The 0.577 Nobody Can Prove Is Irrational

[00:00:00]Start adding up the simplest fractions there are. 1 + 1/2 + 1/3 + 1/4 + 1/5 and keep going, 1 over each whole number in turn. This running total is called the harmonic series and the natural question is where it ends up.

[00:00:17]Here is the first surprise. The pieces you are adding shrink toward nothing. 1 over 100, 1 over 1,000, 1 over 1,000,000, each new term barely moving the needle.

[00:00:29]You might expect the total to settle down and stop somewhere. It does not.

[00:00:33]This sum climbs forever. Add enough terms and you can pass any number you like, >> [music] >> 10, 100, 1,000. It just takes a very long time to get there.

[00:00:43]But it climbs slowly and it slows down as it goes. To get the total past 10, you need more than 12,000 [music] terms.

[00:00:51]To pass 20, you need hundreds of millions. To reach 100, the number of terms is so large it would take longer than the age of the universe to add them one by one. The growth never stops, yet it drags. And that pace, that particular unhurried crawl, is the whole story here because there is a smooth, familiar curve that crawls upward at exactly the same lazy pace and the harmonic series spends its entire life shadowing it. As you add more terms, the total keeps closing the distance to that curve, then holds a steady step behind it, matching its stride for stride out to infinity.

[00:01:29]The two of them rise together, side by side, never quite touching. So the question we are really chasing is simple to state. How far apart do these two things stay? Let me show you the harmonic series as a picture instead of a list. For each whole number, draw a rectangle one unit wide and as tall as that term.

[00:01:49]So over the interval from one to two, a block of height one. From two to three, a block of height one half. From three to four, a block of height 1/3, and so on.

[00:02:00]Line them up and they make a staircase, marching to the right, each step a little shorter than the last.

[00:02:07]The total area of all those steps is exactly the harmonic sum. Now, lay a smooth curve over that staircase, the curve 1/x. It sweeps down from the upper left, high near the start, then flattening out as it stretches to the right, hugging the axis, but never reaching it.

[00:02:24]The area underneath that curve from one out to some point [music] is a quantity you have met before. It is the natural logarithm. The area under 1/x from one to n is the natural log of n.

[00:02:38]So, the smooth curve is the logarithm, and the staircase is the harmonic sum.

[00:02:43]Look at how they sit together. Each step of the staircase starts at its left edge at the full height of the curve there, then holds flat while the curve keeps sliding down beneath it.

[00:02:54]So, every single step pokes up above the curve. The staircase always sits a little higher than the smooth area. The harmonic sum always overshoots the logarithm. That overshoot is the gap, and now we can watch what it does. Take the harmonic sum out to some number of terms, then subtract the natural logarithm of that same number.

[00:03:15]You are measuring exactly how much the staircase beats the curve. Do it and something quietly remarkable happens.

[00:03:23]The difference does not blow up, and it does not fade away.

[00:03:27]It settles. Watch the numbers. Out to 10 [music] terms, the harmonic sum beats the logarithm by about 0.626.

[00:03:35]Out to 100 terms, the lead is 0.582.

[00:03:40]Out to 1,000, 0.5772.

[00:03:43]Out to 10,000, still 0.5772.

[00:03:47]Now, pinned down to more decimal places.

[00:03:50]The gap is not drifting off to infinity like the sum itself, and it is not [music] collapsing to zero. It is homing in on one fixed value. Two things that both run away to infinity, the endless sum and the endless logarithm, differ by a perfectly finite amount that never changes.

[00:04:08]The staircase and the curve both climb without limit forever, [music] and yet the space between them locks onto a single number and stays there.

[00:04:16]That number is what we came here for.

[00:04:19]And to enough decimal places to matter, it begins 0.5772156.

[00:04:25]That fixed gap between the harmonic series and the logarithm has a name and a value.

[00:04:31]It is called the Euler-Mascheroni constant. And it starts 0.5772156649, going on without any pattern anyone has ever found.

[00:04:43]It gets written with the Greek letter gamma. The story behind the name is short.

[00:04:48]Leonhard Euler found this [music] constant back around 1734 while he was studying exactly this kind of sum, and he worked out the first several digits by hand. Then in 1790, an Italian mathematician named Lorenzo Mascheroni pushed the calculation much further, computing many more digits, and the constant ended up carrying both their names.

[00:05:10]The Greek letter came into standard use later. So, this is a genuine mathematical constant sitting right next to pi and e in importance, and it falls straight out of the most basic sum in all of mathematics.

[00:05:24]You do not need anything exotic to meet it. Just add up 1/1, 1/2, 1/3 forever, and compare against the logarithm, and there it is, 0.577 and the rest. There is a cleaner way to see what gamma really is, and it turns the constant into a shape you can picture.

[00:05:43]Go back to the staircase sitting over the curve. Every step pokes up above the curve by a little sliver, a thin overshoot region between the flat top of the step and the curve dipping below it.

[00:05:55]Gamma is the total area of all those slivers, every last one of them added up forever. Measure the first sliver, the one sitting over the interval from one to two.

[00:06:06]Its area works out to 1 minus the natural log of 2, which is about 0.307.

