What is the square root of 2?

The square root of 2 (√2), approximately equal to 1.414, is a famous number in mathematics for several reasons:

• Irrational Number: √2 is an irrational number. This means it cannot be expressed as a simple fraction (a ratio of two integers). Its decimal representation goes on forever without repeating. This was a significant discovery in ancient Greece, shattering the belief that all numbers could be expressed as ratios.

• Pythagorean Theorem: √2 arises naturally from the Pythagorean theorem. A right-angled triangle with two sides of length 1 has a hypotenuse of length √2. This is easily demonstrated using the theorem: a² + b² = c², where a and b are the lengths of the shorter sides and c is the length of the hypotenuse. In this case, 1² + 1² = c², so c = √2.

• Geometric Construction: √2 can be constructed geometrically using only a compass and straightedge. This is directly related to its connection to the Pythagorean theorem and the unit square diagonal.

• Continued Fraction: √2 has a simple and elegant continued fraction representation: 1 + 1/(2 + 1/(2 + 1/(2 + ...))). This infinite continued fraction provides a way to approximate √2 with increasing accuracy.

• Approximations: While its exact value is impossible to write down, √2 has been approximated to trillions of decimal places using sophisticated computational methods. Simple approximations include 1.414 or 7/5 (1.4).

• Significance in various fields: √2 appears in many areas of mathematics, physics, and engineering, often relating to geometric problems, trigonometry, and calculations involving distances and angles. For example, it's related to the diagonal of a square and the calculation of the length of the diagonal.

In short, √2 is a deceptively simple number with profound mathematical significance, highlighting the beauty and complexity of irrational numbers.