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A Square and a Cube – Class 8 Maths Chapter 1 Notes & NCERT PDF

Class 8MathematicsChapter 1NCERT book: Ganita Prakash - Mathematics Class 8

Notes and a simple summary of Chapter 1 of the NCERT Class 8 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 1: A Square and a Cube

Introduction

The chapter opens with a puzzle from Queen Ratnamanjuri's will. Her son Khoisnam and 99 relatives are taken to a room with 100 lockers. Person 1 opens every locker, Person 2 toggles every 2nd locker, Person 3 every 3rd, and so on until all 100 have had a turn. Khoisnam knew the answer before anyone started.

The trick: a locker is toggled once for every factor of its number. Factors come in partner pairs (6 = 1 × 6 = 2 × 3), so most numbers have an even number of factors and their lockers end up closed. Only numbers where a factor pairs with itself — 1 × 1, 2 × 2, 3 × 3 … — have an odd number of factors. So the lockers left open are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. The lockers touched exactly twice are the primes, so the passcode was 2-3-5-7-11.

Key Ideas

1.1 Square Numbers

  • A number that is a number multiplied by itself is a square: n × n = n² ("n squared"). The name comes from the area of a square: side 5 → area 5² = 25 sq units.
  • Squares of natural numbers are perfect squares: 1, 4, 9, 16, 25, …
  • Squares of fractions and decimals also exist, e.g. (2.5)² = 6.25.

    • Patterns in Perfect Squares

  • A perfect square always ends in 0, 1, 4, 5, 6 or 9 — never 2, 3, 7 or 8. The last digit can prove a number is not a square, but can't prove it is one (26 ends in 6 but is not a square).
  • If a number ends in 1 or 9, its square ends in 1; squares ending in 6 come from numbers ending in 4 or 6 (16 = 4², 196 = 14², 576 = 24²).
  • The number of zeros at the end of a square is double the zeros of the number (100² = 10000), so squares only have an even number of end zeros.
  • Even numbers have even squares; odd numbers have odd squares.

    • Squares and Odd Numbers

    The sum of the first n odd numbers is n².

    1 = 1, 1 + 3 = 4, 1 + 3 + 5 = 9, 1 + 3 + 5 + 7 = 16 … The picture of "inverted L" shapes around a square shows why each step adds the next odd number. The nth odd number is 2n − 1.

  • Test for a square: keep subtracting 1, 3, 5, … If you land exactly on 0, the number is a perfect square (25 − 1 − 3 − 5 − 7 − 9 = 0, five steps, so 25 = 5²). If you cross below 0 (as with 38), it is not.
  • Between two consecutive squares p and q there are q − p − 1 numbers.
  • Two consecutive triangular numbers add up to a square: 1 + 3 = 4, 3 + 6 = 9, 6 + 10 = 16.

    • Square Roots

  • If y = x², then x is a square root of y. Every perfect square has two integer roots: √64 = ±8. In this chapter we use the positive root.
  • Prime factorisation method: if the prime factors can be split into two identical groups, the number is a square. 324 = 2 × 2 × 3 × 3 × 3 × 3 = (2 × 3 × 3)² = 18², so √324 = 18. But 156 = 2 × 2 × 3 × 13 cannot be paired, so it is not a square.
  • Estimating: for √1936 — it lies between 40² = 1600 and 50² = 2500; the last digit 6 means the root ends in 4 or 6; 45² = 2025 is too big, so the answer is 44.

    • 1.2 Cubic Numbers

  • A number multiplied by itself three times is a cube: n × n × n = n³. 1, 8, 27, 64, 125 are perfect cubes. A cube of edge 4 is made of 4 layers of 4 × 4 = 64 unit cubes.
  • Cubes can end in any digit, and a cube can never end in exactly two zeros.
  • Cubes of negatives and fractions work too: (−6)³ = −216.
  • Consecutive odd numbers again: 1 = 1³, 3 + 5 = 2³, 7 + 9 + 11 = 3³, 13 + 15 + 17 + 19 = 4³ … and 91 + 93 + … + 109 = 10³ = 1000.

    • Taxicab Numbers

    1729 = 1³ + 12³ = 9³ + 10³

    When G. H. Hardy called his taxi number 1729 "rather a dull number", Srinivasa Ramanujan replied that it is the smallest number that is a sum of two cubes in two different ways. Such numbers are called taxicab numbers; the next two are 4104 and 13832.

    Cube Roots

  • If y = x³, then x = ∛y. ∛8 = 2, ∛1000 = 10.
  • Prime factorisation method: split the factors into three identical groups. 3375 = (3 × 5) × (3 × 5) × (3 × 5) = 15³, so ∛3375 = 15. 500 = 2 × 2 × 5 × 5 × 5 cannot be split, so it is not a cube.

    • Successive Differences

    For squares, the differences 3, 5, 7, 9 … have a second-level difference that is always 2. For cubes, the differences become constant (always 6) only at the third level.

