Chapter 10 The Other Side Of Zero

Yes, there can be numbers less than 0. These numbers are called negative numbers.

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We can definitely think of ways to have "less than 0" of something. While you cannot physically have a negative count of discrete objects (like -3 apples), negative numbers are used to represent quantities that can go below a reference point, which is often set as zero. Here are some examples:

• Temperature: Temperatures can go below $0^\circ$ Celsius (freezing point of water) or $0^\circ$ Fahrenheit. For example, $-5^\circ$C represents a temperature 5 degrees below freezing.
• Money: In a bank account, a negative balance (like -₹500) means you owe the bank money; you have less than zero rupees in your account. Similarly, incurring a debt represents a negative amount of money.
• Elevation/Altitude: Heights can be measured relative to sea level (defined as 0). Being below sea level is represented by a negative elevation, e.g., -10 meters means 10 meters below sea level.
• Depth: Similar to elevation, the depth of a mine or a submarine can be represented by a negative number relative to ground level or sea level.
• Golf Scores: In golf, a score below par (often considered the reference 0) is represented by a negative number (e.g., -2 means two strokes under par).
• Time: Time can be measured relative to an event. Time before the event can be represented by negative numbers (e.g., T-minus 10 seconds before launch is represented as -10 seconds).
• Change in a quantity: If a quantity decreases, the change can be negative. For example, a temperature drop of $5^\circ$C can be represented as a change of $-5^\circ$C.

In these contexts, "less than 0" represents a state below a chosen reference point, indicating deficit, debt, position below a standard level, etc.

The buttons shown in the image represent the floors in the building that the elevator can go to. The numbered buttons (1 through 12) represent floors above the ground level, and the 'B' button represents a basement level.

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Assuming that floor '1' is the ground floor or main entry level, we can determine which button to press based on the desired movement relative to this floor.

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To go four floors up from floor 1:

Starting from floor 1, going four floors up means reaching floor $1 + 4 = 5$.

The button corresponding to floor 5 is labeled '5'.

You press the button labeled 5.

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To go three floors down from floor 1:

Starting from floor 1, going three floors down means reaching floor $1 - 3 = -2$.

Looking at the available buttons, there is no button labeled '-2'. The only button for a floor below '1' is 'B'. In the context of this question, it is implied that the button 'B' represents the floor that is three levels below floor 1.

You press the button labeled B.

The image shows a building with a ground floor and several floors above and below it. The ground floor is typically designated as 0.

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Floors above the ground floor are numbered using positive integers.

Floors below the ground floor (basement floors) are numbered using negative integers.

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Based on the labels provided in the image:

• The ground floor is labeled '0'.
• The floors above the ground floor are labeled '1', '2', '3', '4'. In the standard numbering system, these correspond to $+1, +2, +3, +4$.
• The basement floors are labeled 'B1', 'B2', 'B3'. In the standard numbering system, these correspond to $-1, -2, -3$.

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Therefore, numbering all the floors in the Building of Fun from the highest floor to the lowest floor:

Floor 4: +4

Floor 3: +3

Floor 2: +2

Floor 1: +1

Ground Floor: 0

Basement 1 (B1): -1

Basement 2 (B2): -2

Basement 3 (B3): -3

Based on the previous question and the image of the Building of Fun:

The Food Court is located on floor B3.

From the floor numbering in the previous answer (Page 244), floor B3 corresponds to the integer value -3.

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The instruction is to "press + 2 in the lift". This means moving 2 floors upwards from the current floor.

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Starting from floor -3, and moving +2 floors:

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The destination floor is -1.

Referring again to the floor numbering (Page 244), floor -1 is labeled B1, and it is the Parking floor.

You will reach the Parking (Floor B1).

We are starting from Floor +2 and moving -3 floors in the lift of the Building of Fun.

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The starting position is Floor +2, which corresponds to the integer +2.

Pressing -3 in the lift means moving 3 floors downwards, which corresponds to adding the integer -3 to the current floor number.

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To find the destination floor, we add the movement to the starting floor:

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Based on the floor numbering of the Building of Fun, Floor -1 is the Parking floor (B1).

You will reach the Parking (Floor B1).

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An expression for this movement is:

$(+2) + (-3)$ or simply $2 + (-3)$

We evaluate each expression by performing the addition of the given integers. Thinking of them as starting floor + movement in the Building of Fun can help visualize the process.

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a. (+ 1) + (+ 4)

Starting at Floor +1, move 4 floors up. $1 + 4 = 5$.

(+ 1) + (+ 4) = +5

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b. (+ 4) + (+ 1)

Starting at Floor +4, move 1 floor up. $4 + 1 = 5$.

(+ 4) + (+ 1) = +5

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c. (+ 4) + (– 3)

Starting at Floor +4, move 3 floors down. $4 - 3 = 1$.

(+ 4) + (– 3) = +1

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d. (– 1) + (+ 2)

Starting at Floor -1 (B1), move 2 floors up. $-1 + 2 = 1$.

(– 1) + (+ 2) = +1

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e. (– 1) + (+ 1)

Starting at Floor -1 (B1), move 1 floor up. $-1 + 1 = 0$.

(– 1) + (+ 1) = 0

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f. 0 + (+ 2)

Starting at Floor 0 (Ground Floor), move 2 floors up. $0 + 2 = 2$.

0 + (+ 2) = +2

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g. 0 + (– 2)

Starting at Floor 0 (Ground Floor), move 2 floors down. $0 - 2 = -2$.

0 + (– 2) = -2

We are looking for starting floors and the corresponding movements required to reach Floor -5. Let the starting floor be $S$ and the movement be $M$. The expression is $S + M = -5$. We need to find pairs of $S$ and $M$ such that their sum is -5.

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Here are a few examples of starting positions, the movement needed, and the corresponding expression:

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Example given in the question: Start at Floor +2, press -7. Expression: $(+2) + (-7) = -5$.

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Example 1:

Start at Floor +5.

To reach -5 from +5, the required movement $M$ is such that $5 + M = -5$.

Movement needed: -10 (10 floors down).

Expression: $(+5) + (-10) = -5$

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Example 2:

Start at Floor 0 (Ground Floor).

To reach -5 from 0, the required movement $M$ is such that $0 + M = -5$.

Movement needed: -5 (5 floors down).

Expression: $0 + (-5) = -5$

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Example 3:

Start at Floor -2 (Basement 2).

To reach -5 from -2, the required movement $M$ is such that $-2 + M = -5$.

Movement needed: -3 (3 floors down).

Expression: $(-2) + (-3) = -5$

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Example 4:

Start at Floor -8.

To reach -5 from -8, the required movement $M$ is such that $-8 + M = -5$.

Movement needed: +3 (3 floors up).

Expression: $(-8) + (+3) = -5$

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Example 5:

Start at Floor -5.

To reach -5 from -5, the required movement $M$ is such that $-5 + M = -5$.

Movement needed: 0 (no movement).

Expression: $(-5) + 0 = -5$

We evaluate each expression by combining the movements represented by the button presses. We can think of starting at 0 (Ground Floor) and applying the sequence of movements.

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a. (+ 1) + (+ 4)

Starting at 0, move +1 floor (up 1), then move +4 floors (up 4).

(+ 1) + (+ 4) = +5

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b. (+ 4) + (+ 1)

Starting at 0, move +4 floors (up 4), then move +1 floor (up 1).

(+ 4) + (+ 1) = +5

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c. (+ 4) + (– 3) + (– 2)

Starting at 0, move +4 floors (up 4), then move -3 floors (down 3), then move -2 floors (down 2).

(+ 4) + (– 3) + (– 2) = -1

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d. (– 1) + (+ 2) + (– 3)

Starting at 0, move -1 floor (down 1), then move +2 floors (up 2), then move -3 floors (down 3).

(– 1) + (+ 2) + (– 3) = -2

The additive inverse of a number is the number that, when added to the original number, results in a sum of zero. For any integer $a$, its additive inverse is $-a$, such that $a + (-a) = 0$ and $(-a) + a = 0$.

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We need to find the additive inverse for each of the given numbers:

• The number is +4. The additive inverse of +4 is the number that sums with +4 to give 0. $(+4) + (-4) = 0$. The inverse of +4 is -4.
• The number is -4. The additive inverse of -4 is the number that sums with -4 to give 0. $(-4) + (+4) = 0$. The inverse of -4 is +4.
• The number is -3. The additive inverse of -3 is the number that sums with -3 to give 0. $(-3) + (+3) = 0$. The inverse of -3 is +3.
• The number is 0. The additive inverse of 0 is the number that sums with 0 to give 0. $0 + 0 = 0$. The inverse of 0 is 0.
• The number is +2. The additive inverse of +2 is the number that sums with +2 to give 0. $(+2) + (-2) = 0$. The inverse of +2 is -2.
• The number is -1. The additive inverse of -1 is the number that sums with -1 to give 0. $(-1) + (+1) = 0$. The inverse of -1 is +1.

