Chapter Guide
Chapter Overview
This chapter develops a clear understanding of square numbers, perfect squares, cubes and perfect cubes. It explores their patterns and properties, including factors, prime factorisation, units digits, consecutive differences and sums of odd numbers. Students learn how to identify perfect squares and cubes and find or estimate their square roots and cube roots using different methods. The chapter also introduces interesting ideas such as taxicab numbers and connects mathematical concepts with their historical development.
Chapter Overview: A Square and A Cube
A Square and A Cube introduces the important ideas of
square numbers, perfect squares, cube numbers, square roots and cube roots.
The chapter begins with an interesting locker puzzle and uses factors to help students
discover why perfect squares have special properties. It then develops patterns in
squares and cubes through numbers, geometry, prime factorisation and logical reasoning.
Students learn that squaring a number means multiplying it by itself:
n2 = n × n,
while cubing a number means multiplying it by itself three times:
n3 = n × n × n.
These ideas are connected with their inverse operations,
square root and cube root.
What You Will Learn in This Chapter
| Topic |
What You Learn |
Example |
| Square Numbers |
A number obtained by multiplying a number by itself is called a square.
Squares of natural numbers are called perfect squares.
|
62 = 6 × 6 = 36
|
| Perfect Squares |
Numbers such as 1, 4, 9, 16, 25, ... that are squares of natural numbers.
|
25 = 52
|
| Square Roots |
The inverse operation of squaring. The square root gives the number
which produces a given perfect square.
|
√49 = 7
|
| Cube Numbers |
A number obtained by multiplying a number by itself three times.
|
43 = 4 × 4 × 4 = 64
|
| Perfect Cubes |
Cubes obtained by cubing natural numbers.
|
1, 8, 27, 64, 125, ...
|
| Cube Roots |
The inverse operation of cubing. It gives the number whose cube
produces the given perfect cube.
|
∛125 = 5
|
| Prime Factorisation |
Prime factors can be grouped to determine whether a number is a
perfect square or a perfect cube and to find its root.
|
324 = 22 × 34
|
Important Patterns of Perfect Squares
The chapter encourages students to observe patterns instead of simply memorising
results. Perfect squares have several useful properties that help in identifying
and comparing numbers.
| Property |
Important Observation |
Example |
| Units Digit |
A perfect square can end only in 0, 1, 4, 5, 6 or 9.
|
16, 25, 36, 49, 100
|
| Impossible Units Digits |
A natural number ending in 2, 3, 7 or 8 cannot be a perfect square.
|
128, 243 and 357 are not perfect squares.
|
| Trailing Zeros |
A perfect square has an even number of zeros at the end.
|
100 = 102, so it has two trailing zeros.
|
| Odd and Even Numbers |
The square of an even number is even, while the square of an odd number is odd.
|
82 = 64 and 72 = 49
|
| Consecutive Squares |
The difference between consecutive squares forms consecutive odd numbers.
|
4 − 1 = 3, 9 − 4 = 5, 16 − 9 = 7
|
| Sum of Odd Numbers |
The sum of the first n odd numbers is the square of n.
|
1 + 3 + 5 + 7 = 16 = 42
|
| Number of Factors |
Perfect squares have an odd number of factors because one factor
occurs as a repeated factor.
|
36 has 9 factors.
|
Square Roots
Square root is introduced as the inverse of squaring. If
x2 = y,
then x is a square root of
y.
For example:
72 = 49
⇒
√49 = 7
A perfect square has two integer square roots, one positive and one negative.
For example:
82 = 64
and
(−8)2 = 64
Therefore, the two integer square roots of 64 are
+8 and −8.
In the chapter, the positive square root is generally considered when the
symbol √ is used.
Methods Used to Find Square Roots
| Method |
Main Idea |
Use |
| Listing Squares |
Compare the given number with known consecutive square numbers.
|
Useful for smaller numbers.
|
| Successive Subtraction |
Subtract consecutive odd numbers starting from 1.
