10. The Other Side of Zero

Yes. Some examples: floors below the ground floor, temperatures below 0°C, places below sea level, and owing money to someone.

Four floors up: + + + + or +4.

Three floors down: − − − or −3.

From the ground floor, we do not press any button to stay there. So it is Floor 0.

(+1) + (+2) = +3. We reach the Book Store.

(+2) + (−3) = −1. We reach the Toy Store.

(a) (+1) + (+4) = +5   (b) (+4) + (+1) = +5   (c) (+4) + (−3) = +1   (d) (−1) + (+2) = +1

(e) (−1) + (+1) = 0   (f) 0 + (+2) = +2   (g) 0 + (−2) = −2

(+1) + (−6) = −5

0 + (−5) = −5

(−2) + (−3) = −5

(+4) + (−9) = −5

(−4) + (−1) = −5

(a) (+1) + (+4) = +5   (b) (+4) + (+1) = +5

(c) (+4) + (−3) + (−2) = −1   (d) (−1) + (+2) + (−3) = −2

(+4) + (−4) = 0. He stays on the ground floor.

+4 → −4   −4 → +4   −3 → +3   0 → 0   +2 → −2   −1 → +1

+5 ↔ −5   −7 ↔ +7   −8 ↔ +8   +9 ↔ −9

Jay: +2 (Art Centre). Asin: +5 (Sports Centre). Binnu: −3 (Cinema). Aman: −1 (Toy Store).

Binnu is on the lowest floor.

(a) −2 < +5   (b) −5 < +4   (c) −5 < −3

(d) +6 > −6   (e) 0 > −4   (f) 0 < +4

(a) −10 > −12   (b) +17 > −10   (c) 0 > −20

(d) +9 > −9   (e) −25 < −7   (f) +15 > −17

B = −9   C = −6   F = +2   G = +6   H = +11

Count the marks from A (−12) upwards:

(a) −7 is 5 marks above A. (b) −4 is 8 marks above A (2 marks above C). (c) +3 is 1 mark above F. (d) −10 is 2 marks above A.

5 + ? = 15, so 15 − 5 = 10

10 + ? = 100, so 100 − 10 = 90

34 + ? = 74, so 74 − 34 = 40

(a) (+1) − (+4) = −3   (b) 0 − (+2) = −2   (c) (+4) − (+1) = +3   (d) 0 − (−2) = +2

(e) (+4) − (−3) = +7   (f) (−4) − (−3) = −1   (g) (−1) − (+2) = −3   (h) (−2) − (−2) = 0

(i) (−1) − (+1) = −2   (j) (+3) − (−3) = +6

(a) (+40) + (+160) = +200   (b) (+40) + (−240) = −200

(c) (−50) + (+250) = +200   (d) (−50) + (−150) = −200

(e) (−200) − (−40) = −160   (f) (+200) − (+40) = +160   (g) (−200) − (+40) = −240

(a) −125 + (−30) = −155   (b) +105 − (−55) = +160

(c) +105 + (+55) = +160   (d) +80 − (−150) = +230

(e) +80 + (+150) = +230   (f) −99 − (−200) = +101

(g) −99 + (+200) = +101   (h) +1500 − (−1500) = +3000

Yes. Subtracting a negative number gives the same answer as adding the positive number. The infinite lift looks like a number line standing upright.

5 to 9: 4 steps forward (5 + 4 = 9).

9 to 3: 6 steps backward, that is −6 (9 + (−6) = 3).

3 to −2: 5 steps backward, that is −5 (3 + (−5) = −2).

Positive: 2, 5, 8. Negative: −1, −3, −7.

−7 < −3 < −1

Yes, 2 > −3, because 2 is to the right of −3 on the number line.

Yes, −2 < 3, because −2 is to the left of 3.

(a) −5   (b) 0   (c) 10   (d) −10   (e) 14   (f) 2

(a) −125 + (−30) = −155   (b) +105 − (−55) = +160   (c) +80 − (−150) = +230   (d) −99 − (−200) = +101

(a) 6 green + 4 green = 10 green → (+6) + (+4) = +10

(b) 3 red + 2 red = 5 red → (−3) + (−2) = −5

(c) 5 green + 7 red: 5 zero pairs, 2 red left → (+5) + (−7) = −2

(d) 2 red + 6 green: 2 zero pairs, 4 green left → (−2) + (+6) = +4

(a) 3 green and 5 red: (+3) + (−5) = −2. He is on Floor −2.

(b) 6 green and 3 red: (+6) + (−3) = +3. He is on Floor +3.

Yes. Both are equal to −2.

(a) (+10) − (+7) = +3   (b) (−8) − (−4) = −4   (c) (−9) − (−4) = −5

(d) (+9) − (+12) = −3 (add 3 zero pairs)   (e) (−5) − (−7) = +2 (add 2 zero pairs)   (f) (−2) − (−6) = +4 (add 4 zero pairs)

(a) (−5) − (−7) = +2   (b) (+10) − (+13) = −3   (c) (−7) − (−9) = +2

(d) (+3) − (+8) = −5   (e) (−2) − (−7) = +5   (f) (+3) − (+15) = −12

Start with 3 red tokens. We need to take out 5 green tokens, so we put in 5 zero pairs.

