What does 1. mean in a mathematica solution (of a sum) -

i'm trying evaluate difficult sum: mathematica seems evaluate it, giving message "solve unable solve system inexact coefficients. answer obtained solving corresponding exact system , numericizing result"

the solution contains expressions "1." such (0.5 + 1.i).

what 1. mean?

you can @ similar question here. mathematica interprets input 0.5 (or input containing 0.5), example, "numerical," , attempts solve numerical in nature, assuming 0.5 real number within whatever relevant level of precision looks it's equal 0.5. though 0.5==1/2 return true, mathematica still treats 2 expressions differently.

if input commands using "numerical" (ie. decimal) numbers, mathematica falls numerical methods (like nintegrate, nsolve, ndsolve, numerical versions of arithmetic operations, etc.) rather apply integers, rationals, etc.

the error occurs due how nsolve (or such algorithm) works. takes step of making equations exact (it know, after all, 0.5=1/2) , gets exact solution, "numericizes" result (hits n command) give numerical equivalent.

type in n[1/2+i] , see get. should 0.5+1.i. means have quantity 1.0000000000000000 in imaginary direction , 0.50000000000000 in real direction.

to see difference explicitly, try:

head[1] head[1.] 

the decimal point indicates mathematica second of 2 "real" number, i.e. floating point arithmetic of sort. first 1 integer, mathematica uses different sorts of algorithms.