Square root of 3

The square root of 3 (written ) is the positive real number that multiplied by itself 3 results. The root 3 is an irrational number. It is a mathematical constant, also known under the name of Theodorus 's constant, named after Theodore of Cyrene.

Approximation, the following applies:

Your continued fraction expansion is [1; 1,2,1,2,1,2,1,2,1,2, ...].

It is also and

Proof of the irrationality

Suppose that would be rational. Then you could write the number as a fraction of two relatively prime integers and:

By squaring the equation obtained

Or

In equation (1 ) the right side is just odd then, if is odd. Similarly, the left side (because 3 is odd ) if odd, when odd. Consequently, and have the same parity. As prime numbers but they can not both be just so and are both odd. Thus, they can be with suitable integers and in the form

. Write Substituting these expressions in (1), it follows

And after elementary transformations ( multiplying out, bring variable terms to one side, divide by 2)

In equation (2) the right side is straight, as it is double an integer, while the one on the left is an odd number. This is a contradiction, so the assumption would be rational, must be wrong. Consequently, irrational.

The first 100 decimal places

1.7320508075 6887729352 7446341505 8723669428 0525381038 0628055806 9794519330 1690880003 7081146186 7572485756

More decimal places are also found among sequence A002194 in OEIS.

Application

• The ratio between the diagonal of a three-dimensional cube and the edge length
• The distance between two opposite sides of a regular hexagon with a side length 1 is
• The chaining factor, the ratio of phase voltage ( 230 V) to line voltage (400 V), is the case of three -phase alternating current
• The height of an equilateral triangle with sides of length 1 is