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The inverses of the given numbers are:

Inverse of +4 is -4.

Inverse of -4 is +4.

Inverse of -3 is +3.

Inverse of 0 is 0.

Inverse of +2 is -2.

Inverse of -1 is +1.

We need to connect each number on the left side of the diagram to its additive inverse on the right side by drawing lines.

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The additive inverse of a number $a$ is $-a$. Let's list the numbers and their inverses:

• -3: Inverse is +3
• +4: Inverse is -4
• 0: Inverse is 0
• -1: Inverse is +1

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Now we connect them based on the list:

• Draw a line connecting -3 (on the left) to +3 (on the right).
• Draw a line connecting +4 (on the left) to -4 (on the right).
• Draw a line connecting 0 (on the left) to 0 (on the right).
• Draw a line connecting -1 (on the left) to +1 (on the right).

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The diagram shows the connections as described above.

To determine who is on the lowest floor, we first need to identify the floor level for each person mentioned and then compare these floor levels.

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We need information about which floor each location (Art Centre, Sports Centre, Cinema Centre, Toys Shop) is on. Looking back at the diagram and floor numbering of the Building of Fun (likely from preceding pages not provided in this excerpt, but implied by the context), we find the locations:

• Art Centre: Floor +2
• Sports Centre: Floor -1 (B1)
• Cinema Centre: Floor +4
• Toys Shop: Floor +3

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Now we can determine the floor level for each person:

1. Jay is in the Art Centre. So, he is on Floor +2.

2. Asin is in the Sports Centre. So, she is on Floor -1 (B1).

3. Binnu is in the Cinema Centre. So, she is on Floor +4.

4. Aman is in the Toys Shop. So, he is on Floor +3.

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The floor levels for the four people are +2, -1, +4, and +3.

To find the lowest floor, we compare these integer values:

+2, -1, +4, +3

The smallest integer among these is -1.

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The person on the lowest floor is the one on Floor -1.

Asin is on Floor -1.

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Therefore, Asin is on the lowest floor.

The perspective referred to is likely standard subtraction within the set of integers, as discussed in the preceding context.

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Evaluation of $15 - 5$:

$15 - 5 = 10$

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Evaluation of $100 - 10$:

$100 - 10 = 90$

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Evaluation of $74 - 34$:

$74 - 34 = 40$

Here, we evaluate each expression by performing subtraction. Thinking in terms of floors, subtracting a number is equivalent to adding its opposite. The result represents the change in floor level (movement) from the starting floor to the target floor.

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a. $(+1) - (+4)$

$+1 - 4 = -3$

Result: -3

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b. $(0) - (+2)$

$0 - 2 = -2$

Result: -2

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c. $(+4) - (+1)$

$+4 - 1 = 3$

Result: +3

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d. $(0) - (-2)$

$0 - (-2) = 0 + 2 = 2$

Result: +2

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e. $(+4) - (-3)$

$+4 - (-3) = 4 + 3 = 7$

Result: +7

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f. $(-4) - (-3)$

$-4 - (-3) = -4 + 3 = -1$

Result: -1

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g. $(-1) - (+2)$

$-1 - 2 = -3$

Result: -3

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h. $(-2) - (-2)$

$-2 - (-2) = -2 + 2 = 0$

Result: 0

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i. $(-1) - (+1)$

$-1 - 1 = -2$

Result: -2

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j. $(+3) - (-3)$

$+3 - (-3) = 3 + 3 = 6$

Result: +6

Regarding the first part of the question:

Yes, in the exercises done above (especially parts b, d, e, f, h, j from Figure it Out on Page 249 and parts b, d, f, h from Intext Questions on Page 251), it was consistently observed that subtracting a negative number resulted in the same value as adding the corresponding positive number.

For example, $(-2) - (-2) = -2 + 2 = 0$ or $+105 - (-55) = 105 + 55 = 160$.

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Regarding the second part of the question:

Yes, the 'infinite lift' (or the representation of floors on a vertical line) does remind us of a number line.

It resembles a number line in the following ways:

1. It has a central reference point, the Ground Floor, which corresponds to 0 on a number line.

2. Floors above the Ground Floor are represented by positive integers, just like numbers to the right (or often above, in a vertical representation) of 0 on a number line are positive.

3. Floors below the Ground Floor (like in a mineshaft) are represented by negative integers, just like numbers to the left (or often below) of 0 on a number line are negative.

4. The sequence of floors represents integers in increasing or decreasing order as one moves up or down, similar to how numbers are ordered on a number line.

We need to select 3 positive numbers and 3 negative numbers and locate their positions on the given number line.

Examples of 3 positive numbers are +2, +5, and +9.

Examples of 3 negative numbers are -3, -7, and -10.

To mark these on the number line:

Locate +2 two units to the right of 0.

Locate +5 five units to the right of 0.

Locate +9 nine units to the right of 0.

Locate -3 three units to the left of 0.

Locate -7 seven units to the left of 0.

Locate -10 ten units to the left of 0.

These points should be clearly indicated on the visual representation of the number line provided in the question's image.

From the previous question, we marked the following 3 negative numbers on the number line: -3, -7, and -10.

We will write these numbers as requested.

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-3

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-7

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-10

First part: Is $2 > -3$?

Yes, $2 > -3$.

Reason: On a number line, numbers located further to the right are always greater than numbers located further to the left. The number $2$ is a positive integer, located $2$ units to the right of $0$. The number $-3$ is a negative integer, located $3$ units to the left of $0$. Since $2$ is located to the right of $-3$ on the number line, $2$ is greater than $-3$.

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Second part: Is $-2 < 3$?

Yes, $-2 < 3$.

Reason: On a number line, numbers located further to the left are always less than numbers located further to the right. The number $-2$ is a negative integer, located $2$ units to the left of $0$. The number $3$ is a positive integer, located $3$ units to the right of $0$. Since $-2$ is located to the left of $3$ on the number line, $-2$ is less than $3$. Additionally, any negative number is always less than any positive number.

We evaluate each of the given expressions:

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(i) $-5 + 0$

Adding zero to any integer results in the same integer.

$-5 + 0 = -5$

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(ii) $7 + (-7)$

Adding an integer to its additive inverse (opposite) results in zero.

$7 + (-7) = 7 - 7 = 0$

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(iii) $-10 + 20$

To add integers with different signs, find the difference between their absolute values and take the sign of the integer with the larger absolute value.

$|-10| = 10$, $|20| = 20$. The difference is $20 - 10 = 10$. Since $|20| > |-10|$, the result is positive.

$-10 + 20 = +10$

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(iv) $10 - 20$

Subtracting a larger number from a smaller number results in a negative number. Alternatively, $10 - 20$ can be written as $10 + (-20)$.

Adding integers with different signs: difference between absolute values is $20 - 10 = 10$. Since $|-20| > |10|$, the result is negative.

$10 - 20 = -10$

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(v) $7 - (-7)$

Subtracting a negative integer is the same as adding its corresponding positive integer.

$7 - (-7) = 7 + 7 = 14$

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(vi) $-8 - (-10)$

Subtracting a negative integer is the same as adding its corresponding positive integer.

$-8 - (-10) = -8 + 10$

Adding integers with different signs: difference between absolute values is $10 - 8 = 2$. Since $|10| > |-8|$, the result is positive.

$-8 + 10 = +2$

We will evaluate each expression by thinking of movement on a number line, starting from the first number and moving according to the operation and the sign of the second number.

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a. Evaluate $-125 + (-30)$:

Start at $-125$ on the number line.

We are adding $(-30)$, which is a negative number. Adding a negative number means moving to the left on the number line.

Move $30$ units to the left from $-125$.

$-125 - 30 = -155$

So, $-125 + (-30) = \mathbf{-155}$.

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b. Evaluate $+105 - (-55)$:

Start at $+105$ on the number line.

We are subtracting $(-55)$. Subtracting a negative number is equivalent to adding the corresponding positive number.

So, $+105 - (-55) = +105 + 55$.

We are adding $+55$, which is a positive number. Adding a positive number means moving to the right on the number line.

Move $55$ units to the right from $+105$.

$105 + 55 = 160$

So, $+105 - (-55) = \mathbf{+160}$.

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c. Evaluate $+80 - (-150)$:

Start at $+80$ on the number line.

We are subtracting $(-150)$. Subtracting a negative number is equivalent to adding the corresponding positive number.

So, $+80 - (-150) = +80 + 150$.

