Question

Write all the integers between -4 and 4.

Answer 1

Hint: In this question, we are given two integers and we have to find all the integers between the two integers -4 and 4. Therefore, we should first understand the definition of integers and then find out how many integers lie between -4 and 4 and those integers will be the answer to this question.

Complete step-by-step answer:

Integers are defined as those numbers which can be written in the decimal form without any non-zero digit after the decimal point. Thus, all the integers can be obtained by zero and the numbers obtained by adding or subtracting any natural number (the set of natural numbers is the set {1,2,3…}) from zero………(1.0)
Thus, any the integer z should be of the form
$z=0+n................(1.1)$
Or $z=0-n....................(1.2)$
where n is some natural number.
Now, we need to find the integers between -4 and 4. Therefore, if the integer is positive, it should be of the form in (1.1) and it should be less than 4. Thus, all positive integers -4 and 4 should satisfy
$\begin{align}
  & z<4\Rightarrow z=0+n<4 \\
 & \Rightarrow n<4 \\
 & \Rightarrow n=1,2,3................(1.3) \\
\end{align}$
Thus from (1.2) and (1.3), we find that the positive integers -4 and 4 should be
$\begin{align}
  & z=0+1,0+2,0+3 \\
 & \Rightarrow z=1,2,3....................(1.4) \\
\end{align}$
If the integer is positive, it should be of the form in (1.2) and it should be greater than 4. Thus, all negative integers -4 and 4 should satisfy
$\begin{align}
  & z>4\Rightarrow z=0-n>4 \\
 & \Rightarrow n<4 \\
 & \Rightarrow n=1,2,3................(1.5) \\
\end{align}$
Thus from (1.2) and (1.3), we find that the positive integers -4 and 4 should be
$\begin{align}
  & z=0-1,0-2,0-3 \\
 & \Rightarrow z=-1,-2,-3....................(1.6) \\
\end{align}$
Thus from (1.0), (1.4) and (1.6), we find that all the integers between -4 and 4 should be -3, -2, -1, 0, 1, 2, 3 which is the required answer.

Note: In this question, we should be careful that although in equations (1.3) and (1.5) the values of n are the same the z are different. This is because the form assumed in deriving n in (1.3) was z=0+n while in (1.5) the assumed was z=0-n.