The Other Side of Zero

Given: Starting Floor = +2, Movement = -3.

Using the formula: Starting Floor + Movement = Target Floor

$$( +2) + (-3) = -1$$

You will reach Floor -1.

Using the concept Starting Floor + Movement = Target Floor:

a. $(+1) + (+4) = \mathbf{+5}$

Starting at Floor +1, moving up 4 floors → Floor +5.

b. $(+4) + (+1) = \mathbf{+5}$

Starting at Floor +4, moving up 1 floor → Floor +5.

c. $(+4) + (-3) = \mathbf{+1}$

Starting at Floor +4, moving down 3 floors → Floor +1.

d. $(-1) + (+2) = \mathbf{+1}$

Starting at Floor -1, moving up 2 floors → Floor +1.

e. $(-1) + (+1) = \mathbf{0}$

Starting at Floor -1, moving up 1 floor → Floor 0 (Ground).

f. $0 + (+2) = \mathbf{+2}$

Starting at Ground Floor, moving up 2 floors → Floor +2.

g. $0 + (-2) = \mathbf{-2}$

Starting at Ground Floor, moving down 2 floors → Floor -2.

Using: Movement needed = Target Floor − Starting Floor = $(-5) - \text{Starting Floor}$

Several examples:

1. Starting Floor = $0$: Movement = $(-5) - 0 = -5$
$$0 + (-5) = -5$$

2. Starting Floor = $+3$: Movement = $(-5) - (+3) = -8$
$$(+3) + (-8) = -5$$

3. Starting Floor = $-1$: Movement = $(-5) - (-1) = -4$
$$(-1) + (-4) = -5$$

4. Starting Floor = $-3$: Movement = $(-5) - (-3) = -2$
$$(-3) + (-2) = -5$$

5. Starting Floor = $+5$: Movement = $(-5) - (+5) = -10$
$$(+5) + (-10) = -5$$

6. Starting Floor = $-5$: Movement = $(-5) - (-5) = 0$
$$(-5) + (0) = -5$$

In general, any starting floor $S$ requires a movement of $(-5 - S)$ to reach Floor $-5$.

Combining button presses: pressing +1 and then +4 means moving up 1 floor and then up 4 more floors.

$$(+1) + (+4) = \mathbf{+5}$$

Pressing +4 and then +1 means moving up 4 floors and then up 1 more floor.

$$(+4) + (+1) = \mathbf{+5}$$

Pressing +4, then -3, then -2:

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

Step 2: $(+1) + (-2) = -1$

$$(+4) + (-3) + (-2) = \mathbf{-1}$$

Pressing -1, then +2, then -3:

Step 1: $(-1) + (+2) = +1$

Step 2: $(+1) + (-3) = -2$

$$(-1) + (+2) + (-3) = \mathbf{-2}$$

The inverse (additive inverse) of a number $n$ is the number which when added to $n$ gives 0.

| Number | Inverse |
|--------|--------|
| $+4$ | $-4$ |
| $-4$ | $+4$ |
| $-3$ | $+3$ |
| $0$ | $0$ |
| $+2$ | $-2$ |
| $-1$ | $+1$ |

Verification:
- $(+4) + (-4) = 0$ ✓
- $(-4) + (+4) = 0$ ✓
- $(-3) + (+3) = 0$ ✓
- $0 + 0 = 0$ ✓
- $(+2) + (-2) = 0$ ✓
- $(-1) + (+1) = 0$ ✓

Note: The inverse of 0 is 0 itself. Lines should connect each number to its inverse as listed above.

Based on the Building of Fun described in the chapter (standard layout):

1. Jay is in the Art Centre → Floor +2
2. Asin is in the Sports Centre → Floor -1 (Sports Centre is below ground)
3. Binnu is in the Cinema Centre → Floor -2 (Cinema is further below)
4. Aman is in the Toys Store → Floor +3 (Toys Store is above Art Centre)

*(Note: Exact floor numbers depend on the building diagram. The key concept is that lower floor numbers mean lower positions.)*

The person on the lowest floor is the one with the smallest (most negative) floor number.