We are adding $+150$, which is a positive number. Adding a positive number means moving to the right on the number line.

Move $150$ units to the right from $+80$.

$80 + 150 = 230$

So, $+80 - (-150) = \mathbf{+230}$.

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d. Evaluate $-99 - (-200)$:

Start at $-99$ on the number line.

We are subtracting $(-200)$. Subtracting a negative number is equivalent to adding the corresponding positive number.

So, $-99 - (-200) = -99 + 200$.

We are adding $+200$, which is a positive number. Adding a positive number means moving to the right on the number line.

Move $200$ units to the right from $-99$. This is equivalent to moving $99$ units to the right to reach $0$, and then moving another $200 - 99 = 101$ units to the right.

$-99 + 200 = 101$

So, $-99 - (-200) = \mathbf{+101}$.

We use tokens to represent the integers. Let's use positive tokens for positive numbers and negative tokens for negative numbers. A positive token and a negative token cancel each other out, representing a sum of zero.

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a. $(+6) + (+4)$

Represent +6 with 6 positive tokens.

Represent +4 with 4 positive tokens.

Combine the tokens: 6 positive tokens and 4 positive tokens.

Total number of positive tokens = $6 + 4 = 10$.

$(+6) + (+4) = \mathbf{+10}$.

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b. $(–3) + (–2)$

Represent -3 with 3 negative tokens.

Represent -2 with 2 negative tokens.

Combine the tokens: 3 negative tokens and 2 negative tokens.

Total number of negative tokens = $3 + 2 = 5$.

$(–3) + (–2) = \mathbf{-5}$.

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c. $(+5) + (–7)$

Represent +5 with 5 positive tokens.

Represent -7 with 7 negative tokens.

Combine the tokens: 5 positive tokens and 7 negative tokens.

Form pairs of one positive and one negative token. 5 positive tokens will cancel out 5 negative tokens.

Remaining tokens = $7 - 5 = 2$ negative tokens.

$(+5) + (–7) = \mathbf{-2}$.

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d. $(–2) + (+6)$

Represent -2 with 2 negative tokens.

Represent +6 with 6 positive tokens.

Combine the tokens: 2 negative tokens and 6 positive tokens.

Form pairs of one positive and one negative token. 2 negative tokens will cancel out 2 positive tokens.

Remaining tokens = $6 - 2 = 4$ positive tokens.

$(–2) + (+6) = \mathbf{+4}$.

We examine the sets of tokens and form zero pairs (one positive and one negative token) to find the sum.

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Set 1:

Count the positive tokens (+): There are 5 positive tokens.

Count the negative tokens (–): There are 3 negative tokens.

Form zero pairs: We can form 3 zero pairs (one positive and one negative token). These pairs cancel out.

Remaining tokens: After cancelling 3 positive tokens with 3 negative tokens, we are left with $5 - 3 = 2$ positive tokens.

On what floor is the lift attendant? The remaining tokens represent the final position. Since 2 positive tokens remain, the lift attendant is on floor +2.

Corresponding addition statement: The initial tokens represented $(+5) + (-3)$. The result is +2.

The addition statement is: $\mathbf{(+5) + (–3) = +2}$.

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Set 2:

Count the positive tokens (+): There is 1 positive token.

Count the negative tokens (–): There are 4 negative tokens.

Form zero pairs: We can form 1 zero pair (one positive and one negative token). This pair cancels out.

Remaining tokens: After cancelling 1 positive token with 1 negative token, we are left with $4 - 1 = 3$ negative tokens.

On what floor is the lift attendant? The remaining tokens represent the final position. Since 3 negative tokens remain, the lift attendant is on floor –3.

Corresponding addition statement: The initial tokens represented $(+1) + (-4)$. The result is -3.

The addition statement is: $\mathbf{(+1) + (–4) = –3}$.

We evaluate each difference using the token method. Subtracting a number means taking away that number of tokens. If we need to take away tokens that are not present, we add zero pairs until we can perform the removal.

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a. $(+10) – (+7)$

Start with 10 positive tokens.

We need to subtract +7, which means removing 7 positive tokens.

Remove 7 positive tokens from the initial 10 positive tokens.

Remaining tokens: $10 - 7 = 3$ positive tokens.

Result: $\mathbf{+3}$.

Check by regular subtraction: $10 - 7 = 3$. The results match.

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b. $(–8) – (–4)$

Start with 8 negative tokens.

We need to subtract -4, which means removing 4 negative tokens.

Remove 4 negative tokens from the initial 8 negative tokens.

Remaining tokens: $8 - 4 = 4$ negative tokens.

Result: $\mathbf{-4}$.

Check by regular subtraction: $-8 - (-4) = -8 + 4 = -4$. The results match.

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c. $(–9) – (–4)$

Start with 9 negative tokens.

We need to subtract -4, which means removing 4 negative tokens.

Remove 4 negative tokens from the initial 9 negative tokens.

Remaining tokens: $9 - 4 = 5$ negative tokens.

Result: $\mathbf{-5}$.

Check by regular subtraction: $-9 - (-4) = -9 + 4 = -5$. The results match.

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d. $(+9) – (+12)$

Start with 9 positive tokens.

We need to subtract +12, which means removing 12 positive tokens.

We only have 9 positive tokens. We need to remove 3 more positive tokens than we have.

To be able to remove 12 positive tokens, add zero pairs until there are enough positive tokens. We need 3 more positive tokens, so add 3 zero pairs (3 positive and 3 negative tokens).

Initial tokens: 9 positive.

Add 3 zero pairs: 9 positive + (3 positive + 3 negative) = 12 positive and 3 negative tokens.

Now remove 12 positive tokens.

Remaining tokens: 3 negative tokens.

Result: $\mathbf{-3}$.

Check by regular subtraction: $9 - 12 = 9 + (-12) = -3$. The results match.

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e. $(–5) – (–7)$

Start with 5 negative tokens.

We need to subtract -7, which means removing 7 negative tokens.

We only have 5 negative tokens. We need to remove 2 more negative tokens than we have.

To be able to remove 7 negative tokens, add zero pairs until there are enough negative tokens. We need 2 more negative tokens, so add 2 zero pairs (2 positive and 2 negative tokens).

Initial tokens: 5 negative.

Add 2 zero pairs: 5 negative + (2 positive + 2 negative) = 7 negative and 2 positive tokens.

Now remove 7 negative tokens.

Remaining tokens: 2 positive tokens.

Result: $\mathbf{+2}$.

Check by regular subtraction: $-5 - (-7) = -5 + 7 = 2$. The results match.

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f. $(–2) – (–6)$

Start with 2 negative tokens.

We need to subtract -6, which means removing 6 negative tokens.

We only have 2 negative tokens. We need to remove 4 more negative tokens than we have.

To be able to remove 6 negative tokens, add zero pairs until there are enough negative tokens. We need 4 more negative tokens, so add 4 zero pairs (4 positive and 4 negative tokens).

Initial tokens: 2 negative.

Add 4 zero pairs: 2 negative + (4 positive + 4 negative) = 6 negative and 4 positive tokens.

Now remove 6 negative tokens.

Remaining tokens: 4 positive tokens.

Result: $\mathbf{+4}$.

Check by regular subtraction: $-2 - (-6) = -2 + 6 = 4$. The results match.

We evaluate each subtraction expression. Remember that subtracting a number is the same as adding its opposite.

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a. $(–5) – (–7)$

This is equivalent to $-5 + 7$.

$-5 + 7 = 2$

Result: $\mathbf{+2}$

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b. $(+10) – (+13)$

This is equivalent to $10 - 13$.

$10 - 13 = -3$

Result: $\mathbf{-3}$

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c. $(–7) – (–9)$

This is equivalent to $-7 + 9$.

$-7 + 9 = 2$

Result: $\mathbf{+2}$

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d. $(+3) – (+8)$

This is equivalent to $3 - 8$.

$3 - 8 = -5$

Result: $\mathbf{-5}$

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e. $(–2) – (–7)$

This is equivalent to $-2 + 7$.

$-2 + 7 = 5$

Result: $\mathbf{+5}$

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f. $(+3) – (+15)$

This is equivalent to $3 - 15$.

$3 - 15 = -12$

Result: $\mathbf{-12}$

We want to evaluate $-3 - (+5)$ using tokens.

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Start with the tokens representing the first number, $-3$. This means we have 3 negative tokens.

Initial state: $|||$ (3 negative tokens)

We need to subtract $+5$. This means we need to remove 5 positive tokens.

Looking at the initial state, we have no positive tokens to remove.

To be able to remove 5 positive tokens, we need to introduce zero pairs (one positive and one negative token) until we have at least 5 positive tokens available.