    Worked Examples

  • Find 36² using 35² = 1225. 36² = 1225 + (36th odd number) = 1225 + (2 × 36 − 1) = 1225 + 71 = 1296.
  • Smallest multiplier to make 9408 a square. 9408 = 2⁶ × 3 × 7². The 3 is unpaired, so multiply by 3: 9408 × 3 = 28224 = 168².
  • Largest integer square handkerchief from 125 cm². 11² = 121 ≤ 125 < 144 = 12², so the side is 11 cm.
  • Estimate √250. 15² = 225 and 16² = 256, so √250 lies between 15 and 16, closer to 16.
  • Make 1323 a cube. 1323 = 3³ × 7². One more 7 is needed: 1323 × 7 = 9261 = 21³.

    • Figure it Out — Hints

  • Use last digits first to reject non-squares (2032, 2048 and 1027 are not squares; 1089 = 33²).
  • 126² = 125² + 125 + 126 = 15625 + 251.
  • For "smallest square divisible by 4, 9 and 10", find the LCM (180 = 2² × 3² × 5) and complete the pair for 5: 900.
  • Cube roots of 27000 and 10648 are 30 and 22.
  • For guessing ∛4913, ∛12167, ∛32768: use the last digit of the cube and the size — answers 17, 23, 32.

    • Maths Words

  • Square number (वर्ग संख्या)
  • Perfect square (पूर्ण वर्ग)
  • Square root (वर्गमूल)
  • Cube (घन)
  • Perfect cube (पूर्ण घन)
  • Cube root (घनमूल)
  • Factor (गुणनखंड)
  • Prime factorisation (अभाज्य गुणनखंडन)
  • Odd number (विषम संख्या)
  • Triangular number (त्रिकोणीय संख्या)
  • Estimate (अनुमान)
  • Unit cube (इकाई घन)
  • Taxicab number (टैक्सीकैब संख्या)
  • Successive differences (क्रमिक अंतर)

    • A Pinch of History

  • The Babylonians made the first known lists of squares and cubes on clay tablets around 1700 BCE.
  • In Sanskrit, varga meant both a square figure and a square power; ghana meant a solid cube and a cube power; the fourth power was varga-varga. Aryabhata (499 CE) defined varga.
  • The word "root" comes from the Sanskrit mula (root of a plant): varga-mula = square root, ghana-mula = cube root. This idea passed into Arabic (jidhr) and Latin (radix). Brahmagupta (628 CE) also used pada for root.

    • Common Mistakes

  • Thinking any number ending in 1, 4, 5, 6, 9 or 0 is a square — the last-digit test only rules numbers out.
  • Writing n² as 2n (5² = 25, not 10).
  • Forgetting that every perfect square has a negative root too.
  • Pairing prime factors for cube roots — cubes need triplets.
  • Assuming the cube of an odd number could be even (it is always odd).

    • Maths in Daily Life

    Square roots help find the side of a square room or plot from its area; cube roots find the edge of a cubical tank from its volume. Estimating roots quickly is useful when cutting tiles, cloth or paper to size.

    💡 Key Learning Points

    • ✓Only perfect squares have an odd number of factors — which is why the square-numbered lockers stay open
    • ✓Perfect squares end only in 0, 1, 4, 5, 6 or 9 and have an even number of zeros at the end
    • ✓The sum of the first n odd numbers is n², and the nth odd number is 2n − 1
    • ✓A number is a perfect square if its prime factors form pairs, and a perfect cube if they form triplets
    • ✓1729 = 1³ + 12³ = 9³ + 10³ is the Hardy–Ramanujan (taxicab) number

    👨‍🏫 Teaching Tips

    • →Act out the 100-locker puzzle with 20 students and 20 cards so the class sees which 'lockers' stay open
    • →Build squares with square tiles and cubes with unit blocks so n² and n³ become areas and volumes
    • →Show the 'inverted L' picture of odd numbers building squares; ask students to draw the next step
    • →Let pairs play Aribam and Bijou's square-root game, including numbers that need estimating like 250
    • →Use factor trees on the board and circle pairs (for squares) or triplets (for cubes) in different colours
    • →Tell the Hardy–Ramanujan taxi story and let students hunt for the two cube-sum ways for 4104 and 13832
    • →Have students compute successive differences for squares and cubes in a table to discover the constants 2 and 6

    📋 Assessment Questions

    1. Can the student explain why the lockers numbered 1, 4, 9, …, 100 remain open?
    2. Can the student decide, using prime factorisation, whether a number like 1156 or 2800 is a perfect square?
    3. Can the student find a square root by estimation (e.g., √1936 = 44) and justify each step?
    4. Can the student find cube roots such as ∛3375 and ∛27000 using triplets of prime factors?
    5. Can the student find the smallest number to multiply by to make a number a square or cube (e.g., 9408, 1323)?

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    Keep learning

    All chapters in Ganita Prakash - Mathematics Class 8

    1. 1. A Square and a Cube
    2. 2. Power Play
    3. 3. A Story of Numbers
    4. 4. Quadrilaterals
    5. 5. Number Play
    6. 6. We Distribute, Yet Things Multiply
    7. 7. Proportional Reasoning-1