[00:06:12][music] That single sliver already accounts for more than half of gamma. The next [music] sliver over two to three has area about 0.095.

[00:06:23]The one after that, >> [music] >> about 0.046.

[00:06:27]Each sliver is smaller than the last, shrinking [music] fast. Add them all together, slivers over one to two, two to three, three to four, and onward without end, and the running total climbs and closes in on 0.5772156.

[00:06:44]That is [music] gamma.

[00:06:46]So, the constant is not some abstract limit you have to squint at. It is a concrete area, the total overshoot of the harmonic staircase above the logarithm curve, gathered from here to infinity. And this is why the gap settles [music] instead of running away.

[00:07:01]Each new step of the staircase adds one more sliver.

[00:07:04]But, the slivers shrink so fast that their total stays finite, even though there are infinitely many of them.

[00:07:11]The harmonic sum and the logarithm both march off to infinity, but the pile of overshoots between them adds up to a bounded amount. That bounded amount is exactly 0.577 and the rest, >> [music] >> and it is the same whether you sum the gap directly or stack the slivers one by one. If gamma only appeared in this one [music] sum, it would be a curiosity.

[00:07:33]But, it does not stay put. It surfaces all over mathematics, [music] in places that seem to have nothing to do with harmonic sums. Take the Riemann's zeta function, the object at the center of the most [music] famous open problem about the primes. It is built from a sum a lot like the harmonic one, and it has a single point where it blows up at the value one.

[00:07:55]If you approach that point and carefully subtract off the piece that blows up, what is left behind, the finite remainder sitting right at the trouble spot, is [music] exactly gamma. You can check it numerically. Get close to that point, strip away the infinite part, and the leftover marches straight to 0.5772.

[00:08:13][music] It shows up again in the gamma function, the smooth curve that extends the factorial to every number. Ask how steeply that curve is rising as it passes the value one, its slope right there, and the answer is minus gamma.

[00:08:29]The same constant to the last digit, a quantity born from adding simple fractions, [music] turns out to be woven into the deepest machinery of analysis.

[00:08:38]When a number keeps appearing like that, unbidden, from every direction, [music] mathematicians pay attention. Here is what makes the next part so strange.

[00:08:47]Gamma is not hard to compute. It is one of the [music] easy ones. There are fast, clever methods that spit out gamma to enormous precision, and people have run them hard. The constant has been calculated to trillions of decimal places, trillions. If you want the 10,000th [music] digit or the billionth, it is right there for the taking, computed and checked. We know this number to a precision that dwarfs anything you could ever measure in the physical world. So, we are not in the dark here. [music] We can pin gamma down as far as we please, faster than almost any other constant of its kind. [music] We have its face memorized to a level of detail that is frankly absurd. To put the scale in perspective, the number of atoms in the observable universe is roughly one followed by 80 zeros. We know gamma to a precision thousands of times finer than that. Every digit computed and verified. And with all of that, with trillions of digits in hand, [music] there is one plain childish question about this number that nobody on Earth can answer. The question is this: Is gamma a fraction? Can it be written as one whole number divided by another, the way 1/2 or 3/7 can? Or does its [music] decimal go on forever without ever settling into a repeating pattern, the way pi and e do?

[00:10:03]In other words, is gamma irrational?

[00:10:06]Nobody knows. After nearly 300 years, no one has managed to prove [music] that gamma is irrational. It might be a perfectly ordinary fraction in disguise, and it might not. And the proof has resisted every [music] attempt. That is a shocking thing to have to say about a number we can compute to trillions of digits. There is one thing we can say, and it only [music] sharpens the mystery.

[00:10:30]Suppose, for the sake of argument, that gamma really is a fraction, some whole number over some other whole number.

[00:10:36]Then that bottom number, the denominator, has been proven to be enormous. It would have to be bigger than one followed by more than 240,000 zeros.

[00:10:47]A fraction with a denominator that colossal is, for all practical purposes, indistinguishable from a number that is not a fraction at all. Almost everyone reads this as [music] strong evidence that gamma is irrational.

[00:10:59]But evidence is not proof, >> [music] >> and the proof simply does not exist.

[00:11:03]Step back and take in how odd this is.

[00:11:06]Two of gamma's [music] famous neighbors were tamed long ago.

[00:11:10]The number e was shown to be irrational by Euler, and then proven transcendental, [music] a much stronger statement, by Charles Hermite in 1873. The number pi was proven transcendental by Ferdinand von Lindemann in 1882, which finally settled the ancient problem of squaring the circle.

[00:11:29]Both of those constants [music] live in difficult, elaborate corners of mathematics and both were conquered more than a century ago. Gamma does not live in a difficult [music] corner. It falls out of the very first sum a student ever adds up. 1 + 1/2 + 1/3. [music] It is the gap between that sum and the logarithm. A gap you can draw as a stack of little slivers on a page and that plain homely constant sitting in the most public place in all of mathematics still refuses to tell us the simplest thing about itself.

[00:12:01]We have trillions [music] of its digits.

[00:12:02]We have proofs that it appears in the zeta function and the gamma function and a dozen places besides. What we do not have three centuries in is an answer to the question a child could ask. Is this number a fraction? Yes or no? Gamma is not saying.