Some Important Definitions
Category : JEE Main & Advanced
(1) Real numbers : Real numbers are those which are either rational or irrational. The set of real numbers is denoted by \[R\].
(3) Variable: A variable is a symbol which can assume any value out of a given set of values.
(i) Independent variable : A variable which can take any arbitrary value, is called independent variable.
(ii) Dependent variable : A variable whose value depends upon the independent variable is called dependent variable.
(4) Constant : A constant is a symbol which does not change its value, i.e., retains the same value throughout a set of mathematical operation. These are generally denoted by \[a,\,\,b,\,\,c\] etc. There are two types of constant, absolute constant and arbitrary constant.
(5) Absolute value : The absolute value of a number \[x,\] denoted by \[|x|,\] is a number that satisfies the conditions
\[|x|=\left\{ \begin{matrix} -x \\ \,\,0 \\ \,\,\,x \\ \end{matrix}\,\,\,\begin{matrix} \text{if} \\ \text{if} \\ \text{if} \\ \end{matrix}\,\,\begin{matrix} x<0 \\ x=0 \\ x>0 \\ \end{matrix}. \right.\] We also define \[|x|,\]as follows, \[|x|=\] maximum \[\{x,\,\,-x\}\] or \[|x|=\sqrt{{{x}^{2}}}\].
(6) Fractional part : We know that \[x\ge [x].\] The difference between the number \['x'\] and it’s integral value \['[x]'\] is called the fractional part of \[x\] and is symbolically denoted as \[\{x\}\]. Thus, \[\{x\}=x-[x]\]e.g., if \[x=4.92\] then \[[x]=4\] and \[\{x\}=0.92\].
Fractional part of any number is always non-negative and less than one.
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