Operation 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 zeroone position on the number strip.

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 +-, the player moves their button forward (toward +15) the number of spaces shown on the standard die.

If the custom die shows a -+, the player moves their button backward (toward -15) the number of spaces shown on the standard die.

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.

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

You add when you have two positive integers like (+3) + (+2) = + [= 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 , while a negative number is always indicated with a '-' sign in front of it.

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.

__{m-purple}Hint: Include the appropriate signs(-,+) in the blanks __

a−4++3 =−1+−3++3 =−1+0=

b+4+−3 =+1++3+−3 =+1+0=

You can see that the answer of 4 – 3 is and – 4 + 3 is .

It is not necessary to display the sign, as positive numbers are assumed by default when no sign is present.

Find the Solution: of the following :

Continue

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? -66.

What is the additive inverse of – 7? 7-7.

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 subtractingadding (+2) from 6 i.e. 6 - 2.

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. −3. Now, if we "subtract" or in other words, remove the debt of Rs. −3, we will get a net amount of Rs. 03.

How do we go from −3 to 0? We add +3 to −3 which gives us the "net zero amount".

Removing Rs. −3 = Rs. −3 + Rs. 3 = Rs.

Thus,

In numberlines, we can also think of it this way: Say we solve 3−−4. We start at point 3-4 and since, we are substracting we try to jump to the leftright.

But before we jump to the left, we encounter another − sign which reverses the direction of the jump i.e. the jumps are ultimately made to the rightleft.

Thus, 3−−4 becomes 3+4 wich is equal to . Try it out with the given below numberline.

Instructions

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.

Instructions

Here, taking 2 as the reference point, we will make jumps to the rightleft side.

Since the signs are the same, the numerical values will be addedsubtracted to each other and the result will be

The larger numerical integer has a positivenegative sign.

Thus, the result also has a positive sign.

Continue

(2) Addition of integers with both integers having opposite signs (+A) + (-B) or (-A) + (+B)

Take for eg. −12+7. Try out what happens by putting the appropriate inputs for the icon by following the instructions given earlier. What do we get?

Instructions

Taking −12 as the reference point, we will make 76 jumps to the rightleft side.

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 −5.

Also, note that the larger numerical integer has a negativepositive sign.

Thus, we get the result with a negative sign.

Take another eg. : 6+−2.

Taking 6 as the reference point, we will make jumps to the leftright side (as the − sign reverses the direction ).

This gives us a sum of . _{span.reveal(when="blank-6")}Notice how the result is the same for 6+−2 and 6−2

Since the signs are opposite in this case as well, the numerical values will be subtractedadded to/from each other. Also, the larger numerical integer has a positivenegative sign.

Thus, we get the result as a positive integer. We can conclude that:

In the above example, we see that: 4, which is a subtraction of 6 and 2 and since, the higher integer of 6 is , 4 is also .

In the earlier example of −12+7, the result of −5 is a subtraction of 12 and 7 and since, the higher numerical integer of 12 has a − sign in front of it, so does −5.

Continue

(3) Addition of integers with both integers having negative signs

Solve: −5+−6

Here, taking −5 as the reference point, we will make jumps to the leftright side.

Instructions

Since the signs are the same, the numerical values will be addedsubtracted to each other and the result will be

The larger numerical integer has a negativepositive sign.

Thus, the result also has a negative sign.

Continue

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:

Instructions

9 - 5 + 2 =

6 - 1 + 3 =

3 + 4 - 2 =

Well done!

(b) Following is the number line representing the temperature in degree Celsius. Plot the name of the city against its temperature.

(c) Which is the coolest place?

(d) Write the names of the places where temperatures are above 10°C. , .

6.In each of the following pairs, which number is to the right of the other on the number line?

(a) 2, 9

is to the right of .

(b) – 3, – 8

is to the right of .

(c) 0, – 1

is to the right of .

(d) – 11, 10

is to the right of .

(e) – 6, 6

is to the right of .

(f) 1, – 100

is to the right of .

Write all the integers between the given pairs (write them in the increasing order.)

(a) 0 and – 7 = , -5 , , , -2 , -1.

(b) – 4 and 4 = , , -1 , , 1 , 2 , .

(c) – 8 and – 15 = , , -12 , -11 , , .

(d) – 30 and – 23 = , , , , -25 , -24.

(a) Write four negative integers greater than – 20. = , , -17 , .

(b) Write four integers less than – 10. = , , , -14.

For the following statements, write True (T) or False (F). If the statement is false, correct the statement.

