Question

Write a pair of integers whose difference is $4$.

Answer 1

Hint: We will solve this problem by using the definition of integers. We will also discuss commutative and associative laws while solving this problem. We will also use some variables and algebraic expressions and by solving those, we will solve this problem.

Complete step by step answer:
Let us know about integers. Integers are the numbers which contain both positive and negative values of all the natural numbers including zero. So, the range of integers is,
\[\left( { - \infty ,..... - 4, - 3, - 2, - 1,0,1,2,3,4,....\infty } \right)\]
Integers are commutative under addition and multiplication and also associative. That means, if we add or multiply TWO integers irrespective of order, the sum or the product remains the same (Commutative). And similarly, if we add or multiply THREE integers irrespective of order, the sum or the product remains the same (Associative). And addition or subtraction or multiplication of two integers also give an integer as result.

Now, it is given that, difference between two integers is 4.So, let one integer be \[x\] and another integer be \[y\].So, mathematically, we can write it as \[x \sim y = 4\] which means either \[x - y = 4\] or \[y - x = 4\] Now, let us consider only the first expression which is \[x - y = 4\]. We can also write it as,
\[ \Rightarrow y = x - 4\]
Now, let us substitute different values of \[x\], and find the values of \[y\] which makes a pair of integers whose difference is 4.
Substitute \[x = 0\],\[ \Rightarrow y = 0 - 4 = - 4\]
Substitute \[x = 1\], \[ \Rightarrow y = 1 - 4 = - 3\]
Substitute \[x = - 1\], \[ \Rightarrow y = - 1 - 4 = - 5\]
Substitute \[x = 4\], \[ \Rightarrow y = 4 - 4 = 0\]
Substitute \[x = 6\], \[ \Rightarrow y = 6 - 4 = 2\]

Therefore, some pair of integers whose difference is 4 are \[\left( {0, - 4} \right),\left( {1, - 3} \right),\left( { - 1, - 5} \right),\left( {4,0} \right),\left( {6,2} \right)\].

Note: We can write infinitely many pairs of numbers whose difference is 4. Here, we considered only one equation and found pairs. But we can also find the pairs by taking the second equation too and we will get the same result. Mathematically, commutative law is written as \[a + b = b + a\] and \[a \times b = b \times a\]. And associative law can be written as \[a + (b + c) = (a + b) + c\] and \[a \times (b \times c) = (a \times b) \times c\].