Revision Notes: Class 8 Maths Chapter 1 A Square and A Cube

Class 8 Maths Chapter 1: A Square and A Cube — Quick Revision Notes

• Square number / perfect square: n×n = n²; the squares of natural numbers (1, 4, 9, 16, 25…) are called perfect squares.
• Odd factor count: Every perfect square has an odd number of factors (one factor pairs with itself) — the basis of the 100-lockers puzzle, where only square-numbered lockers stay open.
• Last-digit test: A perfect square can only end in 0, 1, 4, 5, 6 or 9 — never 2, 3, 7 or 8. (Necessary but not sufficient — not every number ending in these digits is a square.)
• Zero-count rule (squares): Squaring doubles the trailing-zero count, so a perfect square always has an even number of trailing zeros.
• Odd/even parity: An even number squares to an even number; an odd number squares to an odd number.
• Consecutive-odd-number pattern: The sum of the first n consecutive odd numbers is n² (1=1²; 1+3=2²; 1+3+5=3²…).
• Numbers between consecutive squares: Exactly 2n numbers lie between n² and (n+1)².
• Square root: If y=x², x is a square root of y, written √y. Methods: repeated subtraction of consecutive odd numbers, prime factorisation (pairing identical primes), or estimation by bracketing between known squares.
• Cube number / perfect cube: n×n×n = n³; numbers like 1, 8, 27, 64, 125… are perfect cubes. A number is a perfect cube only if every prime factor’s exponent is a multiple of 3.
• Cube root: If y=x³, x is the cube root of y, written ∛y. Method: prime factorisation, grouping each prime into triples; or estimate using the last-digit rule (1→1, 8→2, 7→3, 4→4, 5→5, 6→6, 3→7, 2→8, 9→9) plus bracketing between known cubes.
• Zero-count rule (cubes): Cubing multiplies the trailing-zero count by 3, so a perfect cube’s trailing zeros are always a multiple of 3 (never exactly 1 or 2).
• Consecutive-odd-number pattern (cubes): The nth run of n consecutive odd numbers sums to n³ (e.g. 7+9+11=3³).
• Taxicab numbers: A number expressible as a sum of two positive cubes in two different ways. 1729 (Hardy–Ramanujan number) = 1³+12³ = 9³+10³; the next two are 4104 and 13832.
• History: Babylonians compiled square/cube tables on clay tablets (≈1700 BCE). In Sanskrit mathematics, varga = square and ghana = cube; varga-mula/ghana-mula = square/cube root (Aryabhata, Brahmagupta) — the origin of the word ‘root’ traces to Sanskrit mula.

Written by Satish

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