Square root of 2

In mathematics, the square root of 2 is that positive number that multiplied by itself results in the number 2, i.e. the number for which applies. This number is unambiguous, irrational and is represented by . The first digits of their decimal fraction expansion are: = 1.414213562 ... ${\ displaystyle x> 0}$${\ displaystyle x ^ {2} = 2}$${\ displaystyle {\ sqrt {2}}}$${\ displaystyle {\ sqrt {2}}}$

General

Euclid (fictional after André Thevet , 1584)

irrationality

The square root of 2 is irrational like the circle number or Euler's number e. In contrast to the two, however, it is not transcendent , but algebraic . Already around 500 BC The Greek Hippasus of Metapontus knew irrationality. Probably the most famous proof of the irrationality of the square root of 2 was published around 300 BC. The Greek Euclid . ${\ displaystyle \ pi}$

Decimal places

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

${\ displaystyle {\ sqrt {2}} = 1 {,} 41421 \, 35623 \, 73095 \, 04880 \, 16887 \, 24209 \, 69807 \, 85696 \, 71875 \, 37694 \, \ ldots}$ (Follow A002193 in OEIS )

Continued fraction development

Another way to represent real numbers is the continued fraction expansion . The continued fraction representation of root 2 is - in contrast to the circle number  - periodic , because root 2 is a quadratic irrational number . However , this does not apply to the -th root of 2 with . ${\ displaystyle \ pi}$${\ displaystyle n}$${\ displaystyle n> 2}$

${\ displaystyle {\ sqrt {2}} = [1; \, 2, \, 2, \, 2, \, 2, \, 2, \, \ dotsc]}$ (Follow A040000 in OEIS )

This periodic development results from the following simple facts (with the Gaussian rounding function ): ${\ displaystyle x \ mapsto \ lfloor x \ rfloor}$

${\ displaystyle \ lfloor {\ sqrt {2}} \ rfloor = 1}$
${\ displaystyle {\ frac {1} {{\ sqrt {2}} - 1}} = {\ sqrt {2}} + 1}$
${\ displaystyle \ lfloor {\ sqrt {2}} + 1 \ rfloor = 2}$
${\ displaystyle {\ sqrt {2}} + 1-2 = {\ sqrt {2}} - 1}$

The first approximate fractions of the continued fraction expansion of are ${\ displaystyle {\ sqrt {2}}}$

${\ displaystyle {\ frac {1} {1}}, \, {\ frac {3} {2}}, \, {\ frac {7} {5}}, \, {\ frac {17} {12 }}, \, {\ frac {41} {29}}, \, {\ frac {99} {70}}, \, \ dotsc}$

Geometric construction

Construction of the root 2 on the number line

Since irrational numbers have an infinitely long decimal notation, it is impossible to measure such a number precisely with a ruler. But it is possible to construct the number with a pair of compasses and a ruler : the diagonal of a square is times as long as its side. A right-angled, isosceles triangle, in which the catheters are each 1 unit long, is also sufficient. The length of the hypotenuse is then units. To prove this, the Pythagorean theorem is sufficient : The following applies to the length of the diagonal . ${\ displaystyle {\ sqrt {2}}}$ ${\ displaystyle {\ sqrt {2}}}$${\ displaystyle {\ sqrt {2}}}$${\ displaystyle x}$${\ displaystyle x ^ {2} = 1 ^ {2} + 1 ^ {2}}$

The triangle mentioned is also the beginning of the root snail .

history

The ancient civilizations already thought about the root of 2. The ancient Indians estimate = 1, 41421 5686…. This approximation corresponds to the actual value of to five decimal places, the deviation is only +0.0001502 percent. They probably knew nothing of their irrationality. The Babylonians as well as the Sumerians estimated around 1950 BC The root of 2 converted to 1.41. From around 1800 BC Another approximation has been handed down from the Babylonians. In their cuneiform script, they used a place value system for base 60 and also calculated the approximation ${\ displaystyle {\ sqrt {2}} \ approx {\ tfrac {577} {408}}}$${\ displaystyle {\ sqrt {2}}}$

${\ displaystyle 1 \ cdot 60 ^ {0} +24 \ cdot 60 ^ {- 1} +51 \ cdot 60 ^ {- 2} +10 \ cdot 60 ^ {- 3} = {\ tfrac {30547} {21600 }}}$= 1.41 421 2962 ...

This approximation corresponds to the actual value of to five decimal places, the deviation is only −0.0000424 percent. ${\ displaystyle {\ sqrt {2}}}$

In the late 6th or early 5th century BC In BC, Hippasus of Metapontus , a Pythagorean , discovered either on a square or on a regular pentagon that the ratio of the length of the sides to the diagonal cannot be represented with whole numbers. With this he proved the existence of incommensurable quantities. An ancient legend according to which the Pythagoreans regarded the publication of this finding as a betrayal of secrets is, according to the current state of research, unreliable.

Others

• The record has been 10 trillion decimal places since June 19, 2016, achieved by Ron Watkins (as of April 24, 2018).
• The ratio of the two side lengths of a sheet of paper in A format is rounded to the nearest millimeter (and contrary to some assumptions has nothing to do with the golden ratio ). This ensures that if the sheet is cut in half along the longer side, a sheet in DIN A format (with numbering increased by one) is created again.${\ displaystyle {\ tfrac {1} {\ sqrt {2}}}}$ ${\ displaystyle {\ tfrac {1 + {\ sqrt {5}}} {2}}}$
• The root of 2 is the frequency ratio of two tones in music with an equal temperament , which form a tritone , i.e. half an octave .
• In electrical engineering, the relationship between the peak value and the rms value of sinusoidal alternating voltage also contains the constant .${\ displaystyle {\ sqrt {2}}}$

Memory aid for the first decimal places

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

Integer of expressions

• According to the binomial formulas, the following expression is a natural number for all integers :, namely the denominator of the continued fraction expansion of - the Pell sequence .${\ displaystyle n \ geq 0}$${\ displaystyle {\ frac {\ left (1 + {\ sqrt {2}} \ right) ^ {n} - \ left (1 - {\ sqrt {2}} \ right) ^ {n}} {2 { \ sqrt {2}}}}}$${\ displaystyle {\ sqrt {2}}}$