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The Other Side of Zero – Class 6 Maths Chapter 10 Notes & NCERT PDF

Class 6MathematicsChapter 10NCERT book: Ganita Prakash - Mathematics Class 6

Notes and a simple summary of Chapter 10 of the NCERT Class 6 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 10: The Other Side of Zero

Introduction

We know the counting numbers 1, 2, 3 …, then 0, then the fractions. But the "number line" drawn so far is really a number ray: it starts at 0 and goes only to the right. Are there numbers less than 0? Chapter 10 says yes, and meets them in Bela's Building of Fun.

Key Ideas

10.1 Bela's Building of Fun

Bela's building has floors above and below the ground. The lift has two buttons: '+' to go up and '–' to go down.

  • Pressing + + is written +2, and – – – – is –4.
  • The ground floor (Welcome Hall) is Floor 0. Above it: Food Court +1, Art Centre +2, Book Store +3 … Below it: Toy Store –1, Video Games –2 …
  • A number with '+' is positive, and one with '–' is negative.
    • Zero is neither positive nor negative.

    Addition = movement

    Starting Floor + Movement = Target Floor

    Start at Food Court (+1) and press +2: (+1) + (+2) = +3, the Book Store. Combining button presses is also addition. Gurmit pressed +2 by mistake and then –3: (+2) + (–3) = –1, one floor below where he began.

    Back to zero: inverses. (+3) + (–3) = 0. We call –3 the inverse of +3, and +3 the inverse of –3. The inverse of 0 is 0.

    Comparing. Floor –4 is lower than Floor –3, so –4 < –3. Every negative number is less than 0, and every positive number is greater than 0.

    Subtraction = movement needed

    Target Floor – Starting Floor = Movement needed
  • Art Centre (+2) to Sports Centre (+5): (+5) – (+2) = +3
  • (–1) – (–2) = +1 (go one floor up)
  • (–1) – (+3) = –4 (go four floors down)
  • (+2) – (–2) = +4

    • The mine and the "infinite lift"

    In a mine, levels are metres above (+) or below (–) the ground:

    (+40) + (+60) = +100, (–90) + (–55) = –145, (+40) – (–50) = +90, (–90) – (+40) = –130.

    Imagine a lift that goes on forever both ways: (+2000) – (–200) = +2200, which is the same as (+2000) + (+200).

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

    Positive numbers, negative numbers and zero together are the integers: … –3, –2, –1, 0, 1, 2, 3 …

    Back to the number line

    Turn the infinite lift sideways and it becomes a number line. Negatives lie to the left of 0. Smaller numbers are always on the left, so 2 < 5, –3 < 2 and –5 < –3.

  • From 9 to 3: movement –6, so 9 + (–6) = 3 and 3 – 9 = –6.
  • An unmarked number line (only 0 shown) helps with big numbers: 85 + (–60) = 25, and (–100) – (+250) = –350.
  • Converting: subtracting a number = adding its inverse. For example, (+7) – (+5) = (+7) + (–5) and (+8) – (–2) = (+8) + (+2).

    • 10.2 The Token Model

    Green tokens are +1 and red tokens are –1. One green and one red make a zero pair (value 0).

  • (+5) + (–3): remove 3 zero pairs and 2 green are left, so +2.
  • (+5) + (–8): remove 5 zero pairs and 3 red are left, so –3.
  • (+5) – (+6): there aren't 6 greens to take away, so add one zero pair first, then remove 6 greens. The answer is –1.
  • (+4) – (–6): add 6 zero pairs, remove 6 reds, and 10 greens are left, so +10.

    • 10.3 Integers in Other Places

  • Bank: credits are positive and debits are negative. ₹100 → +60 → ₹160 → –30 → ₹130 → –150 → –₹20 (some banks allow this, usually with a fee) → +200 → ₹180.
  • Geography: heights are measured from sea level (0 m). Places below sea level have negative heights.
  • Temperature: 0°C is the freezing point of water. Places like Leh in Ladakh go below 0°C in winter.

    • 10.4 Explorations

  • Hollow integer grids: every border row and column adds up to the same border sum. Example: 4 + (–1) + (–3) = 0.
  • An amazing grid: circle a number, strike out its row and column, repeat, then add the circled numbers. Surprisingly, you get the same sum every time!

    • 10.5 A Pinch of History

  • China's Nine Chapters on Mathematical Art (1st–2nd century CE) used red and black rods for positive and negative numbers.
  • Kautilya's Arthaśhāstra (c. 300 BCE) described credit and debit, including negative balances.
  • The Bakhshālī manuscript (around 300 CE) wrote a negative number with a special symbol placed after the number.
  • Brahmagupta, in the Brāhma-sphuṭa-siddhānta (628 CE), first treated positive numbers, negative numbers and zero as equally valid numbers, and gave clear rules for working with them.
  • These ideas reached the Arab world by the 9th century and Europe by the 13th. Even in the 18th century, the French mathematician Lazare Carnot called negative numbers "absurd"!
    • Brahmagupta's rules (examples): (–2) + (–3) = –5; to add numbers with different signs, subtract the smaller from the larger and keep the sign of the larger, e.g. –5 + 3 = –2; a number plus its inverse is 0; 2 – (–3) = 2 + 3; 0 – (–2) = 2.

