# One minus one plus one minus one – Numberphile

49
3 Discussing the brain-bending Grandi’s Series and Thomson’s Lamp – featuring Dr James Grime.
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1. It is both on and off, therefore it is half on, and half off. The amount of observed time on and off are both equal to one infinitesimal, so the light is in a sense "half way" on and "half way" off

2. I just realised 1/2 is right between 0 and 1, the two first solutions.

I feel smart now.

3. That sounds like quantum mechanics

4. In Thomson's Lamp the answer is both (quantum mechanics). In the Grandi's series the answer is 1/2 (limit). Thomson's Lamp is NOT an example of Grandi's theory.

5. Not solvable, the mathematician has an epileptic seizure due to the flashing light.

6. imagine coming to a place to learn and coming out with more questions

7. It's a 50 percent chance of either one

8. Time to install a switch dimmer

9. if it ends with…1+1-1 it will equal 0. if it ends with… 1-1+1 then it will be 1

10. A series is a functional representation of numbers whether convergent or divergent or infinite.

11. The sum is Infinite, surely no answer until it is no longer infinite?

12. This guy’s grandfather is Kenneth Williams.

13. What about putting the brackets here:

1 – (1 + 1) – (1 + 1) – (1 + 1) – (1 + 1)…

i.e. 1 – 2 – 2 – 2 – 2 – 2 – 2….

Surely this would be minus infinity?

14. The answer to the 1-1+1-1+1-1… equation is the same answer as the answer to the question, "Is infinity even or odd.". It has no answer. The 1-1+1-1+1… does have an average and that's .5 but it doesn't have value.

15. @4:50 huh?!

7/6 does not equal 1.75

16. A still equals 0 here so isn’t technically illegal to do anything to it?

17. I think the light would be on and of at the same time if Grandi´s Series is equal to 1/2

18. Hi James : You end the video with a question. However I have a feeling that the purpose of your question is not clear to you. What are you looking for by this question? That is my reply to your question! Beside this I have a piece of info, which I don't know if know this: It is about the old analog dimmer switches for lights (old tech.) They dimmed the light brightness by switching on and off the power to the bulb. This alternating action is done at a time rate too fast for our perception to notice. What we perceive is a light that is dimmer or brighter. So the result "1/2" correlate to the effect on perception of the alternating "1" and "0". If you would use for example a 100watts bulb, and you switch the power to it on and off fast enough, what you would perceived is the light output of a 50watts bulb (1/2 of a 100watts. …just for your information, in case you did not know this.

19. 1-(1+1)-(1+1)-(1+1)-(1+1)…

20. You could get -∞ with 1-1+1-…
Set it like that 1-((1+1)-(1+1)-(1+1)-…)

22. You can't manipulate infinite sums like that. In fact I can "prove" that S=0 or any number with this logic.
S=1-1+1-1+1…

T=1+1+1+1+1…

S+T=2+0+2+0+2+0…

S+T=2+2+2+2…

S+T=2(1+1+1…)

S+T=2T

S=T

T-S=O+2+0+2+0…

T-S=2+2+2+2…

T-S=2(1+1+1…)

T-S=2T

S=-T

S=T & S=-T ==> S=0

23. That’s the dumbest thing I ever heard! Im listening

24. What if you ry 2-2+2-2 and so on what do you get?

25. The switch is in the halfway position

26. None of this really Mathers anyways

27. In the first equation he uses only (8) 1's which equal (0) –+the second equation he uses (9) 1's that that would of course equal 1 more 1 than (0) 1's 1-1+1-1+1-1+1-1=0
1-1+1-1+1-1+1-1+1=1

28. 1 and 0: all right let's find a solution that satisfy both our needs
1-1+1-1+1-1+….=1/2

29. If one is the loneliest number – shouldn't 0 be lonelier? At least half the time?

30. Duhhhh. Even with 0 brackets, the answer will be either 1 or 0 depending on when u stop the equation. Adding brackets changes the equation, obviously changing the answer.

31. It's both on and off till you measure it 😉

32. The light is half way on at infinity. At that point it cant go any faster.

33. I just saw the first 4 minutes yet but personally I would say it could be both it just depends if the last number is plus or minus one wich is not defined but if we say it’s fifty fifty if the last one is one or minus one it would be 1/2 times 1 plus 1/2 times 0 wich is 1/2 this is the way I would define it

34. One of the interesting things about sound and light is that the perception of it will actually amount to a half.
If you turn on a light for 1 millisecond and turn it on for 1 millisecond and let it repeat like that it will appear as though it's running at 50% brightness.

The same phenomenon occurs with sound. If you have a little speaker playing a tone for a millisecond then being off, then playing a tone again. It will be perceived at half volume.

35. Maybe the answer is 0.5
Edit: I wrote that after a minute of watching lol can't believe I'm not wrong

36. It means that, by that point, the light is occiliating so quickly between on and off that the luminance it produces is exactly on half of the luminance it produces when on.

37. Hold on just a minute. If an infinite sum (not 1-1+1-1+…) converges to 2, then the averages of the partial sums converges to 0. The limit of the averages is lim (Sn / n) as n -> oo where Sn is the partial sum to n terms. Sn -> 2 and n -> oo, so Sn / n -> 0.

38. When it’s at half, the light is off, but the switch is on (something’s wrong with the circuit or the electricity company got rid of all your electricity)

39. Reminds me a little of pulse width modulation, turn something on and off really quickly you actually dim the light to different levels. I found this very interesting!

40. Are you allowed to add and subtract divergent series?

41. infinity can't be imagined, so the question is useless.

42. Anything other than 0 is just intelligent people trying to overthink it. You have to read the problem as is written. If you start adding brackets here and there and "+'s" that aren't originally in the problem, then you're just changing it to support whatever you want it to be. In that case you can add "x's" or "÷'s" as well.

43. If you turn the light on and off fast enough it will only appear half as bright. That's where I thought you were going with your logic.

44. According to the initial statement and the BIDMAS (operation order) the result shouldn’t be 0, then you randomly start play around with brackets and mess up yourself because the initial statement hasn’t got brackets and the operation order is to collect like terms and perform additions before subtract anything, so, the result depends on how many terms you have got…

45. 7/6 is not 1.75. Prove me wrong.

46. The second one couldn’t be 0 vbecause you would multiply where those brackets are. It would still be 1. the third example, s is a finite series. So, he has changed from an infinite sum to the sum of a finite series. With this reasoning you could reproduce S an infinite number of times, so you’d have 1/infinite. If you put the brackets around 1+1, you’d get -infinite. He’s getting 1/2 because it is the average of the alternating sums that would alternate forever.