We need to remove 5 positive tokens, so we must add 5 zero pairs.

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Number of zero pairs to put in: $\mathbf{5}$

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Add 5 zero pairs to the initial 3 negative tokens.

Initial state: 3 negative tokens.

Add 5 zero pairs: 5 positive tokens and 5 negative tokens.

Combined state: 3 negative tokens + 5 positive tokens + 5 negative tokens = 5 positive tokens and 8 negative tokens.

Now, remove the 5 positive tokens as required by the subtraction of $+5$.

Remaining tokens: We started with 5 positive tokens (after adding zero pairs) and we remove all 5. We are left with the 8 negative tokens.

Result: $\mathbf{-8}$

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Check by standard subtraction: $-3 - (+5) = -3 - 5 = -8$. The results match.

We evaluate each expression using the token method. Start with tokens representing the first number. To subtract the second number, remove that many tokens of the corresponding sign. If tokens are insufficient, add zero pairs until enough are available, then remove the required tokens. The remaining tokens give the result.

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a. Evaluate $(-3) - (+10)$:

Start with 3 negative tokens.

Need to remove 10 positive tokens. Since we have none, add 10 zero pairs (10 positive and 10 negative tokens).

Current state: 3 negative tokens + 10 positive tokens + 10 negative tokens = 10 positive tokens and 13 negative tokens.

Remove 10 positive tokens.

Remaining: 13 negative tokens.

Result: $\mathbf{-13}$.

Check: $-3 - (+10) = -3 - 10 = -13$.

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b. Evaluate $(+8) - (-7)$:

Start with 8 positive tokens.

Need to remove 7 negative tokens. Since we have none, add 7 zero pairs (7 positive and 7 negative tokens).

Current state: 8 positive tokens + 7 positive tokens + 7 negative tokens = 15 positive tokens and 7 negative tokens.

Remove 7 negative tokens.

Remaining: 15 positive tokens.

Result: $\mathbf{+15}$.

Check: $+8 - (-7) = 8 + 7 = 15$.

---

c. Evaluate $(-5) - (+9)$:

Start with 5 negative tokens.

Need to remove 9 positive tokens. Since we have none, add 9 zero pairs (9 positive and 9 negative tokens).

Current state: 5 negative tokens + 9 positive tokens + 9 negative tokens = 9 positive tokens and 14 negative tokens.

Remove 9 positive tokens.

Remaining: 14 negative tokens.

Result: $\mathbf{-14}$.

Check: $-5 - (+9) = -5 - 9 = -14$.

---

d. Evaluate $(-9) - (+10)$:

Start with 9 negative tokens.

Need to remove 10 positive tokens. Since we have none, add 10 zero pairs (10 positive and 10 negative tokens).

Current state: 9 negative tokens + 10 positive tokens + 10 negative tokens = 10 positive tokens and 19 negative tokens.

Remove 10 positive tokens.

Remaining: 19 negative tokens.

Result: $\mathbf{-19}$.

Check: $-9 - (+10) = -9 - 10 = -19$.

---

e. Evaluate $(+6) - (-4)$:

Start with 6 positive tokens.

Need to remove 4 negative tokens. Since we have none, add 4 zero pairs (4 positive and 4 negative tokens).

Current state: 6 positive tokens + 4 positive tokens + 4 negative tokens = 10 positive tokens and 4 negative tokens.

Remove 4 negative tokens.

Remaining: 10 positive tokens.

Result: $\mathbf{+10}$.

Check: $+6 - (-4) = 6 + 4 = 10$.

---

f. Evaluate $(-2) - (+7)$:

Start with 2 negative tokens.

Need to remove 7 positive tokens. Since we have none, add 7 zero pairs (7 positive and 7 negative tokens).

Current state: 2 negative tokens + 7 positive tokens + 7 negative tokens = 7 positive tokens and 9 negative tokens.

Remove 7 positive tokens.

Remaining: 9 negative tokens.

Result: $\mathbf{-9}$.

Check: $-2 - (+7) = -2 - 7 = -9$.

Given:

Starting Bank Balance = 0

Credits (Amounts added) = 30, 40, 50

Debits (Amounts removed) = 40, 50, 60

---

To Find:

The final bank account balance.

---

Solution:

We can calculate the total credits and total debits.

Total Credits = 30 + 40 + 50

$30 + 40 + 50 = 120$

Total Credits = 120

---

Total Debits = 40 + 50 + 60

$40 + 50 + 60 = 150$

Total Debits = 150

---

The final balance is the starting balance plus total credits minus total debits.

Final Balance = Starting Balance + Total Credits - Total Debits

Final Balance = 0 + 120 - 150

Final Balance = 120 - 150

Evaluating $120 - 150$:

$120 - 150 = -30$

Final Balance = -30

---

Alternatively, step-by-step calculation:

Start with balance: 0

After credits: 0 + 30 = 30

30 + 40 = 70

70 + 50 = 120

After debits: 120 - 40 = 80

80 - 50 = 30

30 - 60 = -30

---

The bank account balance is -30.

Given:

Starting Bank Balance = 0

Debits (Amounts removed) = 1, 2, 4, 8, 16, 32, 64, 128

Credit (Amount added) = 256

---

To Find:

The final bank account balance.

---

Solution:

First, calculate the total amount of debits.

Total Debits = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128

Let's add these amounts:

$1 + 2 = 3$

$3 + 4 = 7$

$7 + 8 = 15$

$15 + 16 = 31$

$31 + 32 = 63$

$63 + 64 = 127$

$127 + 128 = 255$

Total Debits = 255

---

Total Credit = 256

---

The final balance is the starting balance plus total credits minus total debits.

Final Balance = Starting Balance + Total Credits - Total Debits

Final Balance = 0 + 256 - 255

Final Balance = (256 - 255)

Final Balance = 1

---

The bank account balance is +1 (or 1).

It is generally better to try and maintain a positive balance in your bank account for several reasons:

1. Avoiding Fees: Banks often charge fees, such as overdraft fees, when your account balance goes below zero. Maintaining a positive balance prevents these charges.

2. Avoiding Interest Charges: If a negative balance is allowed (like through an overdraft facility), the bank will charge interest on the borrowed amount. This interest adds to the cost of using the money.

3. Financial Stability: A positive balance indicates that you have money available, providing a cushion for unexpected expenses or emergencies without needing to borrow.

4. Building Financial Discipline: Managing your spending to stay within your available funds encourages responsible financial habits.

---

Circumstances under which it may be worthwhile to temporarily have a negative balance (often through arranged overdrafts or short-term loans) are usually related to situations where the benefit outweighs the cost (fees and interest):

1. Emergencies: To cover essential unexpected expenses (like medical bills or urgent home repairs) when there are no other immediate funds available.

2. Bridging a Temporary Cash Flow Gap: When you know money is coming into the account very soon (e.g., salary payment) but you need to make an urgent payment before it arrives. An overdraft might be cheaper than a late payment fee or penalty.

3. Investments or Opportunities: In some business or personal finance situations, a temporary negative balance (short-term borrowing) might be used to seize an opportunity or make an investment that is expected to yield returns greater than the cost of borrowing.

4. Avoiding Service Disruption: To ensure continuous service for essential utilities or subscriptions where non-payment could lead to disconnection or significant penalties.

It is important to note that having an unplanned or unauthorized negative balance is almost always disadvantageous due to high fees.

Based on the provided geographical cross-section image, we can determine the height or depth of each labeled point relative to sea level (0 meters).

---

Mountain peak = +1500 m

---

Top of tree = +50 m

---

Top of house = +15 m

---

Sea level = 0 m

---

Fish = -100 m

---

Submarine = -200 m

---

Ocean floor = -500 m

From the heights determined in the previous question, we have:

Mountain peak: +1500 m

Top of tree: +50 m

Top of house: +15 m

Sea level: 0 m

Fish: -100 m

Submarine: -200 m

Ocean floor: -500 m

---

The highest point is the one with the greatest value among these numbers.

Comparing the values, the largest value is $+1500$.

The point corresponding to $+1500$ m is the Mountain peak.

---

The lowest point is the one with the smallest value among these numbers.

Comparing the values, the smallest (most negative) value is $-500$.

The point corresponding to $-500$ m is the Ocean floor.

The points A, B, C, D, E, F, G correspond to the following locations and heights:

A: Mountain peak (+1500 m)

B: Top of tree (+50 m)

C: Top of house (+15 m)

D: Sea level (0 m)

E: Fish (-100 m)

F: Submarine (-200 m)

G: Ocean floor (-500 m)

---

Sequence of decreasing order of heights:

We arrange the points from the highest (greatest height) to the lowest (smallest height, which is the most negative value).

