Square root of 2

The square root of 2 is a special number that is about 1.4142. When you multiply this number by itself, you get 2. You can write it as √2 or 2^1/2. This number is very important in math because it is an algebraic number, not a transcendental number.

In geometry, the square root of 2 is the length of the diagonal of a square whose sides are each one unit long. This comes from the Pythagorean theorem. It was one of the first numbers discovered to be irrational, meaning it cannot be written as a simple fraction.

A good approximation for the square root of 2 is the fraction 99/70, which is about 1.4142857 and works well for many calculations. The digits of the square root of 2 go on forever without repeating, and they are listed in a special database called the On-Line Encyclopedia of Integer Sequences.

History

Long ago, people tried to find a number that, when multiplied by itself, equals 2. An old Babylonian tablet from around 1800–1600 BC shows a way to get close to this number using a special counting system. They found an answer that is very close to the true value.

Ancient Indian texts also share a method to get near this number. They used a step-by-step process to improve their guess. Even though their answer was a bit less exact, it was still very good.

The discovery that the square root of 2 cannot be written as a simple fraction surprised many ancient thinkers. Some stories say a man named Hippasus shared this secret and faced serious trouble, but we do not know much about what really happened.

In building, ancient Romans used this number in their designs. They followed a method to create larger squares by using the diagonal of a smaller square. This helped them make beautiful patterns and spaces in their buildings.

Decimal value

Further information: Methods of computing square roots

There are many algorithms to find the square root of 2 as a decimal number. One common way is called the Babylonian method, which is a type of Newton's method. You start with a guess and then improve it step by step. Each step makes the guess better, often doubling how many correct digits you have.

Some simple fractions can also get close to the square root of 2. For example, 99 divided by 70 is about 1.4142, which is very close. Even better guesses can be made with more steps or different starting points.

In 1997, people calculated the square root of 2 to over 137 billion decimal places. Later, in 2010, someone used a home computer to find one trillion decimal places! These big calculations help us study numbers like this more deeply.

Proofs of irrationality

Proof by infinite descent

One way to show that the square root of 2 is not a rational number is through a proof by infinite descent. This means we assume it is rational and then find a contradiction.

1. Assume the square root of 2 can be written as a fraction of two whole numbers.
2. If these two numbers share a common factor, we can simplify them using a method called the Euclidean algorithm.
3. We then write the square root of 2 as a fraction where the top and bottom numbers have no common factors.
4. Squaring both sides shows that the top number squared equals two times the bottom number squared.
5. This means the top number must be even (divisible by 2).
6. Because the top number is even, we can write it as two times another whole number.
7. Substituting this back, we find that the bottom number must also be even.
8. But if both numbers are even, they share a common factor of 2, which contradicts our earlier step.

Since we reached a contradiction, our original assumption must be wrong. Therefore, the square root of 2 cannot be written as a fraction of two whole numbers, proving it is irrational.

This proof was hinted at by Aristotle and first appeared fully in Euclid's Elements.

Proof using reciprocals

Another method assumes, for contradiction, that the square root of 2 can be expressed as a fraction in its simplest form. Using algebraic manipulation and properties of integers, we again reach a contradiction that shows this assumption cannot be true.

Proof by unique factorization

This proof uses the idea that every whole number can be broken down into prime numbers in exactly one way. By examining the prime factors of both sides of the equation involving the square root of 2, we find a contradiction in the number of times the prime number 2 appears, proving the square root of 2 cannot be rational.

Application of the rational root theorem

The rational root theorem helps us understand possible rational solutions to certain equations. Applying this theorem to the equation x² - 2 = 0 shows that the square root of 2 cannot be a simple fraction of whole numbers, confirming it is irrational.

Geometric proofs

Tennenbaum's proof

A geometric proof by Stanley Tennenbaum uses areas of squares to show contradictions when assuming the square root of 2 is rational. By comparing areas and fitting squares together, we find impossible relationships that can only be resolved if the square root of 2 is not a rational number.

Tom M. Apostol's proof

Tom M. Apostol used geometric constructions with triangles and circles to create a smaller version of the original problem. This process can continue indefinitely, leading to a contradiction if we assume the square root of 2 is rational.

Constructive proof

Some proofs show not just that the square root of 2 is irrational, but also give a way to measure how far it is from any fraction, providing a stronger form of proof.

Proof by Pythagorean triples

This proof uses properties of special sets of three whole numbers that form the sides of right-angled triangles. By assuming the square root of 2 is rational and examining these number sets, we find a contradiction, proving it cannot be rational.

Multiplicative inverse

The multiplicative inverse, or reciprocal, of the square root of two is a special number often used in geometry and trigonometry. It is equal to half of the square root of two, and its decimal value starts with 0.707106781...

This number appears when we look at a unit vector that makes a 45° angle with the axes in a plane. The coordinates of this vector are both equal to this special number. This number also equals the sine and cosine of a 45° angle.

Properties

The square root of 2 is a special number. When you multiply it by itself, you get 2. It is written as √2 or 2^1/2.

One fun fact is that if you take 1 divided by (√2 − 1), you get √2 + 1. This works because (√2 + 1) multiplied by (√2 − 1) equals 1.

The square root of 2 also appears in some formulas for circles and angles. For example, it is part of Viète's formula for pi, which helps calculate the value of pi by using many square roots.

Representations

The square root of 2, written as √2 or 2^1/2, is a special number. When you multiply it by itself, you get 2. This number can be shown in many different ways using math.

One way is by using a special math rule that connects angles and these numbers. There are also ways to write it as a long chain of simple fractions that get closer and closer to the real value. These chains are called continued fractions.

Another interesting way is to use nested squares, where you keep putting the same pattern inside itself many times. This also gets closer to the square root of 2.

Applications

Paper size

In 1786, a German professor named Georg Christoph Lichtenberg discovered that if you take any sheet of paper and make the long side √2 times longer than the short side, you can fold it in half and line it up perfectly with the short side. This creates a new sheet with the same shape as the original. This special ratio was used when Germany made standard paper sizes in the early 1900s. Today, common paper sizes like A4 follow this ratio, which is √2.

Physical sciences

The square root of 2 shows up in some cool places in science:

• It is the ratio of frequencies in a special musical interval called a tritone in twelve-tone equal temperament music.
• It helps decide the steps between f-stops in camera lenses, which control how much light comes through.
• It relates to the angle of the Sun in the sky during certain days of the year.
• In your brain, there are special cells that help you understand space, and the way they are arranged changes by the square root of two.