Changing the format of "section" and "equation" counters in document class "book"

Question

I am using document class 'book'. I am using the following code

\documentclass{book}
\usepackage{amsmath, amsthm, amssymb, amsfonts}
\theoremstyle{definition}
\newtheorem{definition}{Def{i}nition}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{proposition}[definition]{Proposition}
\begin{document}
\mainmatter
\chapter{Quadratic Equation}
\section{Introduction}
\begin{definition}
\quad An equation $ax^2 + bx + c = 0,\ a\neq 0,\ a,\ b,\ c \in     \mathbb{R}$ is a called \textbf{quadratic equation}. The values of $x$ which satisfies it, is called its \textbf{roots}.
\end{definition}
\begin{proposition}
\quad Roots of the quadratic $ax^2 + bx + c = 0,\ (a\neq 0,\ a,\ b,\ c \in \mathbb{R}$) are given by\\
\begin{align}
    x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\end{align}
\end{proposition}
\end{document}

I want section numbering '1' instead of '1.1' also the equation number is '1' instead of '1.1'. And the numbering definition and propositions should be 1.1 and 1.2.

I tried much to get this effect I could not do. Please help me.

Answer 1

Assuming your TeX distribution is reasonably up to date, you may achieve your formatting objectives by adding the instructions

\counterwithout{equation}{chapter}
\counterwithout{section}{chapter}

in the preamble. If your TeX distribution is somewhat out of date, you'll have to load the chngcntr package before executing the two preceding instructions.

Some additional comments:

• No need to insert an explicit line break, \\, before the start of the align environment. In fact, I'd go further and say that it's poor practice to insert \\ at that location.

• Since the displayed equation in the proposition consists of a single line, you should replace the align environment with an equation environment.

• Instead of writing a single inline equation, $ax^2 + bx + c = 0,\ a\neq 0,\ a,\ b,\ c \in \mathbb{R}$, I'd write three [3!] separate inline equations: `$ax^2 + bx + c = 0$, $a\neq 0$, $a,\ b,\ c \in \mathbb{R}$. If nothing else, have three short equations rather than one long one will ease TeX's job of finding suitable line breaks.

  Similarly, I would replace $ax^2 + bx + c = 0,\ (a\neq 0,\ a,\ b,\ c \in \mathbb{R}$) with $ax^2 + bx + c = 0$, ($a\neq 0$, $a,b,c\in\mathbb{R}$).

• I would assume that the purpose of {i} in \newtheorem{definition}{Def{i}nition}[section] is to break up the fi ligature. If that is the case, do be aware that using {} to break up ligatures works under pdfLaTeX but doesn't work under XeLaTeX and LuaLaTeX. Better to write \newtheorem{definition}{Def\kern0ptinition}[section].

\documentclass{book}
\usepackage{amsmath, amsthm, amssymb}
\theoremstyle{definition}
\newtheorem{definition}{Def\kern0ptinition}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{proposition}[definition]{Proposition}
\counterwithout{equation}{chapter}
\counterwithout{section}{chapter}
\begin{document}
\mainmatter
\chapter{Quadratic Equation}
\section{Introduction}
\begin{definition}
An equation $ax^2 + bx + c = 0$, $a\neq 0$, $a, b, c \in\mathbb{R}$ is a 
called a \textbf{quadratic equation}. The values of $x$ which satisfies 
it are called its \textbf{roots}.
\end{definition}
\begin{proposition}
The roots of the quadratic $ax^2 + bx + c = 0$, ($a\neq 0$, 
$a,b,c\in\mathbb{R}$) are given by
\begin{equation}
    x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ,.
\end{equation}
\end{proposition}
\end{document}