Concept: On the number line (or building), the number to the left (lower floor) is smaller.

a. $-2 \ \boxed{<} \ +5$

(Floor -2 is below Floor +5; all negative numbers are less than positive numbers)

b. $-5 \ \boxed{<} \ +4$

(Floor -5 is below Floor +4)

c. $-5 \ \boxed{<} \ -3$

(Floor -5 is lower than Floor -3; among negative numbers, the one with larger absolute value is smaller)

d. $+6 \ \boxed{>} \ -6$

(Floor +6 is above Floor -6)

e. $0 \ \boxed{>} \ -4$

(Floor 0 is above all negative floors; all negative numbers are less than 0)

f. $0 \ \boxed{<} \ +4$

(Floor 0 is below Floor +4; all positive numbers are greater than 0)

a. $-10 \ \boxed{>} \ -12$

(Floor -10 is higher than Floor -12; -10 is closer to 0)

b. $+17 \ \boxed{>} \ -10$

(All positive numbers are greater than all negative numbers)

c. $0 \ \boxed{>} \ -20$

(0 is greater than all negative numbers)

d. $+9 \ \boxed{>} \ -9$

(Positive numbers are greater than negative numbers)

e. $-25 \ \boxed{<} \ -7$

(Floor -25 is much lower than Floor -7)

f. $+15 \ \boxed{>} \ -17$

(Positive numbers are greater than negative numbers)

Given information: Floor A = -12, Floor D = -1, Floor E = +1.

The floors are equally spaced on the number line. Between A (-12) and D (-1) there are floors B and C, dividing the interval into equal parts.

From A to D: $-1 - (-12) = 11$ units over 3 intervals → each interval = not equal.

Assuming the floors are marked at consecutive integers or at equal intervals based on the figure (which shows a vertical number line):

If the spacing between consecutive marked floors is consistent:
- A = -12, and moving upward by equal steps to reach D = -1 with B and C in between.
- 3 gaps from A to D: step = $\frac{-1-(-12)}{3} = \frac{11}{3}$ — not integer.

More likely the floors are at integer values and the labels A through H mark specific floors:
- A = -12, B = -9 (or similar), C = -6, D = -1 (or as per diagram spacing).

*Note: The exact answer depends on the figure. Based on a typical equal-spacing assumption with 7 labeled points (A to G) and given A = -12, D = -1, E = +1:*

From D to E: $+1 - (-1) = 2$ units, 1 gap → step between D and E = 2.

Assuming uniform spacing throughout: step = 2 (since D = -1 and E = +1 differ by 2).

Working backwards from D = -1 with step 2:
- C = -1 - 2 = -3
- B = -3 - 2 = -5
- A = -5 - 2 = -7 ≠ -12 (contradiction)

Alternative: step between D and E = 2, but different step elsewhere. Most likely the figure shows a number line where each unit is 1:
- A = -12, B = -8, C = -4, D = -1 (not uniform)

*Since the exact figure is not available, the general method is:*

Use the given anchor points to determine the scale/step size, then apply it to find the remaining floors. With A = -12, D = -1, E = +1:
- Step from D to E = 2 units per division
- F = E + 2 = +3
- G = F + 2 = +5
- H = G + 2 = +7
- C = D - 2 = -3
- B = C - 2 = -5

(Students should read off values directly from their figure.)

To mark these floors on the number line (building shown as a vertical line):

- -7: Mark 7 units below 0 (below ground level).
- -4: Mark 4 units below 0.
- +3: Mark 3 units above 0.
- -10: Mark 10 units below 0.

On the number line, the order from bottom to top is:
$$-10 < -7 < -4 < 0 < +3$$

Students should locate and label these points on their building/number line diagram accordingly.

