Etydhv

I am writing a paper that deals with Gaussian integers and I have a section where I am talking about the norm of the units in the Gaussian integers. Since there are only four units in the Gaussian integers, I'd like to have the calculations of their norms appear neatly in four columns that are centered. Basically so that you can easily look down each of the four columns and see the calculations for each unit. What I have right now sort of does that but its very sloppy and I'm sure there is a much cleaner way to do it. I'm still new to Latex so I realize this code is probably painfully inefficient but I would appreciate any suggestions of how to fix it and what packages I'd need to use to implement those changes.

begincenter 
$N(1)=N(1+0i)$ 
hspace.025cm 
$N(-1)=N(-1+0i)$ 
hspace.025cm 
$N(i)=N(0+1i)$ 
hspace.025cm 
$N(-i)=N(0-1i)$ 
endcenter 

begincenter 
$N(1)=1^2+0$ 
hspace.1cm 
$N(-1)=(-1)^2+0^2$ 
hspace.1cm 
$N(i)=0^2+1^2$ 
hspace.1cm 
$N(-i)=0^2+(-1)^2$ 
endcenter 

begincenter 
$N(1)=1$ 
hspace.5cm 
$N(-1)=1$ 
hspace.5cm 
$N(i)=1$ 
hspace.5cm 
$N(-i)=1$ 
endcenter

Here are three different suggestions using either array or alignedat:

documentclassarticle
usepackagegeometry
usepackageamsmath

begindocument

[ 
beginarrayllll
N(1)=N(1+0i) & N(-1)=N(-1+0i) & N(i)=N(0+1i) & N(-i)=N(0-1i) \
N(1)=1^2+0 & N(-1)=(-1)^2+0^2 & N(i)=0^2+1^2 & N(-i)=0^2+(-1)^2 \
N(1)=1 & N(-1)=1 & N(i)=1 & N(-i)=1
endarray
]

[ 
beginarraycccc
N(1)=N(1+0i) & N(-1)=N(-1+0i) & N(i)=N(0+1i) & N(-i)=N(0-1i) \
N(1)=1^2+0 & N(-1)=(-1)^2+0^2 & N(i)=0^2+1^2 & N(-i)=0^2+(-1)^2 \
N(1)=1 & N(-1)=1 & N(i)=1 & N(-i)=1
endarray
]


beginalignat*4
N(1)&=N(1+0i) &quad N(-1)&=N(-1+0i) &quad N(i)&=N(0+1i) &quad N(-i)&=N(0-1i) \
N(1)&=1^2+0 & N(-1)&=(-1)^2+0^2 & N(i)&=0^2+1^2 & N(-i)&=0^2+(-1)^2 \
N(1)&=1 & N(-1)&=1 & N(i)&=1 & N(-i)&=1
endalignat*

enddocument

For example I would have used the tables in sequence (but it's just a personal taste) to give a touch of vitality.
In this case I have used the booktabs package to have toprule and bottomrule.

documentclass[a4paper,12pt]article
usepackagearray
usepackagebooktabs

begindocument
[
setlengtharraycolsep0pt
renewcommandarraystretch1.25
beginarrayr @= l
toprule
N(1) &N(1+0i) \
N(-1)&N(-1+0i) \
N(i) &N(0+1i) \
N(-i)&N(1+0i) \ 
bottomrule
endarray
quadRightarrowquad
beginarrayr @= l
toprule
N(1) &1^2+0 \
N(-1)&(-1)^2+0^2\
N(i) &0^2+1^2 \
N(-i)&0^2+(-1)^2\ 
bottomrule
endarray
quadRightarrowquad
beginarrayr @= l
toprule
N(1) &1 \
N(-1)&1 \
N(i) &1 \
N(-i)&1 \ 
bottomrule
endarray
]
enddocument

I propose this layout:

documentclassarticle
usepackage[utf8]inputenc%
usepackagegeometry 
usepackage[table, svgnames]xcolor
usepackagemathtools
colorletshadecolorGainsboro!50! Lavender
newcommand*shadebox[1]%
colorboxshadecolorhspace1em$displaystyle #1 $hspace1em
begindocument


beginequation
begingathered
beginaligned
 N(1) & =N(1+0i)\
 & =1^2+0^2
endaligned \[1.5ex]
shadeboxN(1) = 1
endgathered
 qquad
begingathered
beginaligned
 N(-1) & =N(-1+0i) \
& =(-1)^2+0^2
endaligned \[1ex]
 shadeboxN(-1) = 1
endgathered
 qquad
begingathered
beginaligned
 N(i) & =N(0+1i) \
 & =0^2+1^2
endaligned \[1ex]
shadebox N(i) = 1
endgathered
 qquad
begingathered
beginaligned
 N(-i) & =N(0-1i) \
 & =0^2+(-1)^2
endaligned \[1ex]
 shadeboxN(-i) = 1
endgathered
endequation

enddocument