If the result reaches 0, the number is a perfect square.
|
Useful for understanding the pattern behind square numbers.
|
| Prime Factorisation |
Pair equal prime factors. The product of one factor from each pair
gives the square root.
|
Very useful for larger perfect squares.
|
| Estimation |
Locate a number between two nearby perfect squares and estimate its root.
|
Useful when the number is not a perfect square.
|
Prime Factorisation and Perfect Squares
A number is a perfect square when its prime factors can be arranged into
two identical groups. For example:
324 = 2 × 2 × 3 × 3 × 3 × 3
324 = (2 × 3 × 3)(2 × 3 × 3)
= (18)2
Hence,
√324 = 18.
In contrast, the prime factorisation of 156 cannot be divided into two
identical groups, so 156 is not a perfect square.
Cubes and Perfect Cubes
The chapter then extends the idea of squares to cubes. A cube is obtained
when a number is multiplied by itself three times:
n3 = n × n × n
The first few perfect cubes are:
13 = 1,
23 = 8,
33 = 27,
43 = 64,
53 = 125,
63 = 216
The geometric interpretation is also important: a cube of side
n units contains
n3
unit cubes.
Perfect Cubes and Prime Factorisation
Just as the prime factors of a perfect square can be grouped in pairs,
the prime factors of a perfect cube can be grouped in groups of three.
3375 = 3 × 3 × 3 × 5 × 5 × 5
3375 = (3 × 5)3
= 153
Therefore:
∛3375 = 15
If the prime factors cannot be grouped into three identical groups,
the number is not a perfect cube.
Cube Roots
Cube root is the inverse operation of cubing. If
x3 = y,
then x is the cube root of
y.
53 = 125
⇒
∛125 = 5
| Number |
Prime Factorisation |
Root |
| 64 |
26 |
∛64 = 4 |
| 216 |
23 × 33 |
∛216 = 6 |
| 3375 |
33 × 53 |
∛3375 = 15 |
| 1728 |
26 × 33 |
∛1728 = 12 |
Interesting Number Patterns
The chapter goes beyond basic calculations and encourages students to discover
patterns in numbers. It examines successive differences of squares and cubes,
sums involving consecutive odd numbers, and relationships between triangular
numbers and squares.
One particularly interesting idea is the relationship between cubes and
consecutive odd numbers:
1 = 13
3 + 5 = 8 = 23
7 + 9 + 11 = 27 = 33
The chapter also introduces the famous Hardy–Ramanujan number
1729, which can be expressed as the sum of two positive cubes in
two different ways:
1729 = 13 + 123
= 93 + 103
Historical Connection
The chapter also connects mathematics with history. It mentions that the
Babylonians compiled lists of perfect squares and cubes as early as
1700 BCE. It also discusses the historical use of the
Sanskrit terms varga for square and
ghana for cube, along with mula as the
basis for the mathematical idea of a root.
Chapter in One View
| Concept |
Key Formula / Idea |
Example |
| Square |
n2 = n × n |
72 = 49 |
| Square Root |
√(n2) = n for positive n |
√81 = 9 |
| Cube |
n3 = n × n × n |
43 = 64 |
| Cube Root |
∛(n3) = n |
∛125 = 5 |
| Square Test |
Prime factors can be grouped into pairs. |
324 = 182 |
| Cube Test |
Prime factors can be grouped into triples. |
3375 = 153 |
| Square Pattern |
Sum of first n odd numbers = n2 |
1 + 3 + 5 = 9 = 32 |
Why This Chapter Is Important
A Square and A Cube builds the foundation for several
mathematical ideas that students will use later. It strengthens
number sense, factorisation, pattern recognition, estimation and
logical reasoning. Understanding perfect squares and cubes also
makes later topics such as algebraic identities, exponents, geometry,
Pythagorean relationships and numerical problem-solving easier to understand.
For examinations, students should pay special attention to the
properties of perfect squares and cubes, prime-factorisation methods,
square roots, cube roots, number patterns, estimation and reasoning-based
questions.
Learn Clearly
Important Concepts and notes
Important Formulas – A Square and A Cube
This chapter is mainly based on squares, cubes, square roots,
cube roots, prime factorisation and number patterns. The following
formulas and properties are the most important ones to remember for
solving problems and preparing for examinations.