$1500 > 50 > 15 > 0 > -100 > -200 > -500$

Corresponding points:

A (+1500), B (+50), C (+15), D (0), E (-100), F (-200), G (-500)

Sequence: A, B, C, D, E, F, G

---

Sequence of increasing order of heights:

We arrange the points from the lowest (smallest height, which is the most negative value) to the highest (greatest height).

$-500 < -200 < -100 < 0 < 15 < 50 < 1500$

Corresponding points:

G (-500), F (-200), E (-100), D (0), C (+15), B (+50), A (+1500)

Sequence: G, F, E, D, C, B, A

The highest point above sea level on Earth is Mount Everest.

Its official height is 8848.86 meters above sea level.

The lowest point with respect to sea level on Earth is the Challenger Deep in the Mariana Trench, located on the ocean floor.

---

Its height (or depth) with respect to sea level is approximately -10,935 meters.

Yes, there are several places in India where temperatures can go below $0^\circ\text{C}$, especially during the winter months.

---

Some places in India where temperatures sometimes go below $0^\circ\text{C}$ include:

Jammu and Kashmir and Ladakh: Leh, Kargil, Srinagar.

Himachal Pradesh: Manali, Shimla (during severe winters), Keylong.

Uttarakhand: High altitude regions like Mussoorie (sometimes), parts of the Himalayas.

Sikkim and Arunachal Pradesh: High mountain areas.

Parts of the North Indian plains can also experience temperatures slightly below $0^\circ\text{C}$ during severe cold waves, though this is less common and typically for shorter durations compared to the mountainous regions.

---

What is common among the places where temperatures regularly go below $0^\circ\text{C}$ is their high altitude and location in mountainous regions, primarily in the northern parts of India.

---

Reasons why it becomes colder there and not in other places:

1. Altitude: Temperature generally decreases as altitude increases. Higher elevations have thinner air, which is less able to retain heat. This is a primary reason mountainous areas are colder.

2. Latitude: Places located further north (at higher latitudes) receive less direct sunlight, especially during winter, leading to lower temperatures compared to places closer to the equator.

3. Snow Cover: Many of these regions receive snowfall. Snow has a high albedo, meaning it reflects a large portion of incoming solar radiation back into space. Also, processes like melting and sublimation of snow absorb heat from the surroundings, further cooling the air.

4. Prevailing Winds: Cold winds originating from very cold regions to the north, such as Siberia and Central Asia, can reach northern India and contribute significantly to low temperatures.

We match the temperature readings with the appropriate time of the day/night based on the table provided in the image.

---

12 midnight: $-1.5^\circ\text{C}$

---

3 a.m.: $-2.5^\circ\text{C}$

---

6 a.m.: $-4.5^\circ\text{C}$

---

9 a.m.: $-1.0^\circ\text{C}$

---

12 noon: $+2.0^\circ\text{C}$

---

3 p.m.: $+3.0^\circ\text{C}$

---

6 p.m.: $-0.5^\circ\text{C}$

---

9 p.m.: $-1.0^\circ\text{C}$

To perform the calculations and find the border sum for the second grid, we need the specific numbers placed within that grid.

---

Since the content of the second grid is not provided here, I cannot perform the exact calculations or determine the border sum.

---

Assuming the calculation involves adding or performing operations on numbers in the border cells of the grid, you would typically follow these steps:

1. Identify all the numbers located in the cells along the outer edge (border) of the grid.

2. Perform the specific operation indicated (likely addition, based on similar grid exercises) on these border numbers.

3. Sum up the results if multiple operations are performed, or simply sum all the border numbers if that is the required operation.

---

Please provide the numbers within the second grid so that the calculation can be completed.

The question refers to a grid presented earlier (presumably on page 263 or 264). Based on the previous context and typical placement in the textbook, the grid in image 9.jpg seems to be the one intended.

---

The grid in image 9.jpg is:

-1 | +1 | -3

+4 | +2 | -2

+3 | -5 | +6

The border cells are the numbers in the first row, the last row, the first column, and the last column.

Border numbers are: -1, +1, -3 (from the top row); +4, -2 (from the middle sides); +3, -5, +6 (from the bottom row).

The border sum is the sum of these numbers:

Border Sum = $(-1) + (+1) + (-3) + (+4) + (-2) + (+3) + (-5) + (+6)$

Border Sum = $-1 + 1 - 3 + 4 - 2 + 3 - 5 + 6$

Group positive and negative numbers:

Positive sum = $1 + 4 + 3 + 6 = 14$

Negative sum = $-1 - 3 - 2 - 5 = -(1 + 3 + 2 + 5) = -11$

Border Sum = $14 + (-11) = 14 - 11 = 3$

The border sum is +3.

The "last grid above" likely refers to the 3x3 grid structure shown in image 16.jpg (associated with Question 2). We need to fill the eight border cells of this grid such that the sum of the numbers in these cells is -4.

The center cell does not contribute to the border sum, so it can be any integer.

---

Way 1:

Let's fill the border cells with numbers that add up to -4. For example, we can use a mix of positive and negative numbers.

Numbers for the border cells: +1, +1, +1, 0, 0, -2, -3, -2

Sum of these numbers = $(+1) + (+1) + (+1) + (0) + (0) + (-2) + (-3) + (-2) = 3 - 7 = -4$.

We can place these numbers in the border cells of the grid. Let the center cell be 0.

Grid arrangement:

+1	+1	+1
0	0	0
-2	-3	-2

Border sum: $(+1) + (+1) + (+1) + (0) + (0) + (-2) + (-3) + (-2) = 3 - 7 = -4$.

---

Way 2:

Let's use a different set of numbers for the border cells that also sum to -4.

Numbers for the border cells: +2, -1, -1, 0, 0, -1, -1, -2

Sum of these numbers = $(+2) + (-1) + (-1) + (0) + (0) + (-1) + (-1) + (-2) = 2 - 6 = -4$.

We can place these numbers in the border cells of the grid. Let the center cell be any other integer, for example, +5.

Grid arrangement:

+2	-1	-1
0	+5	0
-1	-1	-2

Border sum: $(+2) + (-1) + (-1) + (0) + (0) + (-1) + (-1) + (-2) = 2 - 6 = -4$.

There are many other possible ways to fill the border cells to achieve a sum of -4.

Assuming the grids being referred to are the 3x3 grids presented in the "Figure it Out" section on pages 263-264, where a specific border sum is required:

---

All the grids presented in this format can be filled in multiple ways to achieve the required border sum.

---

Reason:

The border sum is determined by the sum of the numbers placed in the eight outer cells of the 3x3 grid. The number in the center cell does not affect the border sum.

When you need to find a set of numbers that sum up to a specific target value, and there are multiple numbers involved in the sum (in this case, 8 numbers), there are generally infinitely many combinations of integers that can satisfy the condition. For example, if you have two numbers that sum to 10 (like $5+5=10$), you can also use $6+4=10$, $7+3=10$, $10+0=10$, $12+(-2)=10$, etc. You can adjust one number and compensate by adjusting another without changing the sum.

With 8 numbers whose sum is fixed, you have a lot of flexibility to choose values for individual cells, as long as their total equals the required border sum. Additionally, the center cell can be any integer, providing further variation in the complete grid layout.

Here is an example of a border integer square puzzle that you can challenge your classmates with.

---

Puzzle:

Fill in the missing integer in the center cell of the grid such that the sum of the integers in the border cells is $\mathbf{+10}$.

Grid:

+2	-1	+4
+3	?	0
+5	-2	-1

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Solution (for you, not for your classmates initially):

The border cells are the numbers in the first row, last row, and the remaining numbers in the first and last columns.

The border numbers are: $+2, -1, +4, +3, 0, +5, -2, -1$.

The border sum is the sum of these numbers:

Border Sum = $(+2) + (-1) + (+4) + (+3) + (0) + (+5) + (-2) + (-1)$

Border Sum = $2 - 1 + 4 + 3 + 0 + 5 - 2 - 1$

Group positive numbers: $2 + 4 + 3 + 0 + 5 = 14$

Group negative numbers: $-1 - 2 - 1 = -(1 + 2 + 1) = -4$

Border Sum = $14 + (-4) = 14 - 4 = 10$

The required border sum is indeed +10 with these numbers.

The value in the center cell does not affect the border sum. Therefore, the missing integer in the center cell (?) can be any integer.

This question asks us to repeat an activity from earlier on page 265 by choosing different numbers and calculating a sum. Since I am an AI, I cannot perform a physical activity of "choosing numbers" and calculating a sum in real-time like a person.

---

However, based on the phrasing "choose different numbers this time", if you choose a set of numbers that is different from the set you used the "first time", the sum you get will most likely be different.

