Skip to main content

SICP: Broken Math

Here's a great little math trick from SICP:
v = 1 + 2 + 4 + 8 + ...
Let's play around with that equation a bit:
v - 1 = 2 + 4 + 8 + ...
(v - 1) / 2 = 1 + 2 + 4 + 8 + ...
But that's the same as the first line. Hence, we see that:
(v - 1) / 2 = v
Solving for v, we get:
v = -1
-1 = 1 + 2 + 4 + 8 + ...
Crazy, eh? Sussman said:
Arguments that are based on convergence are flaky, if you don't know the convergence beforehand. You can make wrong arguments. You can make deductions as if you know the answer and not be stopped somewhere by some obvious contradiction.
I think the bad assumption in this case is thinking that some v could exist and be a normal integer. Clearly, there is no such normal integer v that equals 1 + 2 + 4 + 8 + ...


kib said…
Funny post,

Sussman is right, we have to be very careful when handling limits.

But for this one, it seems rather natural that v will not be an integer.
stan said…
Although, if you write a C program to evaluate that sum using signed integers, it does converge to -1. :)
Anonymous said…
Why do you call it a trick? Seems more like an anti-trick to me, if anything. As in, no, really, don't try this at home or anywhere else.
kib said…
A nice one was found by Gauss (it is said to, be maybe that's a legend...) when he was a child and asked to find this (finite) sum :

Sn = 1 + 2 + 3 + ... + n

Gauss started by writing the numbers like this :

Sn = 1 2 3 4 5 ... n

then in reversed order :
Sn = n n-1 n-2 2 1

What's the point ? He could now add each column and see that they all give n+1:

2*Sn = n+1 n+1 n+1 ... n+1

As with have n equal terms, 2*Sn is easily computed :
2*Sn = = n*(n+1)

So, Sn = n*(n+1)/2

Nice no ?!
jjinux said…
> Although, if you write a C program to evaluate that sum using signed integers, it does converge to -1. :)

Haha, I thought of that too ;)

> Why do you call it a trick?

I didn't know what else to call it ;)
jjinux said…
> A nice one was found by Gauss

Ah, yes, I know that story. The funny is when I showed my wife this "trick", she came up with the same story ;) So let me see:

n = infinity
(infinity (infinity - 1)) / 2 = infinity

(But note, it's the same "kind" of infinity, since there's a 1-1 mapping between the two.)
Anonymous said…
There is quite a theory about divergent series. Have you found
where they give a bibliography, including a book by G. H. Hardy.

I think this particular sum is due to Euler.

Popular posts from this blog

Ubuntu 20.04 on a 2015 15" MacBook Pro

I decided to give Ubuntu 20.04 a try on my 2015 15" MacBook Pro. I didn't actually install it; I just live booted from a USB thumb drive which was enough to try out everything I wanted. In summary, it's not perfect, and issues with my camera would prevent me from switching, but given the right hardware, I think it's a really viable option. The first thing I wanted to try was what would happen if I plugged in a non-HiDPI screen given that my laptop has a HiDPI screen. Without sub-pixel scaling, whatever scale rate I picked for one screen would apply to the other. However, once I turned on sub-pixel scaling, I was able to pick different scale rates for the internal and external displays. That looked ok. I tried plugging in and unplugging multiple times, and it didn't crash. I doubt it'd work with my Thunderbolt display at work, but it worked fine for my HDMI displays at home. I even plugged it into my TV, and it stuck to the 100% scaling I picked for the othe

ERNOS: Erlang Networked Operating System

I've been reading Dreaming in Code lately, and I really like it. If you're not a dreamer, you may safely skip the rest of this post ;) In Chapter 10, "Engineers and Artists", Alan Kay, John Backus, and Jaron Lanier really got me thinking. I've also been thinking a lot about Minix 3 , Erlang , and the original Lisp machine . The ideas are beginning to synthesize into something cohesive--more than just the sum of their parts. Now, I'm sure that many of these ideas have already been envisioned within , LLVM , Microsoft's Singularity project, or in some other place that I haven't managed to discover or fully read, but I'm going to blog them anyway. Rather than wax philosophical, let me just dump out some ideas: Start with Minix 3. It's a new microkernel, and it's meant for real use, unlike the original Minix. "This new OS is extremely small, with the part that runs in kernel mode under 4000 lines of executable code.&quo

Haskell or Erlang?

I've coded in both Erlang and Haskell. Erlang is practical, efficient, and useful. It's got a wonderful niche in the distributed world, and it has some real success stories such as CouchDB and Haskell is elegant and beautiful. It's been successful in various programming language competitions. I have some experience in both, but I'm thinking it's time to really commit to learning one of them on a professional level. They both have good books out now, and it's probably time I read one of those books cover to cover. My question is which? Back in 2000, Perl had established a real niche for systems administration, CGI, and text processing. The syntax wasn't exactly beautiful (unless you're into that sort of thing), but it was popular and mature. Python hadn't really become popular, nor did it really have a strong niche (at least as far as I could see). I went with Python because of its elegance, but since then, I've coded both p