YouTube Treasures continues with a numerical proof that’s mind boggling. Enjoy! (More)

*Midday Matinee is our people watching, people doing and people being feature. Join the Woodland Creatures for an afternoon break.*

Its mind blowing to watch this, but the results are amazing. The summation of all positive whole numbers yields a result that flies in the face of credulity.

Loved this! It’s like going on an adventure. Even better it bends my mind into a pretzel.

Fascinating! If you stop the sequence [1+2+3+4…] at any point below infinity, you get a positive sum. If you stop at 4, the sum 10. If you stop at 5, the sum is 15. If you stop at 6, the sum is 21. And the sums keep getting bigger the higher you go …

if you stop.But if you

neverstop and instead add all of the natural numbers up to infinity … the sum is -1/12.Weird. Very, very weird….

Then again, “infinity” is a slippery idea. For example, how many positive

evenwhole numbers (2, 4, 6, 8…) are there? The answer, of course, is “infinity” … because no matter how big that an even number is, you canalwaysadd 2 and get yet another even number.Okay, that makes sense. So how many positive whole numbers are there, both odd and even? Again, the answer is “infinity” … because no matter how big a whole number is, you can

alwaysadd 1 and get another whole number.But here’s the thing. The answer to “how many positive, even-only whole numbers are there>” is “infinity” … and the answer to “how many positive whole numbers are there?” is also “infinity” …

… but that second “infinity”

is exactly twice as bigas that first “infinity,” because that second “infinity” includes both the positive,evenwhole numbers and the positive,oddwhole numbers.And yes, the answer to the question “how many whole numbers are there?” is

alsoinfinity … and that third “infinity” is even bigger than the second one, because it includes both the positive and negative whole numbers, plus zero.But we don’t need huge numbers to have mind-bending fun with infinities. How many decimal numbers are there between 0 and 1? The answer, again, is “infinity” … because you can always add more decimal places.

Okay, makes sense. So how many decimal numbers are there between 0 and 2? Again, “infinity” … but the “infinity” of decimal numbers between 0 and 2 is

twice as bigas the “infinity” of decimal numbers between 0 and 1.So yeah, “infinity” is a mind-twister….

Hi Crissie! You ask some important questions. I have a 20 minute video on infinities that will show that in at least some cases, the infinities of positive integers and the infinities of positive even integers is actually the same size. You are correct that the infinity of real numbers is much larger than the infinity of positive integers however.

The video is a mind bender, and if fans want, I’ll queue it up soon.

Looking forward to the video.

OKAY! 🙂