For example, if the activity was simply adding a set of numbers, and the first time you added $2, 3, 5$, the sum was $10$. If you choose different numbers like $1, 4, 6$, the sum is $11$, which is different.

---

The result would only be the same as the first time if, by coincidence, the new set of different numbers you chose happened to have the exact same sum as the previous set.

The purpose of trying a few more times with different numbers is likely to reinforce the idea that changing the numbers in a sum (or calculation) usually changes the result, unless there's a specific pattern or property being explored that makes the sum constant.

---

To answer this question for yourself, you should perform the calculation described in the activity on page 265 using a new set of numbers and compare the sum to the sum you got when you did the activity the "first time".

The game likely involves calculating the border sum of the grids, similar to the exercise on pages 263-264. We will calculate the border sum for each grid shown in the image.

---

First Grid:

The grid is:

-10 | +1 | -3

+4 | +2 | -2

+3 | -5 | +6

The border cells are the numbers in the first row, the last row, and the numbers in the first and last columns that are not in the first or last row.

Border numbers are: -10, +1, -3, +4, -2, +3, -5, +6.

Border Sum = $(-10) + (+1) + (-3) + (+4) + (-2) + (+3) + (-5) + (+6)$

Border Sum = $-10 + 1 - 3 + 4 - 2 + 3 - 5 + 6$

Group positive numbers: $1 + 4 + 3 + 6 = 14$

Group negative numbers: $-10 - 3 - 2 - 5 = -(10 + 3 + 2 + 5) = -20$

Border Sum = $14 + (-20) = 14 - 20 = -6$

The border sum for the first grid is -6.

---

Second Grid:

The grid is:

-10 | +1 | -3

+4 | -7 | -2

+3 | -5 | +6

The border cells are the numbers in the first row, the last row, and the numbers in the first and last columns that are not in the first or last row.

Border numbers are: -10, +1, -3, +4, -2, +3, -5, +6.

These are the exact same border numbers as in the first grid.

Border Sum = $(-10) + (+1) + (-3) + (+4) + (-2) + (+3) + (-5) + (+6)$

Border Sum = $-10 + 1 - 3 + 4 - 2 + 3 - 5 + 6$

Positive sum = $14$

Negative sum = $-20$

Border Sum = $14 + (-20) = 14 - 20 = -6$

The border sum for the second grid is also -6.

The center number changed from +2 to -7, but this does not affect the border sum.

The grids shown in the previous question had the same border numbers, but a different number in the center cell. Despite the difference in the center, the calculated border sum was the same (-6) for both grids.

---

The special aspect of these grids, in the context of calculating the border sum, is that the value in the center cell does not affect the sum of the numbers along the border.

---

The "magic" (the specific result of the border sum) is primarily in the numbers placed in the border cells. The sum is calculated based on the values of these eight numbers. The way they are arranged matters in the sense that only the numbers on the outer edge contribute to the border sum, and the number in the center does not. So, it's a combination of the specific numbers used on the border and the grid structure that defines which numbers are part of the border calculation.

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Yes, you can make more such grids. You can create new grids by:

1. Keeping the same border numbers and changing the number in the center cell. This will result in the same border sum.

2. Choosing a completely different set of eight numbers for the border cells that add up to a desired target sum (like -4, as explored in a previous question, or any other integer). You can then place any integer in the center cell.

We list the integers between the given pairs in increasing order. "Between" means strictly greater than the smaller number and strictly less than the larger number.

---

a. Integers between 0 and –7 (i.e., between –7 and 0) in increasing order:

The integers greater than $-7$ and less than $0$ are $-6, -5, -4, -3, -2, -1$.

-6, -5, -4, -3, -2, -1

---

b. Integers between –4 and 4 in increasing order:

The integers greater than $-4$ and less than $4$ are $-3, -2, -1, 0, 1, 2, 3$.

-3, -2, -1, 0, 1, 2, 3

---

c. Integers between –8 and –15 (i.e., between –15 and –8) in increasing order:

The integers greater than $-15$ and less than $-8$ are $-14, -13, -12, -11, -10, -9$.

-14, -13, -12, -11, -10, -9

---

d. Integers between –30 and –23 in increasing order:

The integers greater than $-30$ and less than $-23$ are $-29, -28, -27, -26, -25, -24$.

-29, -28, -27, -26, -25, -24

We need to find three integers that, when added together, result in a sum of -8.

---

There are many possible combinations. Here is one example:

Let the three numbers be -3, -2, and -3.

Let's find their sum:

$(-3) + (-2) + (-3) = -5 + (-3) = -5 - 3 = -8$

---

Another example:

Let the three numbers be 0, -5, and -3.

Let's find their sum:

$0 + (-5) + (-3) = -5 + (-3) = -5 - 3 = -8$

---

Another example:

Let the three numbers be +2, -10, and 0.

Let's find their sum:

$(+2) + (-10) + 0 = 2 - 10 + 0 = -8$

---

Three numbers such that their sum is -8 are, for example: -3, -2, and -3.

(Any set of three integers that add up to -8 is a valid answer.)

Given:

Numbers on the faces of each die: $\{-1, 2, -3, 4, -5, 6\}$.

Smallest possible sum = $(-5) + (-5) = -10$.

Largest possible sum = $(6) + (6) = 12$.

---

To Find:

Numbers between $-10$ and $12$ that are not possible sums when rolling the two dice.

---

Solution:

We list all possible sums by adding each number from the first die to each number from the second die. The set of numbers on both dice is the same: $\{-1, 2, -3, 4, -5, 6\}$.

Let's construct a table of all possible sums:

+	-1	2	-3	4	-5	6
-1	-2	1	-4	3	-6	5
2	1	4	-1	6	-3	8
-3	-4	-1	-6	1	-8	3
4	3	6	1	8	-1	10
-5	-6	-3	-8	-1	-10	1
6	5	8	3	10	1	12

---

The unique possible sums are: $\{-10, -8, -6, -4, -3, -2, -1, 1, 3, 4, 5, 6, 8, 10, 12\}$.

The integers between $-10$ and $12$ are the integers from $-9$ to $11$ inclusive.

Integers between $-10$ and $12$: $\{-9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$.

Now we compare the list of integers between $-10$ and $12$ with the list of unique possible sums. We want to find the integers that are in the first list but not in the second list.

Integers present in the range $\{-9, ..., 11\}$ but not in the possible sums $\{-10, -8, -6, -4, -3, -2, -1, 1, 3, 4, 5, 6, 8, 10, 12\}$:

-9 (not in possible sums)

-8 (in possible sums)

-7 (not in possible sums)

-6 (in possible sums)

-5 (not in possible sums)

-4 (in possible sums)

-3 (in possible sums)

-2 (in possible sums)

-1 (in possible sums)

0 (not in possible sums)

1 (in possible sums)

2 (not in possible sums)

3 (in possible sums)

4 (in possible sums)

5 (in possible sums)

6 (in possible sums)

7 (not in possible sums)

8 (in possible sums)

9 (not in possible sums)

10 (in possible sums)

11 (not in possible sums)

The numbers between $-10$ and $12$ that are not possible to get as a sum are $\{-9, -7, -5, 0, 2, 7, 9, 11\}$.

We evaluate each of the given expressions.

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a. $-12 + (-3)$

Adding two negative numbers results in a negative number. Add their absolute values and keep the negative sign.

$-12 + (-3) = -12 - 3 = -(12+3) = -15$

Result: -15

---

b. $+10 + (-5)$

Adding integers with different signs. Find the difference between their absolute values and use the sign of the number with the larger absolute value.

$|+10| = 10$, $|-5| = 5$. Difference is $10 - 5 = 5$.

Since $|+10| > |-5|$, the result is positive.

$+10 + (-5) = 10 - 5 = +5$

Result: +5

---

c. $-2 + (+6)$

Adding integers with different signs. Find the difference between their absolute values and use the sign of the number with the larger absolute value.

$|-2| = 2$, $|+6| = 6$. Difference is $6 - 2 = 4$.

Since $|+6| > |-2|$, the result is positive.

$-2 + (+6) = -2 + 6 = +4$

Result: +4

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d. $-5 - (+3)$

Subtracting a positive number is the same as adding the corresponding negative number.

$-5 - (+3) = -5 + (-3)$

Adding two negative numbers results in a negative number. Add their absolute values and keep the negative sign.

$-5 + (-3) = -5 - 3 = -(5+3) = -8$

Result: -8

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e. $+7 - (-4)$

Subtracting a negative number is the same as adding the corresponding positive number.

$+7 - (-4) = +7 + 4 = 7 + 4 = 11$

Result: +11

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f. $-6 - (-2)$

Subtracting a negative number is the same as adding the corresponding positive number.

