How do you center multiple equations that have multiple steps?

Question

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.

\begin{center}  
$N(1)=N(1+0i)$  
\hspace{.025cm}  
$N(-1)=N(-1+0i)$  
\hspace{.025cm}  
$N(i)=N(0+1i)$  
\hspace{.025cm}  
$N(-i)=N(0-1i)$  
\end{center}  

\begin{center}  
$N(1)=1^2+0$  
\hspace{.1cm}  
$N(-1)=(-1)^2+0^2$  
\hspace{.1cm}  
$N(i)=0^2+1^2$  
\hspace{.1cm}  
$N(-i)=0^2+(-1)^2$  
\end{center}  

\begin{center}  
$N(1)=1$  
\hspace{.5cm}  
$N(-1)=1$  
\hspace{.5cm}  
$N(i)=1$  
\hspace{.5cm}  
$N(-i)=1$  
\end{center}

Answer 1

Here are three different suggestions using either array or alignedat:

\documentclass{article}
\usepackage{geometry}
\usepackage{amsmath}

\begin{document}

\[ 
\begin{array}{llll}
N(1)=N(1+0i) & N(-1)=N(-1+0i)   & N(i)=N(0+1i) & N(-i)=N(0-1i)   \\
N(1)=1^2+0   & N(-1)=(-1)^2+0^2 & N(i)=0^2+1^2 & N(-i)=0^2+(-1)^2 \\
N(1)=1       & N(-1)=1          & N(i)=1       & N(-i)=1
\end{array}
\]

\[ 
\begin{array}{cccc}
N(1)=N(1+0i) & N(-1)=N(-1+0i)   & N(i)=N(0+1i) & N(-i)=N(0-1i)   \\
N(1)=1^2+0   & N(-1)=(-1)^2+0^2 & N(i)=0^2+1^2 & N(-i)=0^2+(-1)^2 \\
N(1)=1       & N(-1)=1          & N(i)=1       & N(-i)=1
\end{array}
\]


\begin{alignat*}{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
\end{alignat*}

\end{document}

Answer 2

I propose this layout:

\documentclass{article}
\usepackage[utf8]{inputenc}%
\usepackage{geometry} 
\usepackage[table, svgnames]{xcolor}
\usepackage{mathtools}
\colorlet{shadecolor}{Gainsboro!50! Lavender}
\newcommand*\shadebox[1]{%
\colorbox{shadecolor}{\hspace{1em}$\displaystyle #1 $\hspace{1em}}}
\begin{document}


\begin{equation}
\begin{gathered}
\begin{aligned}
 N(1) & =N(1+0i)\\
  & =1^2+0^2
\end{aligned} \\[1.5ex]
\shadebox{N(1) = 1}
\end{gathered}
  \qquad
\begin{gathered}
\begin{aligned}
 N(-1) & =N(-1+0i) \\
& =(-1)^2+0^2
\end{aligned} \\[1ex]
 \shadebox{N(-1) = 1}
\end{gathered}
 \qquad
\begin{gathered}
\begin{aligned}
 N(i) & =N(0+1i) \\
  & =0^2+1^2
\end{aligned} \\[1ex]
\shadebox{ N(i) = 1}
\end{gathered}
 \qquad
\begin{gathered}
\begin{aligned}
 N(-i) & =N(0-1i) \\
  & =0^2+(-1)^2
\end{aligned} \\[1ex]
 \shadebox{N(-i) = 1}
\end{gathered}
\end{equation}

\end{document} 

Answer 3

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}
\usepackage{array}
\usepackage{booktabs}

\begin{document}
\[
\setlength\arraycolsep{0pt}
\renewcommand\arraystretch{1.25}
\begin{array}{r @{{}={}} l}
\toprule
N(1) &N(1+0i)   \\
N(-1)&N(-1+0i)  \\
N(i) &N(0+1i)   \\
N(-i)&N(1+0i)   \\  
\bottomrule
\end{array}
\quad\Rightarrow\quad
\begin{array}{r @{{}={}} l}
\toprule
N(1) &1^2+0     \\
N(-1)&(-1)^2+0^2\\
N(i) &0^2+1^2   \\
N(-i)&0^2+(-1)^2\\  
\bottomrule
\end{array}
\quad\Rightarrow\quad
\begin{array}{r @{{}={}} l}
\toprule
N(1) &1 \\
N(-1)&1 \\
N(i) &1 \\
N(-i)&1 \\  
\bottomrule
\end{array}
\]
\end{document}