MHB A complex numbers' modulus identity.

Alone
Messages
57
Reaction score
0
I am searching for a shortcut in the calculation of a proof.

The question is as follows:

2.12 Prove that:

$$|z_1|+|z_2| = |\frac{z_1+z_2}{2}-u|+|\frac{z_1+z_2}{2}+u|$$

where $z_1,z_2$ are two complex numbers and $u=\sqrt{z_1z_2}$.

I thought of showing that the squares of both sides of the above identity are equal, in which case since both sides of the above identity are nonnegative we will get the above identity.

The problem that it seems too tedious work, unless there's some trick to be used here?
 
Mathematics news on Phys.org
You have [math]\left|\frac{z_1+ z_2}{2}- \sqrt{z_1z_2}\right|= \left|\frac{z_1- 2\sqrt{z_1z_2}+ z_2}{2}\right|[/math].

Noting the resemblance to [math]a^2+ 2ab+ b^2[/math] I would let [math]a= \sqrt{z_1}[/math] and [math]b= \sqrt{z_2}[/math]. Then [math]\left|\frac{z_1- 2\sqrt{z_1z_2}+ z_2}{2}\right|= \frac{(a- b)^2}{2}[/math]. Do the same thing for [math]\left|\frac{z_1+ z_2}{2}+ \sqrt{z_1z_2}\right|= \left|\frac{z_1+ 2\sqrt{z_1z_2}+ z_2}{2}\right|[/math]
 
Last edited by a moderator:
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.
Back
Top