$-6 - (-2) = -6 + 2$

Adding integers with different signs. Find the difference between their absolute values and use the sign of the number with the larger absolute value.

$|-6| = 6$, $|+2| = 2$. Difference is $6 - 2 = 4$.

Since $|-6| > |+2|$, the result is negative.

$-6 + 2 = -4$

Result: -4

We need to determine the years based on the given information. We will assume the "present year" is 2024 CE (Common Era).

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a. From the present year (2024 CE), which year was it 150 years ago?

To find the year 150 years ago, we subtract 150 from the present year.

$2024 - 150 = 1874$

The year 150 years ago was 1874 CE.

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b. From the present year (2024 CE), which year was it 2200 years ago?

We need to go back 2200 years from 2024 CE. We must account for the transition from CE to BCE (Before Common Era), noting that there is no year 0.

Years from 2024 CE back to 1 CE = $2024 - 1 = 2023$ years.

We need to go back a total of 2200 years. The remaining number of years to go back from 1 CE is $2200 - 2023 = 177$ years.

When going back from 1 CE into the BCE period, going back N years from 1 CE lands you in year N BCE.

So, going back 177 years from 1 CE means the year was 177 BCE.

To check: Years from 177 BCE to 1 CE is 177 years. Years from 1 CE to 2024 CE is $2024 - 1 = 2023$ years. Total years = $177 + 2023 = 2200$ years.

The year 2200 years ago was 177 BCE.

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c. What will be the year 320 years after 680 BCE?

Starting from 680 BCE, we move forward 320 years. Moving forward in BCE means the year number decreases (gets closer to 1 BCE).

We can subtract the number of years moved forward from the starting BCE year number to find the new BCE year number.

$680 - 320 = 360$

We need to check if this calculation crosses the transition from BCE to CE. The years from 680 BCE to 1 BCE is $680 - 1 = 679$ years. Since we are only moving forward 320 years, and $320 < 679$, we do not reach 1 CE; we remain in the BCE period.

The year 320 years after 680 BCE will be 360 BCE.

We analyze the pattern in each sequence to find the missing terms.

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a. $(–40), (–34), (–28), (–22), \_\_\_\_\_, \_\_\_\_\_\_, \_\_\_\_\_\_$

Let's find the difference between consecutive terms:

$-34 - (-40) = -34 + 40 = 6$

$-28 - (-34) = -28 + 34 = 6$

$-22 - (-28) = -22 + 28 = 6$

The pattern is to add 6 to the previous term.

The next terms are:

$-22 + 6 = -16$

$-16 + 6 = -10$

$-10 + 6 = -4$

The completed sequence is: (– 40), (– 34), (– 28), (– 22), –16, –10, –4.

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b. $3, 4, 2, 5, 1, 6, 0, 7, \_\_\_\_\_, \_\_\_\_\_\_, \_\_\_\_\_\_$

This sequence alternates between two patterns. Look at the terms at odd positions and even positions separately.

Odd positions: $3, 2, 1, 0, \_\_\_\_\_$ (Subtracting 1 each time)

Even positions: $4, 5, 6, 7, \_\_\_\_\_$ (Adding 1 each time)

The next term is at the 9th position, which is an odd position. The term after 0 in the odd sequence is $0 - 1 = -1$.

The next term is at the 10th position, which is an even position. The term after 7 in the even sequence is $7 + 1 = 8$.

The next term is at the 11th position, which is an odd position. The term after -1 in the odd sequence is $-1 - 1 = -2$.

The completed sequence is: 3, 4, 2, 5, 1, 6, 0, 7, –1, 8, –2.

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c. $\_\_\_\_\_, \_\_\_\_\_\_, 12, 6, 1, (–3), (–6), \_\_\_\_\_, \_\_\_\_\_\_, \_\_\_\_\_\_$

Let's find the difference between consecutive known terms:

$6 - 12 = -6$

$1 - 6 = -5$

$-3 - 1 = -4$

$-6 - (-3) = -6 + 3 = -3$

The difference is increasing by 1 each time: -6, -5, -4, -3. This is an arithmetic progression of the difference.

To find the next terms, the differences should continue this pattern by adding 1:

Next difference = $-3 + 1 = -2$. The term after -6 is $-6 + (-2) = -8$.

Next difference = $-2 + 1 = -1$. The term after -8 is $-8 + (-1) = -9$.

Next difference = $-1 + 1 = 0$. The term after -9 is $-9 + 0 = -9$.

To find the terms before 12, we work backwards with the differences. The difference before -6 was $-6 - 1 = -7$. The term before 12 is $12 - (-7) = 12 + 7 = 19$.

The difference before -7 was $-7 - 1 = -8$. The term before 19 is $19 - (-8) = 19 + 8 = 27$.

Let's re-check the difference before 12. $12 - 19 = -7$, not -6. My backward calculation of the difference was correct, but applying it backward is $Term_{previous} - Term_{current} = Difference_{previous}$. So $Term_{previous} = Term_{current} + Difference_{previous}$.

Let's restart the backward calculation. The difference before -6 is -7. So the term before 12 should be $12 - (-7) = 12 + 7 = 19$. Ah, applying it the other way: $Term_{current} - Term_{previous} = Difference$. $12 - x = -6$. $x = 12 - (-6) = 18$. Yes, the difference before -6 is -7. $18 - y = -7$. $y = 18 - (-7) = 25$.

Differences going left: ..., -8, -7, -6, -5, -4, -3, -2, -1, 0.

Let $a_n$ be the terms. $a_{n+1} - a_n = d_n$, where $d_n$ is the sequence of differences.

Known terms: ..., $a_3=12$, $a_4=6$, $a_5=1$, $a_6=-3$, $a_7=-6$, ...

$d_3 = a_4 - a_3 = 6 - 12 = -6$

$d_4 = a_5 - a_4 = 1 - 6 = -5$

$d_5 = a_6 - a_5 = -3 - 1 = -4$

$d_6 = a_7 - a_6 = -6 - (-3) = -3$

The sequence of differences is ..., $d_2$, $d_3=-6$, $d_4=-5$, $d_5=-4$, $d_6=-3$, $d_7=-2$, $d_8=-1$, $d_9=0$, ...

The difference pattern $d_{n+1} = d_n + 1$. So $d_2 = d_3 - 1 = -6 - 1 = -7$. And $d_1 = d_2 - 1 = -7 - 1 = -8$.

Now find the terms using $a_n = a_{n-1} + d_{n-1}$.

$a_2 = a_3 - d_2 = 12 - (-7) = 12 + 7 = 19$. (Check: $a_3 - a_2 = 12 - 19 = -7 = d_2$).

$a_1 = a_2 - d_1 = 19 - (-8) = 19 + 8 = 27$. (Check: $a_2 - a_1 = 19 - 27 = -8 = d_1$).

The terms before 12 are 27 and 19.

The terms after -6 are:

$a_8 = a_7 + d_7 = -6 + (-2) = -8$.

$a_9 = a_8 + d_8 = -8 + (-1) = -9$.

$a_{10} = a_9 + d_9 = -9 + 0 = -9$.

The completed sequence is: 27, 19, 12, 6, 1, –3, –6, –8, –9, –9.

Given Cards: $(+1), (+7), (+18), (–5), (–2), (–9)$.

Target Value: To find an expression whose value is close to $–30$.

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We need to choose a subset of the given cards and combine them using addition and subtraction operators to get a value near $-30$. To get a value close to a negative number like $-30$, we should likely include negative cards and potentially subtract positive cards.

Let's try combining some of the negative cards and subtracting some positive cards.

Consider the cards $(–5)$, $(+7)$, and $(+18)$.

Let's form the expression $(–5) – (+7) – (+18)$.

Evaluating this expression:

$(–5) – (+7) – (+18) = -5 - 7 - 18$

$-5 - 7 = -12$

$-12 - 18 = -30$

The value of the expression $(–5) – (+7) – (+18)$ is $-30$.

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This value, $-30$, is exactly the target value. Thus, an expression with this value is the closest possible to $-30$.

Another expression that is close is $(-5) - (+1) - (+7) - (+18) = -5 - 1 - 7 - 18 = -31$, which is also very close (difference of 1).

Another expression that is also close is $(-9) - (+7) - (+18) = -9 - 7 - 18 = -34$, which has a difference of 4.

The expression $(–5) – (+7) – (+18)$ gives exactly $-30$, so its value is the closest to $-30$.

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An expression whose value is closer to $–30$ is:

$(–5) – (+7) – (+18)$

The value of this expression is $-30$.

We examine the result of different operations involving positive and negative integers.