Using: Target Floor − Starting Floor = Movement needed

a. $(+1) - (+4)$: To go from Floor +4 to Floor +1, move down 3.
$$(+1) - (+4) = \mathbf{-3}$$

b. $(0) - (+2)$: To go from Floor +2 to Floor 0, move down 2.
$$(0) - (+2) = \mathbf{-2}$$

c. $(+4) - (+1)$: To go from Floor +1 to Floor +4, move up 3.
$$(+4) - (+1) = \mathbf{+3}$$

d. $(0) - (-2)$: To go from Floor -2 to Floor 0, move up 2.
$$(0) - (-2) = \mathbf{+2}$$

e. $(+4) - (-3)$: To go from Floor -3 to Floor +4, move up 7 (3 to reach 0, then 4 more).
$$(+4) - (-3) = \mathbf{+7}$$

f. $(-4) - (-3)$: To go from Floor -3 to Floor -4, move down 1.
$$(-4) - (-3) = \mathbf{-1}$$

g. $(-1) - (+2)$: To go from Floor +2 to Floor -1, move down 3.
$$(-1) - (+2) = \mathbf{-3}$$

h. $(-2) - (-2)$: To go from Floor -2 to Floor -2, no movement needed.
$$(-2) - (-2) = \mathbf{0}$$

i. $(-1) - (+1)$: To go from Floor +1 to Floor -1, move down 2.
$$(-1) - (+1) = \mathbf{-2}$$

j. $(+3) - (-3)$: To go from Floor -3 to Floor +3, move up 6 (3 to reach 0, then 3 more).
$$(+3) - (-3) = \mathbf{+6}$$

Using: Missing addend = Target − Starting; and subtraction rules.

a. $(+40) + \_ = +200$

Missing = $200 - 40 = +160$
$$(+40) + \mathbf{(+160)} = +200$$

b. $(+40) + \_ = -200$

Missing = $-200 - 40 = -240$
$$(+40) + \mathbf{(-240)} = -200$$

c. $(-50) + \_ = +200$

Missing = $200 - (-50) = 200 + 50 = +250$
$$(-50) + \mathbf{(+250)} = +200$$

d. $(-50) + \_ = -200$

Missing = $-200 - (-50) = -200 + 50 = -150$
$$(-50) + \mathbf{(-150)} = -200$$

e. $(-200) - (-40) = ?$

Subtracting a negative = adding its positive:
$$(-200) - (-40) = (-200) + (+40) = \mathbf{-160}$$

f. $(+200) - (+40) = ?$

$$(+200) - (+40) = \mathbf{+160}$$

g. $(-200) - (+40) = ?$

$$(-200) - (+40) = -200 - 40 = \mathbf{-240}$$

Using the infinite lift concept: Starting Level + Movement = Target Level, and Target − Starting = Movement.

a. $-125 + (-30)$

Start at -125, move down 30:
$$-125 + (-30) = \mathbf{-155}$$

b. $+105 - (-55)$

Target = +105, Starting = -55. Movement = $105 - (-55) = 105 + 55$:
$$+105 - (-55) = \mathbf{+160}$$

c. $+105 + (+55)$

Start at +105, move up 55:
$$+105 + (+55) = \mathbf{+160}$$

*(Note: (b) and (c) give the same answer — subtracting a negative equals adding a positive.)*

d. $+80 - (-150)$

$= 80 + 150$:
$$+80 - (-150) = \mathbf{+230}$$

e. $+80 + (+150)$

$$+80 + (+150) = \mathbf{+230}$$

*(Note: (d) and (e) give the same answer.)*

f. $-99 - (-200)$

$= -99 + 200 = 200 - 99$:
$$-99 - (-200) = \mathbf{+101}$$

g. $-99 + (+200)$

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

*(Note: (f) and (g) give the same answer.)*

h. $+1500 - (-1500)$

$= 1500 + 1500$:
$$+1500 - (-1500) = \mathbf{+3000}$$

On the number line, mark:

Positive numbers (to the right of 0): For example, $+1, +3, +5$

Negative numbers (to the left of 0): For example, $-1, -3, -5$

These are placed at their respective positions on the number line, with positive numbers to the right of 0 and negative numbers to the left of 0.

