 hours).
Inches in a kilometer. (39,370.1, compared to 35,840 ounces in a long ton, which is the heaviest ton.)
$0^{\circ}C$ is "hotter" since it is equal to $32^{\circ}F$
For #10 you could use the Taylor series expansion for e^x = 1 + x + x^2/2! + ... and notice that e^x > 1 + x for x not 0. Substitute x -> pi/e - 1 to get e^(pi/e - 1) > pi/e. Multiplying by e gives e^(pi/e) > pi. Finally, raise both sides to the power e and you have e^pi > pi^e.
3 comments:
8/10. I messed up the ton one, and pi^e vs e^pi is brutal. I'm not sure I could get anywhere with that one given pencil and paper, and unlimited time.
Tim,
I thought #5 was tricky, since they're so close. You didn't just use Newton's method for #10? ;)
For #10 you could use the Taylor series expansion for e^x = 1 + x + x^2/2! + ... and notice that e^x > 1 + x for x not 0. Substitute x -> pi/e - 1 to get e^(pi/e - 1) > pi/e. Multiplying by e gives e^(pi/e) > pi. Finally, raise both sides to the power e and you have e^pi > pi^e.
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