The Irrational Integer
1.1 Introduction.
Some expressions in mathematics just seem to defy logic… until just one very clever pattern is found… or revealed…
Today we have a classic from the Indian math legend Srinivasa Ramanujan; this was originally published in the Indian mathematics journal — — where nobody solved it for 6 months until Ramanujan himself gave the answer!
And yet when I explain the logic to you its gonna be so simple that after the first step you can do it yourself.
Thus, it is a great exercise in the realm of unconventional but hella clever mathematics — — Enjoy!
1.2 Set up
It all starts from the simple identity we all know:
Despite being so obvious, pay attention to the next manipulation, which is where the real magic happens.
And going just one step further,
as it is, this is an identity for all positive real numbers.
Don’t believe me?
Well lets try it with your favorite number. Putting x=17, we get:
which is perfectly correct!
This is the main idea of notice in this problem— this super hidden and sly pattern.
Ever thought x + 1 could be written as that kind of square root?
These kinds of concealed equalities are what competitive math questions love to exploit the most.
Moreover, the opposite direction is what you’d see in a ‘good’ question.
i.e. they might give you something like this to simplify:
If you had this fact in the back of your mind, this simplification would strike in seconds instead of minutes
Anyhow, to continue with Ramanujan's real mischief, notice: we have written the formula for x, but if it is an identity, it should hold for x +1 as well right?
Thus, putting x +1 in the formula, we get:
Which simplifies to:
Nothing, out of the ordinary right?
Well, here lies the secret.
In true mathematician fashion he asked : “But x+1= root(1+x(x+2)) right?”
Thus, he created the first iteration of this infinite pattern.
What’s stopping you from finding and expression for x +3 and plugging that in too? Nothing!
So keep going he did:
And you must see the pattern by now right? Continuing this infinitely, our boy obtained:
Yep, that is also an identity for all positive numbers — can you imagine that?
Get OreoPlushie’s stories in your inbox
Join Medium for free to get updates from this writer.
But, however beautifully it may be, there lies the question: “Are you allowed to do this infinitely?”
Like for finite terms we checked earlier and it held fine, but won’t going to infinity break something?
Well, luckily for us, this pattern doesn’t, and that was proved by Ramanujan himself.
So while I won’t be covering said proof here, its important to keep in mind how volatile infinity can be!
Just so we can take a look at some numbers for once, lets bring our favorite number back, see if you can evaluate this in your mind.
If you got the answer as 17, you’re perfectly right!!
1.3 The Concept of Beauty
The truth is, we are inherently uncomfortable with square roots and things we cannot mentally visualize.
Take a simple example: when you see root(4)or 2, which one feels more familiar? The 2, of course, even though it is deeply wired into our brains that they are one and the same.
This is where the concept of mathematical beauty comes in.
The force that drives mathematicians to pursue elegant proofs, solutions that others might dismiss as unnecessary. If 3.14 works fine, why bother?
But at the end of the day, this concept is what keeps Maths fun, interesting, and rejuvenating for all of us.
The strange beauty, the conflicting (maybe even paradoxical) sights it reveals to us, and the eccentric patterns hidden everywhere.
You might find a brutal, computer-generated proof for the Four-Color Theorem, but a mathematician will always hold out hope for that one elegant, general solution that fits on a single napkin.
I hope you get the point I want to make here.
Maths is a universal language but is far from only ours. Thus, it’s our job to find the beauty in every piece we come across, be it an infinite radical or a spiral.
And if you asked me, I doubt there was someone better than Srinivasa Ramanujan at this.
So, please, if you have time, look into his other mind-bending works and formulas; he was a masterclass of someone who lived and breathed in the dimension of math — someone who could see the beauty nearly physically.
1.4 BOSS FIGHT
I’m putting this here cause its customary so here we go, this is the original question put forward by Ramanujan:
At this point, it should be obvious to you what the answers are.
(ii) Uses a bit of shifting logic, try going back to where we started, and changing the initial condition a bit!
And to complement it, here’s the original solution provided by Ramanujan as well:
2.0 Conclusion
Wanna see another recursive pattern like this be fore you go?
Note: x²-x-1=0 has the Golden ratio(φ) and its -ve reciprocal as roots.
After a simple rearrangement, we get:
Wait but x is the same as 1 + 1/x, so:
And then continue this pattern forever:
We know this value is equal to the positive answer not the negative one cause well…obviously, but also there’s a pretty cool method of this system of solving quadratics, lets talk of it another day.
Ultimately, whether it is an infinite radical or a nested fraction, math is far more than cold calculation.
It is an art form hidden in plain sight.
Ramanujan lived in this exact dimension, seeing an equation not as data, but as a masterpiece.
To truly study math is not just for the sake of it, but to witness its profound, hidden beauty.
See ya’ll next week!