(a) – 8 is to the right of – 10 on a number line. TrueFalse

(b) – 100 is to the right of – 50 on a number line. FalseTrue

-100 is to the left of -50 on a number line since, -50 > -100.

(c) Smallest negative integer is – 1. FalseTrue

We cannot find the value of the smallest negative integer. It goes till infinity.

(d) – 26 is greater than – 25. FalseTrue

-26 is smaller than -25.

10. Answer the questions for the following numberline.

(a) Which number will we reach if we move 4 numbers to the right of – 2?

(b) Which number will we reach if we move 5 numbers to the left of 1?

(c) If we are at – 8 on the number line, in which direction should we move to reach – 13?

(d) If we are at – 6 on the number line, in which direction should we move to reach – 1?

Multiplication Of Integers

We can add and subtract integers. Let us now learn how to multiply integers.

Multiplication of a Positive and a Negative Integer

We know that multiplication of whole numbers is repeated addition.For example,

Can you represent addition of integers in the same way?

We have from the following number line, (–5) + (–5) + (–5) = –15

Now, try this out yourself.

Instruction

But we can also write:

Similarly

(- 4) + (- 4) + (- 4) + (- 4) + (- 4) = 5 × (- 4) =

Now, try it again by yourself.

Instruction

We see that:

When a number is added to itself, a certain number of times - we get the resulting number to be equal to:

Resulting Number = Number of repetitions × Number Value

So:

−3+−3+−3+−3 = × =

(Enter the number of repetitions first)

Again,

(–7) + (–7) + (–7) = × =

(Enter the number of repetitions first)

Now, let us see how to find the product of a positive integer and a negative integer without using number line.

Let us find 3 × (–5) in a different way. First, find 3 × 5 and then put minus sign (–) before the product obtained. We get –151410.

Similarly, 5 × (– 4) = –(5 × 4) = -20-19-21

Till now we multiplied integers as (positive integer) × (negative integer). Now, let's try multiplying them as (negative integer) × (positive integer).

We first find: –3 × 5.

Observe the pattern as we move along.

Instruction

So, we get −3×5 = –15 =3×−5

Using such patterns, we also find that:

−5×4= =5×−4

−4×8= = 8×−4

−3×7 = = 7×−3

−6×5 = = 5×−6

−2×9 = = 9×−2

We thus find that while multiplying a positive integer and a negative integer,

multiply them as whole numbers and put a minus sign (–) before the product. We thus, get a negativepositive integer.

Multiplication of two Negative Integer

Similarly, let's try to understand the pattern when dealing with the multiplication of two negative integers.

What is the product of (–3) × (–2) ?

Let's find out.

Observe the below pattern:

Instruction

Do you see any pattern? Observe how the products change. Based on this observation, complete the following:

Finding the product of -4 x -3:

Instruction

Instruction

Instruction

From these patterns we can find out that,

(–3) × (–1) = = 3 × 1

(–3) × (–2) = = 3 × 2

(–3) × (–3) = = 3 × 3

Similarly,

(– 4) × (–1) = = 4 × 1

So,

(– 4) × (–2) = 4 × 2 =

(– 4) × (–3) = 4 × 3 =

So, observing these products we can say that:

The product of two negative integers is a positive integer. We multiply the two negative integers as whole numbers and put the positive sign before the product.

Thus, we have

(–10) × (–12) =

(–15) × (– 6) =

Division Of Integers

We know that division is the inverse operation of multiplication. Let us see an example for whole numbers.

We can thus, say: for each multiplication statement of whole numbers there are two division statements.

Can you write multiplication statement and its corresponding divison statements for integers?

From the above we observe that :

We observe that when we divide a negative integer by a positive integer, we divide them as whole numbers and then put a minus sign (–) before the quotient.

Continue

Lastly, we observe that:

−12÷−6=2

−20÷−4=5

−32÷−8=4

−45÷−9=5

So, we can say that when we divide a negative integer by a negative integer, we first divide them as whole numbers and then put a positive sign (+).

Can we say that (– 48) ÷ 8 = 48 ÷ (– 8)?

Let us check.

We know that (– 48) ÷ 8 = – 6 and 48 ÷ (– 8) = – 6

So (– 48) ÷ 8 = 48 ÷ (– 8) TrueFalse.

Check this for (i) 90 ÷ (– 45) and (–90) ÷ 45 TrueFalse

(ii) (–136) ÷ 4 and 136 ÷ (– 4) TrueFalse