Based on the example in Question 1, the three negative numbers marked are:

$$\boxed{-1} \quad \boxed{-3} \quad \boxed{-5}$$

(Students should write the three negative numbers they actually marked on their number line.)

**Is $2 > -3$?**

Yes, $2 > -3$.

Reason: On the number line, 2 lies to the right of -3. All positive numbers are greater than all negative numbers. Since 2 is positive and -3 is negative, $2 > -3$.

**Is $-2 < 3$?**

Yes, $-2 < 3$.

Reason: On the number line, -2 lies to the left of 3. Since -2 is negative and 3 is positive, and all negative numbers are less than all positive numbers, $-2 < 3$.

a. $-5 + 0$

Adding 0 to any number gives the same number:
$$-5 + 0 = \mathbf{-5}$$

b. $7 + (-7)$

A number plus its additive inverse equals 0:
$$7 + (-7) = \mathbf{0}$$

c. $-10 + 20$

Start at -10, move up 20: $20 - 10 = 10$ (positive, since 20 > 10):
$$-10 + 20 = \mathbf{+10}$$

d. $10 - 20$

Start at 10, target is found by going back 20: $10 - 20 = -(20-10)$:
$$10 - 20 = \mathbf{-10}$$

e. $7 - (-7)$

Subtracting a negative = adding its positive:
$$7 - (-7) = 7 + 7 = \mathbf{14}$$

f. $-8 - (-10)$

Subtracting a negative = adding its positive:
$$-8 - (-10) = -8 + 10 = \mathbf{+2}$$

Using the token method: positive tokens (+) and negative tokens (-); zero pairs cancel out.

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

Place 6 positive tokens and 4 positive tokens. No zero pairs. Total = 10 positive tokens.
$$(+6) + (+4) = \mathbf{+10}$$

b. $(-3) + (-2)$

Place 3 negative tokens and 2 negative tokens. No zero pairs. Total = 5 negative tokens.
$$(-3) + (-2) = \mathbf{-5}$$

c. $(+5) + (-7)$

Place 5 positive and 7 negative tokens. Cancel 5 zero pairs. Remaining: 2 negative tokens.
$$(+5) + (-7) = \mathbf{-2}$$

d. $(-2) + (+6)$

Place 2 negative and 6 positive tokens. Cancel 2 zero pairs. Remaining: 4 positive tokens.
$$(-2) + (+6) = \mathbf{+4}$$

*(Note: The exact token images are not visible, but the method is described below.)*

Method: Count the positive tokens (green/+) and negative tokens (red/-). Cancel equal numbers of positive and negative tokens (zero pairs). The remaining tokens give the floor.

a. After cancelling zero pairs, if $p$ positive tokens remain → Floor $+p$; if $n$ negative tokens remain → Floor $-n$.

For example, if there are 5 positive and 3 negative tokens:
- Cancel 3 zero pairs → 2 positive tokens remain
- Floor = +2
- Addition statement: $(+5) + (-3) = +2$

b. Similarly, if there are 2 positive and 6 negative tokens:
- Cancel 2 zero pairs → 4 negative tokens remain
- Floor = -4
- Addition statement: $(+2) + (-6) = -4$

*(Students should apply this method to their actual token images.)*

Using the token method for subtraction: place tokens for the first number, then remove tokens of the second number (adding zero pairs if needed).