    Worked Examples

  • (–125) + (–30) = –155. Both are negative: add 125 + 30 and put a minus sign.
  • (+105) – (–55) = 105 + 55 = 160.
  • (–99) – (–200) = –99 + 200 = 101.
  • Bank: credits ₹30, ₹40, ₹50 and debits ₹40, ₹50, ₹60, starting from ₹0: 120 – 150 = –₹30.
  • Debits ₹1, 2, 4, 8, 16, 32, 64, 128 (total ₹255), then a credit of ₹256: balance +₹1.
  • Dice with faces –1, 2, –3, 4, –5, 6: sums from –10 to 12 are possible, but –9, –7, –5, 0, 2, 7, 9, 11 cannot be made.

    • Maths Words

  • Integer (पूर्णांक)
  • Positive number (धनात्मक संख्या), Negative number (ऋणात्मक संख्या)
  • Zero (शून्य)
  • Number line (संख्या रेखा), Number ray (संख्या किरण)
  • Additive inverse (योज्य प्रतिलोम)
  • Zero pair (शून्य युग्म)
  • Credit (जमा), Debit (नामे/निकासी)
  • Balance (शेष राशि)
  • Sea level (समुद्र तल)
  • Temperature (तापमान)

    • Common Mistakes

  • Thinking –3 < –4 because 3 < 4. On the number line, –4 is further left, so –4 < –3.
  • Calling 0 positive (or negative). It is neither.
  • (–8) – (–13): students often write –21. The correct answer is –8 + 13 = +5.
  • Forgetting to add zero pairs in the token model when there aren't enough tokens to take away.
  • Keeping the wrong sign when adding numbers with different signs (e.g. –5 + 3 is –2, not +2).

    • Maths in Daily Life

    Lift buttons in malls, basement parking (B1, B2), bank passbooks, weather reports from Leh or Drass, sea depths and mountain heights, and profit and loss in a shop all use integers.

    💡 Key Learning Points

    • ✓There are numbers less than zero (negative numbers); together with zero and positive numbers they form the integers, and zero is neither positive nor negative.
    • ✓Addition means Starting Position + Movement = Target Position; subtraction means Target Position − Starting Position = Movement needed.
    • ✓Every number has an inverse that adds to it to give zero; subtracting a number is the same as adding its inverse, so subtracting a negative is adding a positive.
    • ✓Integers can be compared on a number line: smaller numbers lie to the left, so −4 < −3 < 0 < 2.
    • ✓Integers describe real situations — floors, mine levels, bank credits/debits, heights from sea level and temperatures below 0°C — and Brahmagupta (628 CE) first gave rules for them.

    👨‍🏫 Teaching Tips

    • →Draw Bela's Building of Fun on the board with floor names; let children act as the lift, moving up/down as classmates call out '+2', '−3'.
    • →Write the two sentence frames on a chart: 'Starting + Movement = Target' and 'Target − Starting = Movement needed'; refer to them for every sum.
    • →Use green and red paper tokens in pairs; always make children physically add zero pairs when they cannot take away.
    • →Turn the 'infinite lift' sideways to become a number line — ask 'Which way is smaller now?'
    • →Bring a bank passbook sample or make a mock one: credits (+) and debits (−), and let a balance go negative to discuss fees.
    • →Show a laboratory thermometer with markings below 0°C; discuss Leh and Drass temperatures.
    • →Play Integers Snakes and Ladders with one +1…+6 and one −1…−6 die.
    • →Close with the history: red/black rods in China, Kautilya's accounts, the Bakhshali symbol, and Brahmagupta's rules in 628 CE.

    📋 Assessment Questions

    1. Can the student compare integers such as −4 and −3 and explain using floors or the number line?
    2. Can the student evaluate sums and differences like (−8) − (−13) and 13 − (−8) correctly?
    3. Can the student use tokens and zero pairs to show (+5) − (+6) and (+4) − (−6)?
    4. Can the student model a bank balance with credits and debits, including a negative balance?
    5. Can the student state and illustrate at least three of Brahmagupta's rules?

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    Keep learning

    All chapters in Ganita Prakash - Mathematics Class 6

    1. 1. Patterns in Mathematics
    2. 2. Lines and Angles
    3. 3. Number Play
    4. 4. Data Handling and Presentation
    5. 5. Prime Time
    6. 6. Perimeter and Area
    7. 7. Fractions
    8. 8. Playing with Constructions
    9. 9. Symmetry
    10. 10. The Other Side of Zero