Addition of Integers
Let us play a game to understand addition of integers:
Name of the game: Plus-Minus Pursuit
Objective:
The goal is to be the first player to reach the + 15 position on the number strip without hitting -15.
Materials:
A number strip extending from -15 to +15.
One standard die numbered 1 to
One custom die with three sides marked with '+' and three sides with '-'.
Coloured buttons or plastic counters for each player.
Setup:
Each player chooses a coloured button or counter and places it at the
Gameplay:
Players decide the order of play, either by rolling the numbered die or through mutual agreement.
On their turn, each player rolls both dice simultaneously.
The player checks the number on the standard die and the sign on the custom die.
If the custom die shows a
If the custom die shows a
The players take turns rolling the dice and moving their buttons accordingly.
Rules:
If a player's counter lands exactly on +15, they win the game immediately.
If a player's counter moves beyond -15, they are out of the game.
The game continues until a player reaches +15 or all but one player is eliminated for reaching -15.
Winning the Game:
The first player to land on +15 wins the game.
If all players except one are eliminated, the remaining player is declared the winner.
Take a number strip marked with integers from + 15 to – 15.
Player 1
43
Player 2
24
You can play the same game with 12 cards marked with + 1, + 2, + 3, + 4,- 5 and + 6 and –1, – 2, ...– 6. Shuffle the cards after every attempt.
Kamla, Reshma and Meenu are playing this game.
Kamla got + 3, + 2, + 6 in three successive attempts. She kept her counter at the mark +11.
Reshma got – 5, + 3, + 1. She kept her counter at –1. Meenu got + 4, – 3, –2 in three successive attempts; at what position will her counter be
DO THESE
Take two different coloured buttons like white and black. Let us denote one white button by (+ 1) and one black button by (– 1).
A pair of one white button (+ 1) and one black button (– 1) will denote zero i.e. [1 + (– 1) =
In the following table, integers are shown with the help of coloured buttons
Coloured Button | Integers |
---|---|
Let us perform additions with the help of the coloured buttons.
You add when you have two positive integers like (+3) + (+2) = +
You also add when you have two negative integers, but the answer will take a minus (–) sign like (–2) + (–1) = – (2+1) =
In Integers, a positive number result from an addition is represented without the '+' sign, so +3 is written simply as
Find the answers of the following additions
Now add one positive integer with one negative integer with the help of these buttons.
Remove buttons in pairs i.e. a white button with a black button [since (+ 1) + (– 1) = 0]. Check the remaining buttons.
You can see that the answer of 4 – 3 is
It is not necessary to display the sign, as positive numbers are assumed by default when no sign is present.
So, when you have one positive and one negative integer, you must subtract, but answer will take the sign of the bigger integer (Ignoring the signs of the numbers decide which is bigger).
Find the solution of the following :
Addition of integers on a number line
It is not always easy to add integers using coloured buttons.
Shall we use number line for additions?
We now know that integers include both positive and negative numbers. This tells us that when dealing with addition of integers, we will encounter four different cases:
(1) (+A) + (+B)
(2) (+A) + (-B)
(3) (-A) + (+B)
(4) (-A) + (-B)
Before looking into the four case, let's get accustomed to the concept of an additive inverse.
Additive Inverse
Let's "add" 3 and – 3. What do we get?
Similarly, if we "add" 2 and – 2 i.e. (+2) + (-2), we obtain the sum as
Numbers such as 3 and – 3, 2 and – 2, when added to each other give the sum zero. They are called additive inverse of each other.
What is the additive inverse of 6?
What is the additive inverse of – 7?
Consider this operation in another way. Say, we know that additive inverse of (–2) is
Thus, if we add the additive inverse of –2 to 6 i.e (-2) + 6 is the same as
Hence, to subtract an integer from another integer it is enough to add the additive inverse of the integer that is being subtracted, to the other integer.
Similarly, if we try to subtract the additive inverse i.e. 6 – (–2), it is the same as 6 + 2. But how so ?
Think about it this way: When we "subtract something", it means that "that something is being removed". Say we have a debt of Rs.
How do we go from
Removing Rs.
Thus,
Subtracting a negative amount/number is the same as adding the positive counterpart of that amount/number.
In numberlines, we can also think of it this way: Say we solve
But before we jump to the left, we encounter another
Thus,
Having learnt about additive inverses and 'subtracting negative numbers', let's look at the four cases using numberlines now.
(1)Addition of integers with both integers having positive signs (+A) + (+B)
Now, let's see what happens for addition of integers with the same signs. We already know the addition of two integers with positive signs. i.e. 2 + 10 =
But how will we represent it on the numberline? Using the above instructions, try to implement it on the numberline given below.
Here, taking
Since the signs are the same, the numerical values will be
The larger numerical integer has a
Thus, the result also has a positive sign.
Note here that when a positive integer is added to another positive integer, the resulting integer becomes more than both the given integers and has a postive sign in front of it.
We see that
(2) Addition of integers with both integers having opposite signs (+A) + (-B) or (-A) + (+B)
Take for eg.
Taking
What value do we obtain?
We see that since the signs of the two numbers are opposite, the numerical values will be subtracted from each other and the resulting number is
Also, note that the larger numerical integer has a
Thus, we get the result with a negative sign.
Take another eg. :
Taking
This gives us a sum of
Since the signs are opposite in this case as well, the numerical values will be
Thus, we get the result as a positive integer. We can conclude that:
When we add two integers with opposite signs, the result is a subtraction of the two integers involved.
The resulting integer has the same sign as the integer with the higher numerical value involved in the addition.
In the above example, we see that:
In the earlier example of
(3) Addition of integers with both integers having negative signs
Solve:
Here, taking
Since the signs are the same, the numerical values will be
The larger numerical integer has a
Thus, the result also has a negative sign.
When two negative integers are added, the resulting integer is a sum of the given integers with the sign being
We also see that
Now, let's try to get some results using the numberline.
Represent the following algebraic addition/subtraction expressions using the numberline and arrows. Enter the resulting value that the arrows point to:
9 - 5 + 2 =
6 - 1 + 3 =
3 + 4 - 2 =
Well done!