After taking out 5 green, 8 red are left. So −3 − (+5) = −8

(a) (−3) − (+10) = −13   (b) (+8) − (−7) = +15   (c) (−5) − (+9) = −14

(d) (−9) − (+10) = −19   (e) (+6) − (−4) = +10   (f) (−2) − (+7) = −9

After the credit of ₹60: ₹100 + ₹60 = ₹160

After the electric bill of ₹30: ₹160 − ₹30 = ₹130

After the purchase of ₹150: ₹130 − ₹150 = −₹20 (yes, some banks allow this for a short time)

After earning ₹200: −₹20 + ₹200 = ₹180

(+30) + (+40) + (+50) + (−40) + (−50) + (−60) = 120 − 150 = −₹30

Total debits = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 255

Balance = 256 − 255 = ₹1

With a positive balance, we do not owe money to the bank and we do not pay extra interest or fees.

A short negative balance may be worthwhile for an important need, like buying goods for a business that will earn more money soon.

(a) A ≈ +1450 m   (b) B ≈ −500 m   (c) C ≈ +300 m   (d) D ≈ −1200 m

(e) E ≈ +1200 m   (f) F ≈ −200 m   (g) G ≈ +100 m

Highest: A. Lowest: D.

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

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

Mount Everest, about 8849 m above sea level.

On the ocean floor: the Challenger Deep in the Mariana Trench (Pacific Ocean), about −11,000 m.

On land: the shore of the Dead Sea, about −430 m.

Leh, Kargil and Dras (Ladakh), Srinagar and Gulmarg (Jammu and Kashmir), Shimla and Manali (Himachal Pradesh), Tawang (Arunachal Pradesh), and high places in Sikkim and Uttarakhand.

They are all high up in the Himalayan mountains. The higher we go, the colder it becomes.

It is warmest in the afternoon and coldest late at night.

Top row: 5 + (−3) + (−5) = −3

Bottom row: (−8) + (−2) + 7 = −3

Left column: 5 + 0 + (−8) = −3

Right column: (−5) + (−5) + 7 = −3

Border sum = −3

Border sum +4

Border sum −2

Border sum −4

Another way:

Check: 7 − 11 + 0 = −4; 0 − 5 + 1 = −4; −5 + 0 + 1 = −4; 7 − 6 − 5 = −4.

The first grid (border sum +4) can also be filled in many ways. Only the two right-hand corners have to add up to 9.

The second grid has only one answer, because each row and column already has enough numbers given.

Reason: When fewer numbers are given in a row or column, we have more freedom to choose.

Example: Give 4 in the top-left corner and −1 in the middle of the right column. Ask for border sum 0.

The sum is always −1 for the first grid, whichever numbers we choose.

Green grid: −8. Purple grid: −14.

Each row is the row above with the same number added or subtracted, and each column also changes by the same amount. So whichever cells we pick (one from each row and column), the total stays the same.

To make such a grid, write a row like 1, 2, 3, 4 and add the same number (say 5) to get each next row.

(a) −6, −5, −4, −3, −2, −1

(b) −3, −2, −1, 0, 1, 2, 3

(c) −14, −13, −12, −11, −10, −9

(d) −29, −28, −27, −26, −25, −24

(−2) + (−3) + (−3) = −8   or   5 + (−6) + (−7) = −8

−9, −7, −5, 0, 2, 7, 9, 11

8 − 13 = −5   (−8) − 13 = −21   (−13) − (−8) = −5   (−13) + (−8) = −21

8 + (−13) = −5   (−8) − (−13) = 5   13 − 8 = 5   13 − (−8) = 21

(a) 2026 − 150 = 1876

(b) 2026 years back takes us to 1 CE. One more year back is 1 BCE (there is no year 0). 2200 − 2026 = 174 years more, so it is 175 BCE

(c) 680 − 320 = 360 BCE

(a) −40, −34, −28, −22, −16, −10, −4 (add 6)

(b) 3, 4, 2, 5, 1, 6, 0, 7, −1, 8, −2

(c) 27, 19, 12, 6, 1, −3, −6, −8, −9, −9 (subtract 8, 7, 6, 5, 4, 3, 2, 1, 0)

(−2) + (−9) − (+18) − (+1) = −30

(a) (positive) − (negative): always positive

(b) (positive) + (negative): can be positive or negative

(c) (negative) + (negative): always negative

(d) (negative) − (negative): can be positive or negative

(e) (negative) − (positive): always negative

(f) (negative) + (positive): can be positive or negative

The pattern repeats every 5 tokens: 3 positive and 2 negative. Each group = +3 + (−2) = +1.

Number of groups = 100 ÷ 5 = 20. Value = 20 × (+1) = +20

• Two positives: going up and then up again takes you higher. 2 + 3 = 5.
• Two negatives: going down and then down again takes you lower. (−2) + (−3) = −5.
• A positive and a negative: the up and down presses cancel in pairs; the bigger one decides the direction. −5 + 3 = −2.
• A number and its inverse: going up and then coming down the same floors brings you back to 0.
• Adding zero: pressing no button keeps you on the same floor.
• Subtracting a negative: the movement from −3 to 2 is 5 floors up, so 2 − (−3) = 2 + 3.

Addition: 4 + 6 = 10; (−4) + (−6) = −10; (−9) + 4 = −5; 7 + (−7) = 0; −8 + 0 = −8.

Subtraction: 9 − 4 = 5; 4 − 9 = −5; 6 − (−2) = 8; −5 − (−5) = 0; 0 − 7 = −7.