It is just a number. It is roughly 1.41. But the square root of two is actually a bit of a nightmare if you value your sanity and like things to be neat and tidy. Legend has it—and historians like Kurt von Fritz have dug into this—that a guy named Hippasus was supposedly tossed overboard by his fellow Pythagoreans just for proving this number exists. They were a cult that believed the whole universe was built on clean, whole ratios. Then comes $\sqrt{2}$, an irrational monster that never ends and never repeats, and suddenly their perfect world falls apart.
You’ve probably seen it on a calculator. 1.41421356... and it just keeps going. It doesn't stop. It can't be written as a simple fraction like 3/4 or 22/7. That's what makes it "irrational." It literally defies the logic of basic counting numbers.
The Geometry of Your Printer Paper
Believe it or not, you probably hold the square root of two in your hand every time you grab a piece of A4 paper. If you aren't in the US using "Letter" size, your paper is built on this exact ratio.
Think about it.
If you take an A4 sheet and fold it in half, you get an A5 sheet. The magic part? The proportions stay exactly the same. The ratio of the long side to the short side is always $\sqrt{2}$ to 1. This isn't just a random design choice made by bored bureaucrats in a committee room. It’s incredibly efficient for scaling. If you want to enlarge a drawing from A4 to A3, you just scale it by 141%. Because the ratio is constant, nothing gets distorted, and no paper is wasted during the cutting process at the factory.
In the US, we use 8.5 by 11 inches. If you fold that in half, the shape changes completely. It gets squat and fat. We’re basically living in a geometric mess while the rest of the world uses the elegance of irrational numbers to keep their stationery consistent.
The Pythagorean Headache
The math behind it is deceptively simple. Imagine a square where every side is exactly one unit long. If you draw a line from one corner to the opposite corner—the diagonal—how long is that line?
According to Pythagoras, $a^2 + b^2 = c^2$.
So, $1^2 + 1^2 = c^2$.
$1 + 1 = 2$.
Therefore, $c = \sqrt{2}$.
This is where the ancient Greeks hit a wall. They could see the line. They could draw the line. But they couldn't measure it with a ruler that used whole numbers or fractions. It was a physical reality that their math couldn't explain. It’s called Pythagoras’ Constant.
Honestly, it’s kind of wild that something as "real" as the distance across a square can't be expressed as a normal fraction. It’s a gap in the universe. If you try to write it out, you’ll be typing digits until the sun burns out. Even with the fastest supercomputers we have in 2026, we’re just finding more digits, but we’re never getting to "the end." There is no end.
It Is Not Just for Math Nerds
You’ll find the square root of two hiding in your camera bag too. Ever wonder why f-stops on a lens have those weird numbers? f/1.4, f/2, f/2.8, f/4, f/5.6.
It looks like a random sequence, but it's not. Each step is a multiple of the square root of two.
When you move from f/2 to f/1.4, you are doubling the area of the aperture hole. To double the area of a circle, you don't double the diameter—that would quadruple the area. You multiply the diameter by $\sqrt{2}$. This allows exactly twice as much light to hit the sensor. Photographers use this every day without realizing they are performing high-level irrational geometry just to get a clear shot of their lunch.
Why We Can't Just Round It Off
You might think, "Why don't we just call it 1.41 and move on with our lives?"
In the real world, 1.41 is usually fine. If you’re building a bookshelf, 1.41 is plenty of precision. But if you’re doing GPS calculations or high-frequency trading algorithms, those missing decimals start to matter.
Errors compound.
If you use a rounded version of the square root of two in a complex structural engineering simulation, the tiny discrepancy at the beginning can lead to a bridge collapsing or a satellite missing its orbit by miles. We need the "idea" of the absolute square root, even if we can't write it down.
A Quick Reality Check on Ratios
People often confuse $\sqrt{2}$ with other famous constants. It isn't Pi. Pi ($\pi$) is about circles. $\sqrt{2}$ is about squares and triangles. It also isn't the Golden Ratio ($\phi$), which is about 1.618 and shows up in flowers and shells.
The square root of two is grittier. It’s the foundational block of trigonometry. Without it, we wouldn't have the sine and cosine of 45-degree angles, which are both based on $1/\sqrt{2}$. Basically, half of all modern construction and navigation would be a guessing game without this number.
Practical Ways to Use This Knowledge
Knowing about the square root of two isn't just for winning trivia nights. It changes how you look at the world’s proportions.
- When scaling graphics: If you need to double the area of a square or circle in Photoshop, multiply the dimensions by 1.414. It works every time.
- Buying tech: TV sizes are measured diagonally. If you have a square screen (which is rare now, but bear with me), a 50-inch diagonal means the width is only about 35 inches. That $\sqrt{2}$ ratio is what dictates how much "wall space" you actually lose.
- Paper folding: If you’re into origami or DIY zines, understanding the A4 ratio helps you design layouts that scale perfectly from a tiny booklet to a massive poster without awkward cropping.
The square root of two is a reminder that the world isn't as simple as we want it to be. It’s messy. It’s infinite. But it’s also incredibly consistent. Next time you see a diagonal line, just remember that you’re looking at a number that once got a man killed just for talking about it.
Actionable Next Steps
To truly wrap your head around how this works in the real world, try these three things:
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- Check your camera: Look at your lens f-stops. Multiply 1.414 by 1.414. You get 2. Multiply 2 by 1.414. You get 2.8. See the pattern? It’s the secret language of light.
- Test the A4 trick: Take any ISO-standard paper (A4, A5, A3). Measure the long side and divide it by the short side. You’ll get something very close to 1.414.
- Calculate a diagonal: Measure a square table in your house. Multiply the side length by 1.414. Then measure the diagonal with a tape measure. It's a weirdly satisfying way to see math actually "working" in your living room.
Understanding this number gives you a better grasp of scaling, light, and the physical limits of measurement. It’s the simplest "complicated" thing in the universe.