1. Square of a Number
The square of a number is obtained by multiplying the number by itself.
Square of n:
n2 = n × n
Examples:
- 52 = 5 × 5 = 25
- 122 = 12 × 12 = 144
- 252 = 25 × 25 = 625
2. Area of a Square
If the side of a square is s, then its area is:
Area of square:
A = s2
Therefore, if the area of a square is known, its side can be found by
taking the square root.
Side of square:
s = √A
Example:
If the area is 441 m2,
s = √441 = 21 m
3. Square Root
Square root is the inverse operation of squaring.
If:
x2 = y
⇒
x = √y
For positive numbers:
√(n2) = n
Every positive perfect square has two integer square roots:
x2 = n
⇒
x = ±√n
Example:
82 = 64
Therefore, the integer square roots of 64 are
+8 and −8.
4. Square of Positive and Negative Numbers
(+n)2 = n2
(−n)2 = n2
Hence, the square of a positive number and its negative counterpart is
always the same.
62 = 36
(−6)2 = 36
5. Square of Fractions and Decimals
(a/b)2 = a2/b2
(a.b)2 = a.b × a.b
(3/5)2 = 9/25
(2.5)2 = 6.25
6. Square of a Sum
The chapter uses the expansion of a square while estimating and comparing
square numbers.
(a + b)2
= a2 + 2ab + b2
(40 + 5)2
= 402 + 2 × 40 × 5 + 52
= 2025
7. Square of a Difference
(a − b)2
= a2 − 2ab + b2
8. Difference of Two Squares
a2 − b2
= (a − b)(a + b)
This is particularly useful when comparing consecutive squares and
simplifying calculations.
9. Difference Between Consecutive Squares
The difference between two consecutive square numbers is an odd number.
(n + 1)2 − n2
= 2n + 1
52 − 42
= 25 − 16
= 9
Thus, the differences between consecutive squares are:
3, 5, 7, 9, 11, 13, ...
10. Sum of Consecutive Odd Numbers
The sum of the first n odd natural numbers is the square
of n.
1 + 3 + 5 + … + (2n − 1) = n2
1 + 3 + 5 + 7 + 9 = 25 = 52
11. Perfect Square Test Using Odd Numbers
A natural number is a perfect square if it can be expressed as the sum
of consecutive odd natural numbers beginning with 1.
n2 = 1 + 3 + 5 + … + (2n − 1)
36 = 1 + 3 + 5 + 7 + 9 + 11
Therefore, 36 is a perfect square and:
√36 = 6
12. Numbers Between Two Consecutive Squares
If n2 and
(n + 1)2 are consecutive squares, then the
number of natural numbers between them is:
(n + 1)2 − n2 − 1
= 2n
Between 162 and 172:
2 × 16 = 32
13. Possible Last Digits of Perfect Squares
A perfect square can have only the following digits in its units place:
Possible units digits:
0, 1, 4, 5, 6, 9
Therefore, a number ending in 2, 3, 7 or 8 cannot be
a perfect square.
14. Zeros at the End of a Square
If a number has n zeros at the end, its square has
2n zeros at the end.
Number of trailing zeros in n2
= 2 × (number of trailing zeros in n)
1000 has 3 trailing zeros.
10002 = 1,000,000
Therefore, the square has 6 trailing zeros.
15. Parity of a Square
(2n)2 = 4n2
Therefore, the square of an even number is always even.
(2n + 1)2 = 4n2 + 4n + 1
Therefore, the square of an odd number is always odd.
16. Factors of a Perfect Square
A perfect square has an odd number of factors. This happens because one
factor pair contains equal factors.
For n = m2:
the factor m is paired with itself.
Factors of 36:
1, 2, 3, 4, 6, 9, 12, 18, 36
Total = 9 factors
17. Perfect Square Using Prime Factorisation
A number is a perfect square when all its prime factors can be grouped
into pairs of identical factors.
