Chapter 1 Notes: A SQUARE AND A CUBE

A square number is obtained by multiplying a number by itself. The squares of natural numbers (1, 4, 9, 16, 25, ...) are called perfect squares.

Key Properties:

• Perfect squares always end in 0, 1, 4, 5, 6, or 9 (never 2, 3, 7, or 8)
• Squares can only have an even number of zeros at the end
• A number is a perfect square if its prime factors can be split into two identical groups

Common Mistake: A number ending in 0, 1, 4, 5, 6, or 9 is NOT necessarily a perfect square (e.g., 26 ends in 6 but is not a perfect square).

Consecutive Squares Pattern: The difference between consecutive squares gives odd numbers. $4-1=3,\quad 9-4=5,\quad 16-9=7, \ldots$

Sum of Odd Numbers: The sum of first n odd numbers equals n². $1+3+5+\ldots+(2n-1) = n^2$

Between Consecutive Squares: Between n² and (n+1)², there are exactly 2n numbers.

Quick Tip: To find (n+1)², use $(n+1)^2 = n^2 + (2n+1)$ For example, $36^2 = 35^2 + 71 = 1225 + 71 = 1296$

Square root is the inverse operation of squaring. If $y = x^2$, then x is the square root of y.

Finding Square Roots using Prime Factorisation:

1. Find the prime factorisation of the number
2. Pair up the prime factors
3. Take one factor from each pair — their product is the square root

Example: $324 = 2\times2\times3\times3\times3\times3 = (2\times3\times3)^2$, so $\sqrt{324}=18$

Every perfect square has two integer square roots — one positive and one negative.

Common Mistake: Don't forget the negative square root! $\sqrt{64} = \pm 8$, not just 8.

A cube number is obtained by multiplying a number by itself three times. Numbers like 1, 8, 27, 64, 125, ... are perfect cubes.

Key Properties:

• A cube can end in any digit (0–9), unlike squares
• A cube can never end in exactly two zeros (00) — it must end in 0, 3 zeros, etc.
• Cubes of odd numbers are odd; cubes of even numbers are even
• A number is a perfect cube if its prime factors can be split into three identical groups

Cubes and Odd Numbers Pattern: $1 = 1^3,\quad 3+5 = 2^3,\quad 7+9+11 = 3^3$

Cube root is the inverse of cubing. If $y = x^3$, then x is the cube root of y.

Finding Cube Roots using Prime Factorisation:

1. Find the prime factorisation of the number
2. Group the prime factors into triplets
3. Take one factor from each triplet — their product is the cube root

Example: $3375 = 3\times3\times3\times5\times5\times5 = (3\times5)^3 = 15^3$, so $\sqrt[3]{3375}=15$

Estimating cube roots of large numbers: For a number like 32768, note that units digit 8 → cube root ends in 2; it lies between $30^3=27000$ and $40^3=64000$, so try 32 → $32^3 = 32768$ ✓

Taxicab Numbers are numbers that can be expressed as the sum of two cubes in two different ways.

The most famous is the Hardy-Ramanujan Number: 1729 $1729 = 1^3 + 12^3 = 9^3 + 10^3$

This arose when G. H. Hardy visited Ramanujan in hospital and remarked that his taxi number 1729 was 'dull.' Ramanujan instantly knew it was special!

The next taxicab numbers are 4104 and 13832 — try expressing them as sums of two cubes in two ways.

Property	Perfect Square	Perfect Cube
Definition	$n \times n = n^2$	$n \times n \times n = n^3$
Possible last digits	0,1,4,5,6,9	0,1,2,3,4,5,6,7,8,9
Prime factor test	Split into 2 equal groups	Split into 3 equal groups
Symbol	$\sqrt{}$	$\sqrt[3]{}$
Example	$\sqrt{144}=12$	$\sqrt[3]{216}=6$

History Note: Ancient Indians used varga for square, ghana for cube, and mula (meaning root of a plant) for square root — the origin of the word 'root' in mathematics!

Common Mistakes to Avoid:

• Numbers ending in 2, 3, 7, or 8 are never perfect squares
• A perfect square always has an odd number of factors (because one factor pairs with itself)
• The cube of a 2-digit number has at least 3 digits and at most 6 digits