a. $(+10) - (+7)$

Place 10 positive tokens. Remove 7 positive tokens. Remaining: 3 positive.
$$(+10) - (+7) = \mathbf{+3}$$

b. $(-8) - (-4)$

Place 8 negative tokens. Remove 4 negative tokens. Remaining: 4 negative.
$$(-8) - (-4) = \mathbf{-4}$$

c. $(-9) - (-4)$

Place 9 negative tokens. Remove 4 negative tokens. Remaining: 5 negative.
$$(-9) - (-4) = \mathbf{-5}$$

d. $(+9) - (+12)$

Place 9 positive tokens. Need to remove 12 positive but only have 9. Add 3 zero pairs (3 positive + 3 negative). Now have 12 positive and 3 negative. Remove 12 positive. Remaining: 3 negative.
$$(+9) - (+12) = \mathbf{-3}$$

e. $(-5) - (-7)$

Place 5 negative tokens. Need to remove 7 negative but only have 5. Add 2 zero pairs. Now have 7 negative and 2 positive. Remove 7 negative. Remaining: 2 positive.
$$(-5) - (-7) = \mathbf{+2}$$

f. $(-2) - (-6)$

Place 2 negative tokens. Need to remove 6 negative but only have 2. Add 4 zero pairs. Now have 6 negative and 4 positive. Remove 6 negative. Remaining: 4 positive.
$$(-2) - (-6) = \mathbf{+4}$$

Using the rule: subtracting a number = adding its additive inverse.

a. $(-5) - (-7) = (-5) + (+7) = \mathbf{+2}$

b. $(+10) - (+13) = 10 - 13 = \mathbf{-3}$

c. $(-7) - (-9) = (-7) + (+9) = \mathbf{+2}$

d. $(+3) - (+8) = 3 - 8 = \mathbf{-5}$

e. $(-2) - (-7) = (-2) + (+7) = \mathbf{+5}$

f. $(+3) - (+15) = 3 - 15 = \mathbf{-12}$

Given: $-3 - (+5)$

Step 1: Place 3 negative tokens to represent $-3$.

Step 2: We need to remove 5 positive tokens, but we have none.

Step 3: Add 5 zero pairs (5 positive + 5 negative tokens). This does not change the value.

Now we have: 5 positive tokens and $(3 + 5) = 8$ negative tokens.

Step 4: Remove 5 positive tokens.

Remaining: 8 negative tokens.

$$-3 - (+5) = \mathbf{-8}$$

We had to put in 5 zero pairs.

Using the rule: $a - (+b) = a + (-b)$ and $a - (-b) = a + (+b)$.

a. $(-3) - (+10) = (-3) + (-10) = \mathbf{-13}$

*Token method: Place 3 negative. Add 10 zero pairs. Remove 10 positive. Left with 13 negative.*

b. $(+8) - (-7) = (+8) + (+7) = \mathbf{+15}$

*Token method: Place 8 positive. Add 7 zero pairs. Remove 7 negative. Left with 15 positive.*

c. $(-5) - (+9) = (-5) + (-9) = \mathbf{-14}$

*Token method: Place 5 negative. Add 9 zero pairs. Remove 9 positive. Left with 14 negative.*

d. $(-9) - (+10) = (-9) + (-10) = \mathbf{-19}$

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

f. $(-2) - (+7) = (-2) + (-7) = \mathbf{-9}$

Given:
- Starting balance = ₹0
- Credits (positive): ₹30, ₹40, ₹50
- Debits (negative): ₹40, ₹50, ₹60

Total credits = $30 + 40 + 50 = ₹120$

Total debits = $40 + 50 + 60 = ₹150$

Final balance = Starting balance + Total credits − Total debits
$$= 0 + 120 - 150 = \mathbf{-₹30}$$

The bank account balance is −₹30 (i.e., ₹30 in debt/overdraft).

Given:
- Starting balance = ₹0
- Debits: ₹1, 2, 4, 8, 16, 32, 64, 128
- Credit: ₹256

Total debits = $1 + 2 + 4 + 8 + 16 + 32 + 64 + 128$

This is a geometric series: $= 2^0 + 2^1 + 2^2 + \cdots + 2^7 = 2^8 - 1 = 256 - 1 = 255$

Final balance = $0 - 255 + 256 = \mathbf{+₹1}$

The bank account balance is ₹1 (positive).