What is this Craziness? – Illogical Sums

As a young kid  just starting school I had to learn two primary things, English and Numbers and believe me, at that young age, numbers did make lot more sense than English grammar. Maybe because every permutation and combination of just 9 numbers is valid but not of the 26 alphabets. Coming a long way after that I did keep the same notion until very recently. A few weeks ago I stumbled upon some Numbers which defied Rationality and Math. Curiosity helped me find many more such Numbers and so I thought of creating a series of posts titled “What is this Craziness?” discussing each one at a time.  Let’s start then:

Look at this:


Okay, the above series is an infinitely long series in which the next number is double that of the previous. The mathematical jargon for this is ‘ The Geometric series’.Logically speaking if one asks you what will be the last number to get added, you may say it is infinity. What about the sum? Infinity again? What if I say:

1+2+4+8+……= -1

Quiet baffling to see the sum not being infinity. More baffling to see  that the sum of all positive numbers is a negative number.Before saying anything let me prove that I am not bluffing.

Say,      x=1+2+4+8…….

Also,    x=1+2(1+2+4+8…..)

Then,     x=1+2x

Solving this linear equation you get x=-1.

Now you may ask “What is this Craziness?” . There is no absolute explanation but a number of  informal one’s do try to.Two of them caught my eye. One says that the real line is circular with 0 and infinity being the ends of its diameter (Both +infinity and -infinity is at a single point). When the above sum is added it moves from 0 towards infinity , crosses over infinity and comes back from the negative axes to end up at -1!!. A crazy explanation. The other one is more subtle: It says that removing a common like 2( in step 2) outside changes the series and so can’t be replaced by original series. When asked why the above method works for series whose numbers decrease along the series (Convergent Series)  the explanation reasons that as the numbers decrease the last term will be very small (infinitesimally small) and so removing a common and replacing with the original sum will have no significant effect. So according to this approach the sum is infinity. Aah! some relief finally.There are more complex explanations available but the second approach truly provide a consolation at the least.


4 thoughts on “What is this Craziness? – Illogical Sums

  1. I don’t think that you can apply algebraic operations to divergent series. Their sum, as in this case, is undefined (infinity). Infinity + Infinity = infinity, or infinity +/- 1 = infinity. Its a good trick though if you want to rattle friends 🙂 It relies on the misconception that algebra is universally applicable. The misconception is accurate for the most part, except for infinity which seems to be the bane of mathematicians.

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