Square root of 2

Under root 2 ( square root of 2 ) is understood in mathematics that positive number whose square gives the number 2, so the number that applies. This number is determined uniquely irrational and is represented by. The first digits of its decimal expansion are: = 1.414213562 ...

• 4.1 Mnemonic for the first decimal

General

Irrationality

The square root of 2 is π as the circle number or Euler's number e is irrational. However, in contrast to the two, it is not transcendental but algebraically. As early as 500 BC, the Greeks Hippasos of Metapontum the irrationality known. The best known evidence comes from the living in the 4th century BC Greeks Euclid. He is considered the first known proof by contradiction in the history of mathematics. Even the Greek Plato is said to have delivered a proof.

Decimal places

Since square root of 2 is irrational, the number has infinitely many non-periodic decimal places in each place value system and therefore can be represented only approximately in the decimal system. The first 50 decimal digits are:

Continued fraction expansion

Another way to represent real numbers, the continued fraction expansion. The continued fraction of square root of 2 is - in contrast to the circular constant π - periodic because root 2 is a quadratic irrational number. For the n - th root of 2 for n > 2, however, this is not true.

This period is calculated from the fact that:

Geometric construction

Since irrational numbers have an infinitely long decimal, it is impossible to gauge exactly such a number with a ruler. However, it is possible the number to construct with ruler and compass: The diagonal of a square is times as long as its side length. So you could do a right-angled isosceles triangle with the other sides are each 1 unit long. The length of the hypotenuse is then units. To prove this, the Pythagorean theorem is sufficient: For the length of the diagonal.

The said triangle is also the beginning of the root worm.

History

Even the ancient civilizations have been thinking about the square root of 2. The ancient Indians appreciate = 1.414215686 .... This approximation agrees to five decimal places with the actual value of the deviation is only 0.0001502 percent. From their irrationality they probably knew nothing. The Babylonians, as well as the Sumerians estimated around 1950 BC, the square root of 2 converted yet to 1.41. Dating from around 1800 BC, a further approximation is handed down by the Babylonians. They used in their cuneiform a place value system to the base 60 and calculated the approximation

This approximation agrees to five decimal places with the actual value of the deviation is only -0.0000424 percent.

In the late 6th or early 5th century BC discovered Hippasos of Metapontum, a Pythagorean, on either a square or a regular pentagon that the ratio of side length to diagonal can not be represented with integers. He proved the existence of incommensurable sizes. An ancient legend that the publication of this knowledge was considered by the Pythagoreans as a secret betrayal, is untrustworthy according to the current state of research.

Others

• Shigeru Kondo published in 2009 the first 200 billion decimal places of the root 2 The 2010 achieved record is 1 trillion decimal places ( as at 23 March 2010).
• The ratio of the two side lengths of a sheet of DIN - A format is with rounding to the nearest millimeter and contrary to popular assumption is not the golden section. This ensures that when dividing the sheet along the longer side again produced a leaf in the DIN -A format ( with increased by one numbering).
• The square root of 2 is the frequency ratio of two tones in music with equal temperament, which form a tritone, so half an octave.

A reminder for the first decimal

The first four two blocks 14, 14, 21, and 35 decimal places of root 2 are interpreted as two-digit numbers, all by seven divisible. The next four digits can be divided into seven blocks divisible by 623 and 7.