Square root of 3
The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as √. It is more precisely called the principal square root of 3, to distinguish it from the negative number with the same property. The square root of 3 is an irrational number. It is also known as Theodorus' constant, after Theodorus of Cyrene, who proved its irrationality.
The fraction 97/ (1.732142857...) can be used as an approximation. Despite having a denominator of only 56, it differs from the correct value by less than 1/ (approximately 9.2×10−5). The rounded value of 1.732 is correct to within 0.01% of the actual value.
So it's true to say:
then when :
It can also be expressed by generalized continued fractions such as
which is [1; 1, 2, 1, 2, 1, 2, 1, …] evaluated at every second term.
The following nested square expressions converge to √:
Proof of irrationality
Suppose that √ is rational, and express it in lowest possible terms (i.e., as a fully reduced fraction) as m/ for natural numbers m and n.
Therefore, multiplying by 1 will give an equal expression:
where q is the largest integer smaller than √. Note that both the numerator and the denominator have been multiplied by a number smaller than 1.
Through this, and by multiplying out both the numerator and the denominator, we get:
It follows that m can be replaced with √n:
Then, √ can also be replaced with m/ in the denominator:
The square of √ can be replaced by 3. As m/ is multiplied by n, their product equals m:
Then √ can be expressed in lower terms than m/ (since the first step reduced the sizes of both the numerator and the denominator, and subsequent steps did not change them) as 3n − mq/, which is a contradiction to the hypothesis that m/ was in lowest terms.
An alternate proof of this is, assuming √ = m/ with m/ being a fully reduced fraction:
Multiplying by n both terms, and then squaring both gives
Since the left side is divisible by 3, so is the right side, requiring that m be divisible by 3. Then, m can be expressed as 3k:
Therefore, dividing both terms by 3 gives:
Since the right side is divisible by 3, so is the left side and hence so is n. Thus, as both n and m are divisible by 3, they have a common factor and m/ is not a fully reduced fraction, contradicting the original premise.
Geometry and trigonometry
The square root of 3 can be found as the leg length of an equilateral triangle that encompasses a circle with a diameter of 1.
If an equilateral triangle with sides of length 1 is cut into two equal halves, by bisecting an internal angle across to make a right angle with one side, the right angle triangle's hypotenuse is length one and the sides are of length 1/ and √/. From this the trigonometric function tangent of 60° equals √, and the sine of 60° and the cosine of 30° both equal √/.
The square root of 3 also appears in algebraic expressions for various other trigonometric constants, including the sines of 3°, 12°, 15°, 21°, 24°, 33°, 39°, 48°, 51°, 57°, 66°, 69°, 75°, 78°, 84°, and 87°.
The vesica piscis has a major axis to minor axis ratio equal to 1:√, this can be shown by constructing two equilateral triangles within it.
Square root of −3
In power engineering, the voltage between two phases in a three-phase system equals √ times the line to neutral voltage. This is because any two phases are 120° apart, and two points on a circle 120 degrees apart are separated by √ times the radius (see geometry examples above).
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