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a. (positive) – (negative)

Subtracting a negative number is equivalent to adding a positive number. So, (positive) – (negative) is the same as (positive) + (positive).

Example: $+5 - (-3) = 5 + 3 = +8$.

The result is always positive.

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b. (positive) + (negative)

The result depends on the magnitudes of the positive and negative numbers.

If the absolute value of the positive number is greater, the result is positive. Example: $+5 + (-3) = 5 - 3 = +2$.

If the absolute value of the negative number is greater, the result is negative. Example: $+3 + (-5) = 3 - 5 = -2$.

If the absolute values are equal, the result is zero. Example: $+5 + (-5) = 0$.

The result can be positive, negative, or zero.

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c. (negative) + (negative)

Adding two negative numbers. The absolute values are added, and the result is negative.

Example: $-5 + (-3) = -5 - 3 = -8$.

The result is always negative.

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d. (negative) – (negative)

Subtracting a negative number is equivalent to adding a positive number. So, (negative) – (negative) is the same as (negative) + (positive).

Example: $-5 - (-3) = -5 + 3 = -2$.

Example: $-3 - (-5) = -3 + 5 = +2$.

Example: $-5 - (-5) = -5 + 5 = 0$.

The result can be positive, negative, or zero.

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e. (negative) – (positive)

Subtracting a positive number is the same as adding the corresponding negative number. So, (negative) – (positive) is the same as (negative) + (negative).

Example: $-5 - (+3) = -5 - 3 = -8$.

The result is always negative.

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f. (negative) + (positive)

Adding a negative number to a positive number. This is the same operation as (positive) + (negative), just with the order of the numbers reversed.

The result depends on the magnitudes of the negative and positive numbers.

Example: $-3 + (+5) = -3 + 5 = +2$.

Example: $-5 + (+3) = -5 + 3 = -2$.

Example: $-5 + (+5) = -5 + 5 = 0$.

The result can be positive, negative, or zero.

Given:

Total number of tokens in the string = 100.

The pattern of tokens is visible in the image.

Let's assume a '+' token has a value of $+1$ and a '–' token has a value of $-1$.

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To Find:

The total value of the string.

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Solution:

Observing the pattern in the string of tokens provided in the image, we can see a repeating unit:

The pattern is + + - + - -

This repeating unit consists of 6 tokens.

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Let's calculate the value of one repeating unit:

Value of one unit = $(+1) + (+1) + (-1) + (+1) + (-1) + (-1)$

Value of one unit = $1 + 1 - 1 + 1 - 1 - 1$

Value of one unit = $(1+1+1) + (-1-1-1)$

Value of one unit = $3 + (-3)$

Value of one unit = $0$

Each repeating unit has a value of 0.

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The total number of tokens in the string is 100.

We need to find how many times the repeating unit of 6 tokens fits into 100 tokens, and how many tokens are left over.

Number of full units = $\frac{100}{6}$

$100 = 16 \times 6 + 4$

There are 16 full repeating units and 4 remaining tokens.

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The total value of the 16 full repeating units is:

Value of full units = $16 \times (\text{Value of one unit})$

Value of full units = $16 \times 0 = 0$

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The remaining 4 tokens are the first 4 tokens of the repeating pattern '+ + - + - -'.

The remaining tokens are + + - +.

Let's calculate the value of these 4 remaining tokens:

Value of remaining tokens = $(+1) + (+1) + (-1) + (+1)$

Value of remaining tokens = $1 + 1 - 1 + 1$

Value of remaining tokens = $(1+1+1) - 1$

Value of remaining tokens = $3 - 1 = 2$

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The total value of the string is the sum of the value of the full units and the value of the remaining tokens.

Total Value = Value of full units + Value of remaining tokens

Total Value = $0 + 2 = 2$

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The value of the string is +2.

Brahmagupta's rules for arithmetic with positive (fortunes) and negative (debts) numbers can be understood using the analogies of Bela's Building of Fun (movement on floors relative to Ground) and a number line (movement from 0).

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Let's explain the operations using both analogies:

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1. Adding a Positive Number (+a):

This is like adding a 'fortune'.

Building Analogy: Starting at any floor, adding a positive number means going up 'a' number of floors.

Example: Start at Floor -3. Add +5. Go up 5 floors from -3. You reach Floor +2. ($-3 + 5 = 2$).

Number Line Analogy: Starting at any number, adding a positive number means moving right 'a' units.

Example: Start at -3. Add +5. Move right 5 units from -3. You reach +2. ($-3 + 5 = 2$).

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2. Adding a Negative Number (–a):

This is like adding a 'debt'.

Building Analogy: Starting at any floor, adding a negative number means going down 'a' number of floors.

Example: Start at Floor +5. Add -3. Go down 3 floors from +5. You reach Floor +2. ($+5 + (-3) = 2$).

Example: Start at Floor -2. Add -4. Go down 4 floors from -2. You reach Floor -6. ($-2 + (-4) = -6$).

Number Line Analogy: Starting at any number, adding a negative number means moving left 'a' units.

Example: Start at +5. Add -3. Move left 3 units from +5. You reach +2. ($+5 + (-3) = 2$).

Example: Start at -2. Add -4. Move left 4 units from -2. You reach -6. ($-2 + (-4) = -6$).

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3. Subtracting a Positive Number (+a):

This is like subtracting a 'fortune'. Subtracting a positive number is the opposite of adding a positive number.

Building Analogy: Starting at any floor, subtracting a positive number means doing the opposite of going up 'a' floors, which is going down 'a' floors.

Example: Start at Floor +5. Subtract +3. Go down 3 floors from +5. You reach Floor +2. ($+5 - (+3) = 2$).

Example: Start at Floor +2. Subtract +5. Go down 5 floors from +2. You reach Floor -3. ($+2 - (+5) = -3$).

Number Line Analogy: Starting at any number, subtracting a positive number means doing the opposite of moving right 'a' units, which is moving left 'a' units.

Example: Start at +5. Subtract +3. Move left 3 units from +5. You reach +2. ($+5 - (+3) = 2$).

Example: Start at +2. Subtract +5. Move left 5 units from +2. You reach -3. ($+2 - (+5) = -3$).

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4. Subtracting a Negative Number (–a):

This is like subtracting a 'debt'. Subtracting a negative number is the opposite of adding a negative number.

Building Analogy: Starting at any floor, subtracting a negative number means doing the opposite of going down 'a' floors, which is going up 'a' floors.

Example: Start at Floor +2. Subtract -3. Go up 3 floors from +2. You reach Floor +5. ($+2 - (-3) = 5$).

Example: Start at Floor -3. Subtract -5. Go up 5 floors from -3. You reach Floor +2. ($-3 - (-5) = 2$).

Number Line Analogy: Starting at any number, subtracting a negative number means doing the opposite of moving left 'a' units, which is moving right 'a' units.

Example: Start at +2. Subtract -3. Move right 3 units from +2. You reach +5. ($+2 - (-3) = 5$).

Example: Start at -3. Subtract -5. Move right 5 units from -3. You reach +2. ($-3 - (-5) = 2$).

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These explanations cover Brahmagupta's rules for adding and subtracting positive and negative numbers, showing how the result depends on the operation and the signs and magnitudes of the numbers, consistent with movements on a building's floors or a number line.

Let's provide examples for each of the basic rules of integer arithmetic (corresponding to Brahmagupta's rules as explained in the previous answer).

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1. Adding a Positive Number: (+a)

Starting with a positive number, adding a positive: $+7 + (+4) = 7 + 4 = 11$

Starting with a negative number, adding a positive: $-10 + (+6) = -10 + 6 = -4$

Starting with zero, adding a positive: $0 + (+9) = 0 + 9 = 9$

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2. Adding a Negative Number: (+ (-a))

Starting with a positive number, adding a negative: $+12 + (-5) = 12 - 5 = 7$

Starting with a negative number, adding a negative: $-8 + (-3) = -8 - 3 = -11$

Starting with zero, adding a negative: $0 + (-7) = 0 - 7 = -7$

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3. Subtracting a Positive Number: (- (+a))

Starting with a positive number, subtracting a positive: $+15 - (+5) = 15 - 5 = 10$

Starting with a negative number, subtracting a positive: $-6 - (+8) = -6 - 8 = -14$

Starting with zero, subtracting a positive: $0 - (+11) = 0 - 11 = -11$

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4. Subtracting a Negative Number: (- (-a))

Starting with a positive number, subtracting a negative: $+9 - (-2) = 9 + 2 = 11$

Starting with a negative number, subtracting a negative: $-4 - (-7) = -4 + 7 = 3$

Starting with zero, subtracting a negative: $0 - (-13) = 0 + 13 = 13$