Question

Write a pair of integers whose sum gives
(i) Zero
(ii) A negative integer
(iii) An integer smaller than both integers
(iv) An integer greater than both integers
(v) An integer smaller than only one of the integers.

Answer 1

Hint: We use the definition of integers to find the possible pair of integers in each part. Take any two variables which exist in a set of integers and try to find relations between the two variables if possible.
* The set of integers consist of positive natural numbers and their additive inverses (Called negative integers) and the number 0. Set can be represented as \[\left\{ {... - 3, - 2, - 1,0,1,2,3...} \right\}\]

Complete step-by-step answer:
Let us assume two variables as two integers.
Let \[x,y\] be two integers.
Now we solve each part separately.
(i) Zero:
Sum of two integers is given as zero
\[ \Rightarrow x + y = 0\]
Shift one integer to RHS of the equation
\[ \Rightarrow x = - y\]
Put \[y = 2\] then
\[ \Rightarrow x = - 2\]
So, the pair of integers is \[ - 2,2\]
(ii) A negative integer
Sum of two integers is a negative integer.
If we take one integer greater than the other integer, the greater integer should be a negative integer.
Let \[x = - 4,y = 2\]
\[ \Rightarrow x + y = - 4 + 2\]
\[ \Rightarrow x + y = - 2\]
\[\therefore \]The pair of integers is \[ - 4,2\]
(iii) An integer smaller than both integers
Sum of two integers is given to be smaller than both integers
We take both the integers as negative integers as on the addition we will move to the left side of the number line. As we move left on the number line the value decreases.
Let \[x = - 3,y = - 2\]
\[ \Rightarrow x + y = ( - 3) + ( - 2)\]
As negative sign multiplied with positive sign gives negative sign
\[ \Rightarrow x + y = - 3 - 2\]
\[ \Rightarrow x + y = - 5\]
We know \[ - 5 < - 3, - 5 < - 2\]
\[\therefore \]The pair of integers is \[ - 3, - 2\]
(iv) An integer greater than both integers
Sum of two integers is given to be greater than both integers
We take both the integers as positive integers as on the addition we will move to the right side of the number line. As we move right on the number line the value increases.
Let \[x = 3,y = 2\]
\[ \Rightarrow x + y = 3 + 2\]
\[ \Rightarrow x + y = 5\]
We know \[5 > 3,5 > 2\]
\[\therefore \]The pair of integers is \[3,2\]
(v) An integer smaller than only one of the integers.
Sum of two integers is given to be smaller than one of the integers
We take one integer as negative and the second integer as positive.
Let \[x = - 3,y = 4\]
\[ \Rightarrow x + y = ( - 3) + 4\]
\[ \Rightarrow x + y = 1\]
We know \[1 < 4,1 > - 3\]
\[\therefore \]The pair of integers is \[ - 3,4\]

Note: Students can many times make mistakes in selecting the pair of integers in the (v)th part as they don’t check the relation after assuming the integers. Keep in mind, after finding the sum always check if the sum satisfies the condition or not. Also, don’t assign any rational number instead of integers as some student are habitual in making pairs for easy numbers like 1 and \[\dfrac{1}{2}\], but \[\dfrac{1}{2}\]is a rational number so it cannot be included.