| Continued fraction | 1 + \cfrac{1}{1 + \cfrac{1}{2 + \cfrac{1}{1 + \cfrac{1}{2 + \cfrac{1}{1 + \ddots}}}}} |
|---|---|
| Decimal | 1.73205 08075 68877 2935... |
The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as \sqrt {3} or 3^{1/2}. 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's constant, after Theodorus of Cyrene, who proved its irrationality.[1]
In 2013, its numerical value in decimal notation was computed to ten billion digits.[2] Its decimal expansion, written here to 60 decimal places, is given by :
- 1.73205 08075 68877 29352 74463 41505 87236 69428 05253 81038 06280 55806
Archimedes reported a range for its value: (\frac{1351}{780})^{2}>3>(\frac{265}{153})^{2}.[3]
The upper limit \frac {1351}{780} is an accurate approximation for \sqrt {3} to \frac {1}{608,400} (six decimal places, relative error 3 \times 10^{-7}) and the lower limit \frac {265}{153} to \frac {2}{23,409} (four decimal places, relative error 1\times 10^{-5}).
Rational approximations
The square root of 3 is an irrational number, meaning it can not be exactly represented as a fraction x/y where x and y are integers. However, it can be approximated arbitrarily closely by such rational numbers.
Particularly good approximations are the integer solutions of Pell's equations,
x^2 - 3y^2 = 1
which can be algebraically rearranged into the form
\frac{x}{y} = \sqrt{3 + \frac{1}{y^2}} .
The first several solutions are given below:
\boldsymbol n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | \ldots |
|---|---|---|---|---|---|---|---|---|---|---|
\frac{\boldsymbol{x_n}\vphantom{t} }{\boldsymbol {y_n} } | \frac{2}{1} | \frac{7}{4} | \frac{26}{15} | \frac{97}{56} | \frac{362}{209} | \frac{1351}{980} | \frac{5042}{2911} | \frac{18817}{10864} | \frac{70226}{40545} | \ldots |
These approximations also appear among the convergents of its continued fraction.
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 \frac{1}{2} and \frac{\sqrt{3}}{2}. From this, \tan{60^\circ}=\sqrt{3}, \sin{60^\circ}=\frac {\sqrt{3}}{2}, and \cos{30^\circ}=\frac {\sqrt{3}}{2}.
The square root of 3 also appears in algebraic expressions for various other trigonometric constants, including[4] the sines of other angles. For example, \tan{15^\circ}=2-\sqrt{3} and \tan{75^\circ}=2+\sqrt{3}.
It is the distance between parallel sides of a regular hexagon with sides of length 1. It is also the length of the longest side of a triangle formed from two adjacent sides of a regular hexagon; following from the law of cosines:
\begin{align}
c^2 &= a^2 + b^2 - 2ab\cos\gamma, \\[3mu]
\end{align}
Since each angle of a regular hexagon is 120°, we can substitute 120° for \gamma in the equation above.
\begin{align}
c^2 &= 1^2 + 1^2 - 2ab\cos 120^\circ \\
&= 1 + 1 - 2(-1/2) \\
&= 2 - (-1) \\
&= 3 \\
c = \sqrt 3
\end{align}
It is the length of the space diagonal of a unit cube.
The vesica piscis has a major axis to minor axis ratio equal to \sqrt{3}:1. This can be shown by constructing two equilateral triangles within it.
Applications
Electrical engineering
The square root of 3 plays a pivotal role in studies of three-phase electric power. [5][6]
In the delta circuit, loads are connected across the lines, and so loads see line-to-line voltages:[7]
\begin{align} V_{12} &= V_1 - V_2 = (V_\text{LN}\angle 0^\circ) - (V_\text{LN}\angle {-120}^\circ) \\ &= \sqrt{3}V_\text{LN}\angle 30^\circ = \sqrt{3}V_{1}\angle (\phi_{V_1} + 30^\circ), \\ V_{23} &= V_2 - V_3 = (V_\text{LN}\angle {-120}^\circ) - (V_\text{LN}\angle 120^\circ) \\ &= \sqrt{3}V_\text{LN}\angle {-90}^\circ = \sqrt{3}V_{2}\angle (\phi_{V_2} + 30^\circ), \\ V_{31} &= V_3 - V_1 = (V_\text{LN}\angle 120^\circ) - (V_\text{LN}\angle 0^\circ) \\ &= \sqrt{3}V_\text{LN}\angle 150^\circ = \sqrt{3}V_{3}\angle (\phi_{V_3} + 30^\circ). \end{align}
(Φv1 is the phase shift for the first voltage, commonly taken to be 0°; in this case, Φv2 = −120° and Φv3 = −240° or 120°.)
References
- ^ "square root of 3". planetmath.org. Retrieved 2025-07-23.
- ^ Komsta (December 2013). "Computations | Łukasz Komsta". komsta.net. WordPress. Archived from the original on 2023-10-02. Retrieved September 24, 2016.
- ^ Knorr, Wilbur R. (June 1976). "Archimedes and the measurement of the circle: a new interpretation". Archive for History of Exact Sciences. 15 (2): 115–140. doi:10.1007/bf00348496. JSTOR 41133444. MR 0497462. S2CID 120954547. Retrieved November 15, 2022. – via SpringerLink.
- ^ Wiseman, Julian D. A. (June 2008). "Sin and Cos in Surds". JDAWiseman.com. Retrieved November 15, 2022.
- ^ "The Complete Guide to the Square Root of Three in Power Calculations". Relay Training.
- ^ "Why the √3?". Schnackel.
- ^ GloverSarma2011
Further reading
- Podestá, Ricardo A. (2023). "Geometric proofs that
\sqrt3,\sqrt5and\sqrt7are irrational". Mathematics Magazine. 96 (1): 34–39. arXiv:2003.06627. doi:10.1080/0025570X.2023.2168436. MR 4556102. - Wells, D. (1997). The Penguin Dictionary of Curious and Interesting Numbers. Revised ed. London: Penguin Group. p. 23.
External links
- Theodorus' Constant at MathWorld
- Kevin Brown, Archimedes and the Square Root of 3