Square root of 2

Imagine a square. All its sides are the same length. If you draw a line from one corner to another, you make a diagonal.

This line has a special length. It is a number that never ends.

Long ago, people found this out. Some people in Greece even kept it a secret. They found out the number is irrational. This means it cannot be a simple fraction.

Old clay tablets show people knew this too.

They used it to measure things. It is a very famous number in math.

Imagine a square where each side is one unit long. If you draw a line from one corner to the opposite corner, you make a diagonal.

This diagonal has a special length called the square root of 2. It is about 1.4142. If you multiply this number by itself, you get exactly 2.

This number is irrational. This means you cannot write it as a simple fraction. It is a number that never ends and never repeats.

Long ago, a group called the Pythagoreans discovered this. Some stories say they kept it a secret. Legend says a man named Hippasus was even killed for telling people about it.

Many cultures used this number.

An old Babylonian clay tablet shows they knew it well. They used it to make very close guesses. In ancient India, math texts also gave ways to find it. Even Roman builders used it to design buildings. They used the diagonal of a square to make new, larger squares.

Imagine a perfect square where every side is exactly one unit long. If you draw a line from one corner to the opposite corner, you create a diagonal.

This diagonal has a special length called the square root of 2. It is a number that equals about 1.4142 when written as a decimal. If you multiply this number by itself, the result is exactly 2. This number is called an algebraic number. It is also known as the principal square root of 2. This name helps us tell it apart from its negative version.

This number has a very strange property called being irrational. This means you cannot write it as a simple fraction using two whole numbers. It is a decimal that goes on forever without ever repeating a pattern.

Long ago, a group of thinkers called the Pythagoreans discovered this fact. They found that the diagonal of a square is incommensurable with its side. This means the two lengths do not share a common unit of measure. Some legends say a man named Hippasus was killed for sharing this secret. However, there is not much real evidence to prove that story is true.

People have been studying this number for thousands of years.

A Babylonian clay tablet called YBC 7289 shows they knew it well. This tablet dates back to around 1600 BC. It shows a very accurate guess for the number using their own math system. In ancient India, math texts called the Sulbasutras also gave ways to find it. These texts were written around 200 BC. They used a method involving adding parts of the side to find a better length.

Math experts use different ways to find the digits of this number. One common way is called the Babylonian method. This is a type of algorithm used by many modern computers and calculators. Another way is called Newton's method, which uses a guess to get closer to the truth.

People love to see how many digits they can find. In 1997, a team found over 137 billion digits. In 2010, Shigeru Kondo calculated one trillion decimal places. These big tasks help us understand how numbers behave.

We can see this math working in the real world too.

Ancient Roman architects used a technique called ad quadratum. This method used the diagonal of a square to build new shapes. They used it to design beautiful floors and large rooms called atria. This technique is like using a square to grow into a larger square. It shows how a simple shape can lead to much bigger designs. Even today, we use these ideas to understand how shapes and sizes fit together.

The square root of 2 is a unique and fundamental number in mathematics. It is defined as the positive real number that, when multiplied by itself, equals exactly 2. In mathematical notation, it is written as $\sqrt{2}$. Because it is the solution to an algebraic equation, it is classified as an algebraic number. To be precise, mathematicians often call it the principal square root of 2. This specific name distinguishes it from the negative number that also results in 2 when squared.

Geometrically, this number represents a specific physical distance. If you have a square where every side is exactly one unit long, the diagonal across that square is the square root of 2. This relationship is a direct result of the Pythagorean theorem. This connection makes the number vital for understanding how shapes and lengths relate to one another. It is not just an abstract concept; it is a measurable part of geometry.

The most famous property of the square root of 2 is that it is irrational. An irrational number is a number that cannot be expressed as a simple fraction of two integers. Its decimal expansion, which begins approximately as 1.4142, continues forever without ever falling into a repeating pattern. This means you can never write down the exact value using digits alone. This discovery was a turning point in mathematical history.

History shows that many ancient civilizations understood this number deeply. The Babylonian clay tablet YBC 7289, dating to approximately 1600 BC, contains a highly accurate approximation.

Using a sexagesimal system, the tablet shows a value accurate to about six decimal places. In ancient India, the Sulbasutras from around 200 BC provided another method. They suggested increasing the length of a side by one-third, then adding one-fourth of that third, and finally subtracting one-thirty-fourth of that fourth. This method produced a very close approximation of the true value.

The discovery of irrationality is often linked to the Pythagoreans in ancient Greece. They realized that the diagonal of a square was incommensurable with its side. This meant no common unit could measure both lengths perfectly. While the name Hippasus of Metapontum is often associated with this discovery, much of the story is legend. Some tales suggest Hippasus was punished for revealing this secret, though historical evidence for this is thin. Because of this, the number is sometimes called Pythagoras's constant.

Architects have also used the properties of this number for centuries. Ancient Roman architecture utilized a technique known as ad quadratum.

This method involves a geometric progression used to double a square. In this process, the diagonal of one square becomes the side of the next, larger square. The architect Vitruvius attributed this idea to the philosopher Plato. This technique helped designers create balanced proportions for pavements and large rooms called atria.

To find the digits of the square root of 2, mathematicians use various algorithms. One common method is the Babylonian method, which is a specific type of Newton's method. This process starts with an initial guess and uses a recursive formula to improve it. Each step of the calculation roughly doubles the number of correct decimal digits. This efficiency is why the method is often used as a basis for calculations in modern computers and calculators.

Today, calculating the digits of this number is a way to test computing power. In 1997, a team led by Yasumasa Kanada calculated over 137 billion decimal places. Later, in 2010, Shigeru Kondo reached the milestone of one trillion decimal places. These massive computations help mathematicians investigate whether such numbers are "normal." A normal number is one where every digit and sequence of digits appears with the same frequency. This work connects ancient geometry to the cutting edge of modern computer science.