The Other Side of Zero
Integers — complete solutions with step-by-step working
💡 In-text Questions
Conceptual questions asked within the flow of the chapter.
Each + press moves the lift up one floor and each − press moves it down one floor.
Four floors up → press + + + +, written as +4
Three floors down → press − − −, written as −3
Floor 0 is the ground floor (the Welcome Hall) — it is the reference point. Floors above it get positive numbers; floors below it get negative numbers.
| Floor | Shop |
|---|---|
| +3 | Book Store |
| +2 | Art Centre |
| +1 | Food Court |
| 0 | Welcome Hall (Ground) |
| −1 | Toy Store |
| −2 | Video Games |
The additive inverse of a number is the number that, added to it, gives 0.
Each number is joined to its additive inverse (same value, opposite sign).
+5 ↔ −5 · −7 ↔ +7 · −8 ↔ +8 · +9 ↔ −9
- Jay — Art Centre → Floor +2
- Asin — Sports Centre → Floor +5
- Binnu — Cinema Centre → Floor −3
- Aman — Toy Store → Floor −1
Since −3 lies lowest on the number line, Binnu is on the lowest floor (−3).
Subtraction here means: what must be added to the smaller to reach the larger?
\(15-5:\quad 5+?=15 \Rightarrow ?=\) 10
\(100-10:\quad 10+?=100 \Rightarrow ?=\) 90
\(74-34:\quad 34+?=74 \Rightarrow ?=\) 40
Yes — subtracting a negative number is the same as adding the corresponding positive number. The ‘infinite lift’ behaves exactly like a number line.
From 5 to 9 → move +4 (4 steps forward): \(5+4=9\)
From 9 to 3 → move −6 (6 steps backward): \(9+(-6)=3\)
From 3 to −2 → move −5 (5 steps backward): \(3+(-5)=-2\)
Tip: subtracting a negative = adding a positive.
a. \(-125+(-30)=\) −155 (both moves are downward)
b. \(+105-(-55)=105+55=\) +160
c. \(+80-(-150)=80+150=\) +230
d. \(-99-(-200)=-99+200=\) +101
Deposit ₹60 (credit): \(100+60=\) ₹160
Pay electric bill ₹30 (debit): \(160-30=\) ₹130
Business purchase ₹150 (debit): \(130-150=\) −₹20 — yes, this is possible; some banks allow a temporary negative balance (and may charge a fee/interest).
Earn ₹200 the next day (credit): \(-20+200=\) ₹180
📝 Exercise Questions — “Figure it Out”
Every Figure-it-Out problem from the chapter, fully solved.
Starting Floor + Movement = Target Floor
\((+2)+(-3)=-1\) → Floor −1, the Toy Store
Using: Starting Floor + Movement = −5. A few possibilities:
- \((+2)+(-7)=-5\) (the given example)
- \((+1)+(-6)=-5\)
- \((0)+(-5)=-5\)
- \((-2)+(-3)=-5\)
- \((-4)+(-1)=-5\)
A lower floor is a smaller number. All negatives < 0 < all positives.
- B = −9
- C = −6
- F = +2
- G = +6
- H = +11
Read from the tick positions on the vertical line: A(−12), B(−9), C(−6), D(−1), E(+1), F(+2), G(+6), H(+11).
Mark points at these positions (naming them P, Q, R, S):
One possible choice (many answers are correct):
Positive: A = 2, B = 5, C = 8 · Negative: P = −1, Q = −3, R = −7
The three negative numbers: −7, −3, −1
Yes, 2 > −3 — on the number line, 2 lies to the right of −3, so it is larger.
Yes, −2 < 3 — 3 lies to the right of −2, so −2 is the smaller number.
A green (+) and a red (−) token form a “zero pair” and cancel out.
(a) 3 green & 5 red — cancel 3 zero pairs, 2 red remain:
\((+3)+(-5)=\) −2 → attendant on Floor −2
(b) 6 green & 3 red — cancel 3 zero pairs, 3 green remain:
\((+6)+(-3)=\) +3 → attendant on Floor +3
Start with 3 red tokens (−3). To take away +5 you need 5 green tokens, so add 5 zero pairs. Removing 5 green leaves \(3+5=8\) red tokens.
