So, where to go today instead of being productive.
Here will do. It's a mathematically comedic
response to George Vaccaro's horrifying encounters with the Verizon billing department
George Vaccaro's 'horrifying encounters' are worth letting a deadline or two slide as well, if only to be appalled anew at the level of basic inummeracy abroad in the world.
For those who don't get Munroe's joke, there's more distraction to be had. The Mathematics Department at the University of Toronto has a rather nifty Question Corner.
It's not being updated any more but it has a fair swag of questions already answered including a question about e(i?) that makes the middle third of Randall Monroe's cheque explicable.
As for the last third, it's a summation expression. One way of expressing the series in English is
for all whole numbers, n, starting at 1 and going on forever, add together all the fractions, one divided by (2 to the power of n).
Another way, which attempts to lay out the steps involved in calculating the value of each item in the series is as follows.
Step A
Start with 2 to the power of 1
21
Calculate this:
21 = 1*2 = 2
Make this value the denominator of a fraction with 1 as the numerator:
1?2
For convenience's sake, turn the fraction into a decimal:
1?2 = 0.5.
Set this value as the first item in an addition:
0.5 + [more values to come]
Step B
Now go with 2 to the power of 2:
22
Calculate this:
22 = 2*2 = 4
Make this value the denominator of a fraction with 1 as the numerator:
1?4
For convenience's sake, turn the fraction into a decimal
1?4 = 0.25
Set this value as the second item in an addition:
0.5 + 0.25 + [more values to come]
Step C
Next is 2 to the power of 3:
23
Calculate this:
23 = 2*2*2 = 8
Make this value the denominator of a fraction with 1 as the numerator:
1?8
For convenience's sake, turn the fraction into a decimal:
1?4 = 0.125
Set this value as the second item in an addition.
0.5 + 0.25 + 0.125 + [more values to come]
Step D &c
Keep repeating the above steps, increasing the power you raise 2 to by one each time.
Don't stop, ever.
Stepping back from this endless task for a moment. Try the steps above for all values of n from 1 to 10. To really see what's going on, note the intermediate values you get as you increase n.
For example, for n = 2 above, the sum is
- 0.5 + 0.25
For n = 3 above, the sum is
- 0.5 + 0.25 + 0.125
Think for a minute about what happens to the value of this sum as you keep adding together fractions derived from ever larger powers of 2 (hint: it keeps getting closer to a particular number). Now, consider what would happen if you kept going forever.
Now, go back and take another look at Randall Monroe's cheque.
Funny, yes?
/* Funny, no, probably. Nothing like explaining a joke to suck all the life out of it. */
Amused or not, take at least a moment to check out Monroe's webcomic of romance, sarcasm, math, and language. With any luck I'll have a few more people missing deadlines as they spend 'just a few minutes more' reading through the more than 200 strips Munroe has produced as of this posting.
And, don't forget to mouse-over the comics. Most of them have an extra tidbit tucked away in the image's title attribute which will show up only if you hover the pointer over the image for a few seconds.