Szczerski FADEM Rep. We Are the Champions Painting by Andrzej Szczerski

¾∫LAW OF Mathematical βeta ft.
Galileo @ Geo∫ II KinGPIN I |\/\/| |\/\/\/|¼ recursion loop #TIME #LAW #CONTi

Since there’s clearly a discontinuous nature of the square root function in the complex plane, the following laws are not true in general.

• {\displaystyle {\sqrt {zw}}={\sqrt {z}}{\sqrt {w}}} (counterexample for the principal square root: z = −1 and w = −1) This equality is valid only when {\displaystyle -\pi <\theta _{z}+\theta _{w}\leq \pi }
• {\displaystyle {\frac {\sqrt {w}}{\sqrt {z}}}={\sqrt {\frac {w}{z}}}} (counterexample for the principal square root: w = 1 and z = −1)This equality is valid only when {\displaystyle -\pi <\theta _{w}-\theta _{z}\leq \pi }
• {\displaystyle {\sqrt {z^{*}}}=\left({\sqrt {z}}\right)^{*}} (counterexample for the principal square root: z = −1)This equality is valid only when {\displaystyle \theta _{z}\neq \pi }

A similar problem appears with other complex functions with branch cuts, e.g., the complex logarithm and the relations logz + logw = log(zw) or log(z^*) = log(z)^* which are not true in general.

Wrongly assuming one of these laws underlies several faulty “proofs”, for instance the following one showing that −1 = 1:

{\displaystyle {\begin{aligned}-1&=i\cdot i\\&={\sqrt {-1}}\cdot {\sqrt {-1}}\\&={\sqrt {\left(-1\right)\cdot \left(-1\right)}}\\&={\sqrt {1}}\\&=1\end{aligned}}}

The third equality cannot be justified (see invalid proof). It can be made to hold by changing the meaning of √ so that this no longer represents the principal square root (see above) but selects a branch for the square root that contains {\displaystyle {\sqrt {1}}\cdot {\sqrt {-1}}.} The left-hand side becomes either

{\displaystyle {\sqrt {-1}}\cdot {\sqrt {-1}}=i\cdot i=-1}

if the branch includes +i or

{\displaystyle {\sqrt {-1}}\cdot {\sqrt {-1}}=(-i)\cdot (-i)=-1}

if the branch includes −i, while the right-hand side becomes

{\displaystyle {\sqrt {\left(-1\right)\cdot \left(-1\right)}}={\sqrt {1}}=-1,}

where the last equality, {\displaystyle {\sqrt {1}}=-1,} is a consequence of the choice of branch in the redefinition of √.

Nth roots and polynomial roots

The definition of a square root of {\displaystyle x} as a number {\displaystyle y} such that {\displaystyle y^{2}=x} has been generalized in the following way.

A cube root of {\displaystyle x} is a number {\displaystyle y} such that {\displaystyle y^{3}=x}; it is denoted {\displaystyle {\sqrt[{3}]{x}}.}

If n is an integer greater than two, a nth root of {\displaystyle x} is a number {\displaystyle y} such that {\displaystyle y^{n}=x}; it is denoted {\displaystyle {\sqrt[{n}]{x}}.}

Given any polynomial p, a root of p is a number y such that p(y) = 0. For example, the nth roots of x are the roots of the polynomial (in y) {\displaystyle y^{n}-x.}

Abel–Ruffini theorem states that, in general, the roots of a polynomial of degree five or higher cannot be expressed in terms of nth roots.

Square roots of matrices and operators

If A is a positive-definite matrix or operator, then there exists precisely one positive definite matrix or operator B with B² = A; we then define A^1/2 = B. In general matrices may have multiple square roots or even an infinitude of them. For example, the 2 × 2 identity matrix has an infinity of square roots,^[23] though only one of them is positive definite.

References

• Dauben, Joseph W. (2007). “Chinese Mathematics I”. In Katz, Victor J. (ed.). The Mathematics of Egypt, Mesopotamia, China, India, and Islam. Princeton: Princeton University Press. ISBN 978-0-691-11485-9.
• Gel’fand, Izrael M.; Shen, Alexander (1993). Algebra (3rd ed.). Birkhäuser. p. 120. ISBN 0-8176-3677-3.
• Joseph, George (2000). The Crest of the Peacock. Princeton: Princeton University Press. ISBN 0-691-00659-8.
• Smith, David (1958). History of Mathematics. 2. New York: Dover Publications. ISBN 978-0-486-20430-7.
• Selin, Helaine (2008), Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer, Bibcode:2008ehst.book…..S, ISBN 978-1-4020-4559-2.

External links

Wikimedia Commons has media related to Square root.

• Algorithms, implementations, and more – Paul Hsieh’s square roots webpage
• How to manually find a square root
• AMS Featured Column, Galileo’s Arithmetic by Tony Philips – includes a section on how Galileo found square roots