Combining Magnitudes - Binary Stars

The "laws of magnitudes" are

M₂-M₁ = 2.5 log[L₁/L₂]

(on a calculator, enter the brightness ratio, hit the log₁₀ button and multiply by 2.5) and

L₂/L₁=10^{0.4(M₁-M₂)}

(on a calculator, enter 0.4 times the magnitude difference and hit the 10^x button.)

Pretty straightforward to use but what is the magnitude of α Cen which is a binary system with V_A=-0.003 and V_B=+1.333 (ArXiv 0401606)?
We want V_A+B - 0 = 2.5 log[L₀/(L_A+L_B)]
where 0 magnitude corresponds to L₀ which is in fact the magnitude zero point that nobody really seems to agree on... But that is no problem since we work with ratios and L₀/(L_A+L_B) is just the reciprocal of

L_A/L₀=10^0.0-0.4M_A  plus  L_B/L₀=10^0.0-0.4M_B  ie

V_A+B = 2.5 log[1/(10^-0.4M_A+10^-0.4M_B)] = -2.5 log[10^-0.4M_A+10^-0.4M_B+...]

and you can add in as many components as you wish.
Try combining two first magnitude stars: (note that 10^-0.4=1/2.512=0.398~0.4)
M(both)=-2.5log(1/2.512+1/2.512)=0.2474, about 0.25
You could have done that in your head! Two first magnitude stars are twice as bright as magnitude V=1, a factor of two is 0.75 magnitudes (subtract from 1, remember, brighter is smaller) or V=0.25.

For αCenAB we have V_AB=

-2.5log[10^-0.4*(-0.003)+10^-0.4*1.333]=-2.5log(10^0.0012+10^-0.5332)=-2.5log(1.2957)=-0.281

which nudges it ahead of Arcturus (V=-0.04) in the bright star race!

Finally, if the magnitude difference is greater than 2.5 magnitudes you can use the small magnitude difference cheat you use to amaze your astronomer friends: a magnitude difference of 1% is close to 0.01 magnitudes. (Works since the magnitude base is close to e...e^x~1+x....) Try a binary with A=5 and B=8. The V=8 is about 16x fainter than the V=5 so it would add about 100%/16 or 6.25% to the brightness bringing the V=5 down to V=4.9375. Doing the calculation exactly results in V_AB=4.9336.

Try a javascript addition of two magnitudes:

First star: Second star: combined magnitude: <---press

Some examples (Excel cut&paste--hope its right!):

Two brightness Here is Another way Stars: combined: ratios: to do two stars: M(A) M(B) M(A+B) A/B 1+B/A DM M(A)-DM 1 1 0.2474 1 2 0.752575 0.247425 3 3 2.2474 1 2 0.752575 2.247425 3 4 2.6361 2.511 1.398 0.363851 2.636149 7 9 6.8403 6.309 1.158 0.15973 6.84027 2 7 1.9892 100 1.01 0.010803 1.989197 5.7 8.2 5.596518 10 1.1 0.103482 5.596518 3.2 7.9 3.185781 75.86 1.013 0.014219 3.185781