\(-3-(+5)=\) −8
\((+30)+(+40)+(+50)-40-50-60 = 120-150=\) −₹30
Total debits \(=1+2+4+8+16+32+64+128=255\)
\(-255+256=\) ₹1
Keeping a positive balance avoids overdraft fees/interest and keeps money available for spending. A temporary negative balance can be worthwhile for a strategic large purchase or investment — for example, buying stock or equipment for a business that will soon earn back more than the cost.
Approximate heights (positive = above sea level, negative = below):
Highest point = A (+1500 m). Lowest point = D (−1200 m).
Decreasing order: A, E, C, G, F, B, D
Increasing order: D, B, F, G, C, E, A
Highest point on Earth = Mount Everest, height about +8848 m above sea level.
Lowest point = Challenger Deep in the Mariana Trench (Pacific Ocean), depth about −10994 m below sea level.
The warmest reading fits mid-afternoon; the coldest fits the dead of night.
| Temperature | Time |
|---|---|
| 14 °C | 02:00 p.m. |
| 8 °C | 11:00 a.m. |
| −2 °C | 11:00 p.m. |
| −4 °C | 02:00 a.m. |
Each row and each border column must add to the same value.
Top: \(5+(-3)+(-5)=-3\)
Bottom: \((-8)+(-2)+7=-3\)
Left: \(5+0+(-8)=-3\)
Right: \((-5)+(-5)+7=-3\)
Border sum = −3
One valid way for each (other answers are possible):
| −10 | 10 | 4 |
| 5 | −5 | |
| 9 | −10 | 5 |
| 6 | 8 | −16 |
| 11 | −5 | |
| −19 | −2 | 19 |
| 7 | −2 | −9 |
| −3 | −5 | |
| −8 | −6 | 10 |
Grids with an unfilled centre can be filled in many ways, because the centre cell is not part of any border sum — so it is free to be anything.
No matter which numbers you circle (following the rules), the circled numbers always add to the same total for a given grid:
First grid → sum = −8
Second grid → sum = −14
The “magic” comes from how the grid is built: each entry equals a row-number plus a column-number, so the chosen cells always cover every row and column exactly once.
a. 0 and −7 → −6, −5, −4, −3, −2, −1
b. −4 and 4 → −3, −2, −1, 0, 1, 2, 3
c. −8 and −15 → −14, −13, −12, −11, −10, −9
d. −30 and −23 → −29, −28, −27, −26, −25, −24
One combination: −5, 7, −10 (since \(-5+7+(-10)=-8\)).
Others work too, e.g. −2, −3, −3 or 0, −8, 0.
Adding one face of each die can only give even totals in some cases and specific odd ones in others. The sums that cannot be made are:
−9, −7, −5, 0, 2, 7, 9, 11
Remember: there was no year 0 — the year after 1 BCE is 1 CE.
a. Taking the present year as 2026 CE: \(2026-150=\) 1876 CE.
b. \(2026-2200=-174\); with no year 0 this is 175 BCE.
c. 320 years after 680 BCE: \(680-320=360\) → 360 BCE (written as −360 BCE).
a. −40, −34, −28, −22, … (add 6 each time) → −16, −10, −4
b. 3, 4, 2, 5, 1, 6, 0, 7, … (odd spots ↓, even spots ↑) → −1, 8, −2
c. …, 12, 6, 1, −3, −6, … (gaps grow by 1) → before: 27, 19; after: −8, −9, −9
One way that hits exactly −30:
\((-2)+(-9)-(+18)-(+1) = -2-9-18-1 =\) −30
The pattern repeats in blocks of 5: + + + − −, whose value is \(3-2=+1\).
\(100 \div 5 = 20\) blocks → value \(=20\times(+1)=\) +20
Both players start at 0 and race to reach either +50 or −50. Roll two dice (one shows +1 to +6, the other −1 to −6), then add or subtract them in any order. A positive result moves you toward +50; a negative result moves you toward −50 — then follow any snake or ladder you land on.