N = p12a
×
p22b
×
p32c
…
Its square root is obtained by taking one factor from each pair:
√N =
p1a
×
p2b
×
p3c
…
324 = 22 × 34
√324 = 2 × 32 = 18
18. Cube of a Number
Cube of n:
n3 = n × n × n
43 = 4 × 4 × 4 = 64
19. Cube Root
Cube root is the inverse operation of cubing.
x3 = y
⇒
x = ∛y
∛(n3) = n
53 = 125
Therefore, ∛125 = 5
20. Cube of a Negative Number
(−n)3 = −n3
(−6)3 = −216
Unlike a square, the cube of a negative number remains negative.
21. Cube of a Fraction
(a/b)3 =
a3/b3
(4/6)3
= 43/63
= 64/216
22. Perfect Cube Using Prime Factorisation
A number is a perfect cube when its prime factors can be grouped into
groups of three identical factors.
N =
p13a
×
p23b
×
p33c
…
Its cube root is:
∛N =
p1a
×
p2b
×
p3c
…
3375 = 33 × 53
∛3375 = 3 × 5 = 15
23. Prime Factors of a Cube
If:
N = p1a
×
p2b
…
then:
N3 =
p13a
×
p23b
…
12 = 22 × 3
123 = 26 × 33
24. Difference Between Consecutive Cubes
The difference between consecutive cubes follows a quadratic pattern.
(n + 1)3 − n3
= 3n2 + 3n + 1
33 − 23
= 27 − 8
= 19
The successive differences of cubes eventually become constant after
repeated differences.
25. Possible Last Digits of Cubes
The last digit of a cube can be any digit from 0 to 9. The units digit
of the original number determines the units digit of its cube.
| Last Digit of Number |
Last Digit of Cube |
| 0 |
0 |
| 1 |
1 |
| 2 |
8 |
| 3 |
7 |
| 4 |
4 |
| 5 |
5 |
| 6 |
6 |
| 7 |
3 |
| 8 |
2 |
| 9 |
9 |
26. Hardy–Ramanujan Number
An important special number discussed in the chapter is
1729. It is the smallest number that can be expressed
as the sum of two positive cubes in two different ways.
1729 = 13 + 123
1729 = 93 + 103
27. Important Square and Cube Values
| n |
n2 |
n3 |
| 1 |
1 |
1 |
| 2 |
4 |
8 |
| 3 |
9 |
27 |
| 4 |
16 |
64 |
| 5 |
25 |
125 |
| 6 |
36 |
216 |
| 7 |
49 |
343 |
| 8 |
64 |
512 |
| 9 |
81 |
729 |
| 10 |
100 |
1000 |
28. Quick Revision Formulas
| Concept |
Formula / Rule |
| Square |
n2 = n × n |
| Cube |
n3 = n × n × n |
| Square Root |
√(n2) = n |
| Cube Root |
∛(n3) = n |
| Area of Square |
A = s2 |
| Side from Area |
s = √A |
| Square of Sum |
(a + b)2 = a2 + 2ab + b2 |
| Square of Difference |
(a − b)2 = a2 − 2ab + b2 |
| Difference of Squares |
a2 − b2 = (a − b)(a + b) |
| Consecutive Squares |
(n + 1)2 − n2 = 2n + 1 |
| Sum of First n Odd Numbers |
1 + 3 + 5 + … + (2n − 1) = n2 |
| Numbers Between Consecutive Squares |
2n |
| Cube of Negative Number |
(−n)3 = −n3 |
| Consecutive Cubes |
(n + 1)3 − n3 = 3n2 + 3n + 1 |
| Perfect Square Test |
Prime factors occur in pairs. |
| Perfect Cube Test |
Prime factors occur in groups of three. |
| Perfect Square Units Digit |
0, 1, 4, 5, 6 or 9 |
| Trailing Zeros in a Square |
Twice the trailing zeros of the original number. |
| Hardy–Ramanujan Number |
1729 = 13 + 123
= 93 + 103
|
Important Exam Note
Remember these four ideas especially well:
-
Perfect square:
prime factors can be grouped into pairs.
-
Perfect cube:
prime factors can be grouped into triples.
-
Square pattern:
the difference between consecutive squares is an odd number:
2n + 1.
-
Odd-number pattern:
the sum of the first n odd numbers is
n2.