This MIT Integral Looks Impossible

Added: 2026-05-22

[00:00:00]At first glance, this integral looks impossible. We have integral from 0 to 10 of floor function of 1 + square root of 5 over 2 to the power of another floor function x dx.

[00:00:11]So, we have two floor function involved inside of this integrand, but if you remember the meaning of this floor function, everything is pretty simple.

[00:00:18]And at the same time, inside of this parenthesis, 1 + square root of 5 over 2, this is a golden ratio 5.

[00:00:25]So, remember how 1 + square root of 5 over 2, this is the golden ratio 5, right?

[00:00:33]So, let me just call this integral as the I. Then, your integral I [applause] is the same as integral from 0 to 10 of floor function of golden ratio 5 to the power of floor function of the x. Close your floor function, then we have dx.

[00:00:52]Now, let's just talk about this floor function of the x, right?

[00:00:56]So, this floor function of the x, this means the greatest integer that does not exceed the x, right? So, for example, if you have floor function of 1.6, it is just a 1. Floor function of, say, like 2.9, it is just equal to 2, and so on.

[00:01:14]So, that's why this floor function of the x on each interval, first of all, on each interval that is from n to n + [applause] 1.

[00:01:30]Okay.

[00:01:31]Then, floor function of the x is just going to be the same as just the n, which is just an integer, right?

[00:01:38]So, that is why this floor function of golden ratio 5 to the power of floor function of the x, this is going to be the same as just the constant, right?

[00:01:51]Golden ratio 5 to the power of n on each interval.

[00:01:55]We need from 0 to 10, so that's why the end value has to be from zero one, [applause] two, all the way up to nine. This is what we need.

[00:02:08]So, this integrand doesn't seem to be a smooth curve, but it has to be staircase type, right?

[00:02:13]So, that is why this integral I we can just represent I to be summation from n is zero to nine.

[00:02:24]Zero to nine of now floor function of golden ratio phi to the power of n. That's it.

[00:02:32]>> [gasps] >> Then it's all about you plug it in zero to the n, one to the n, all the way up to nine to the n and calculate and add them up together, right? So, let's just calculate this first.

[00:02:45]So, the golden ratio phi to the power of zero is just equal to one. So, that is why floor function of golden ratio phi to the power of zero is just equal to one.

[00:02:57]We already know the approximate value of the golden ratio phi, which has to be 1.618.

[00:03:03]So, golden ratio phi to the power of one it is around 1.618 and so on. So, that is why floor function of that is just going to be equal to one, too.

[00:03:15]And it's all about you keep multiplying this 1.618. We do not need the exact value, but just approximate value, right? So, now golden ratio phi squared is going to be then 2.something.

[00:03:28]So, that's why floor function of that is just going to be equal to two.

[00:03:32]And let's keep doing this.

[00:03:34]If you multiply 1.618 to this 2.something, so the golden ratio phi cubed is going to be then 4.something.

[00:03:42]So, that is why floor function of that is just equal to four, right?

[00:03:48]Okay, then a few more.

[00:03:50]Golden ratio phi to the power of four. You multiply 1.618 to this 4.something, right? It has to be 6.something.

[00:04:00]So, that is why floor function of that is going to be now six.

[00:04:05]And then golden ratio phi to the power of five, multiplying 1.618 to the 6.something, is then going to be then the same as 11.something.

[00:04:16]So, that is why floor function of that is going to be just 11.

[00:04:21]So, a few more. Uh golden ratio phi to the power of six. You multiply 1.618 something to this 11.something. So, approximate value has to be then 17.something.

[00:04:32]So, floor function of that is going to be just uh 17.

[00:04:38]Okay. Then, golden ratio phi to the power of seven, multiplying 1.617 something to the 17.something to get then 29.something.

[00:04:49]So, floor function of that is going to be 29.

[00:04:53]Okay. So, we have two more, right? So, golden ratio phi to the power of eight.

[00:04:58]So, that's going to be then the same as 46.something.

[00:05:02]So, floor function of that is going to be just the 46.

[00:05:08]The last one.

[00:05:10]Golden ratio phi to the power of nine.

[00:05:13]Then, just multiplying 1.6 to this 46.and so on. It is just giving you 76.something.

[00:05:20]So, that is why floor function of that is going to be 76.

[00:05:26]Now, we have those values. All we need to do is to add these values up.

[00:05:38]So, if you go ahead and add them up, then it has to be then the same as 193.

[00:05:45]So, the answer for this question is 193, which is pretty interesting.

[00:05:49]But, what is more interesting is how these numbers, 1 1 2 4 6 11 17 29 46 and 76, resemble some Fibonacci-type sequence that we can call Lucas numbers.

[00:06:02]So, the Lucas numbers >> [applause] >> If you list out some of the Lucas numbers, then it is 2 and 1 and then we have 3 >> [applause] >> 4 and 7 11 18 29 Okay, then we have 47 >> [applause] >> and 76 and so on.

[00:06:32]These are Lucas numbers, and these numbers resemble these numbers that we just calculated, right?

[00:06:37]And Lucas numbers, surprisingly, could be represented using golden ratio phi.

[00:06:41]So, L of n, the Lucas number, is the same as golden ratio phi to the power of n plus parentheses negative of 1 over golden ratio phi to the power of n.

[00:06:54]If you keep increasing the value of the n, this negative 1 over golden ratio phi to the power of n, it's pretty small number. So, that's why these two kind of resemble.

[00:07:04]That's more interesting part about this question, I think.

[00:07:07]So, seemingly impossible integral turns into nice summation of integers only.

[00:07:12]How amazing.