CLASS 8 MATHS · PART 1 · CHAPTER 1 · STUDY NOTES

A Square and a Cube

Understand a square and a cube through short explanations, labelled models and original practice. Read one topic, explain it in your own words, then check.

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A square number is an area you can count

Arrange equal rows of tiles. The number of tiles is the side length multiplied by itself.

Count rows, then multiply

Each small cell is one unit tile. Both side lengths are 4, so this is a square, not a 4 by 2 rectangle.

Square: The result of multiplying a number by itself: n² = n × n.
Perfect square: A whole number that is the square of a whole number, such as 0, 1, 4 or 9.

A 4 by 4 square contains 16 unit tiles. The superscript 2 tells you how many equal factors to multiply; it does not mean “multiply by 2”.

Fractions, decimals and negative numbers can also be squared. A square number is a number; an area needs square units.

WORKED IDEA

A decimal and a fraction

(0.4)² = 0.16; (2/3)² = 4/9

Multiply both factors. For a fraction, multiply the numerators and the denominators.

Let’s look a little closer.

Go deeper: Why squares have an odd number of positive factors

For 49, positive factors form pairs 1 × 49 and 7 × 7. The 7 is counted once, so 49 has three positive factors. A positive nonsquare has different partners in every pair and therefore an even number of positive factors. This argument is about positive integers, not zero.

Take a moment. What do you think?

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Question

A square has side 7 cm. What is its area, and why is the unit cm²?

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NCERT reference: chapter PDF pages 1, 2, 3.

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Use digit clues, but do not mistake a clue for proof

Some last digits are impossible for a perfect square. A possible last digit alone does not make a number a square.

The ten possible last digits

Last digit of n² (digit)

Last digit of n (digit)

Last-digit map · Each point maps a last digit to the last digit of its square.

Only the y-values 0, 1, 4, 5, 6 and 9 appear. Points represent digits, not every number on a continuous curve.

Read the plotted values

Last-digit map

Last digit of n	Last digit of n²
0	0
1	1
2	4
3	9
4	6
5	5
6	6
7	9
8	4
9	1

Necessary condition: Something that must be true, although it may not be enough to prove a claim.
Parity: Whether a whole number is even or odd.

The last digit of an integer square is 0, 1, 4, 5, 6 or 9. It cannot be 2, 3, 7 or 8.

Squaring keeps parity: an even number gives an even square; an odd number gives an odd square. A nonzero integer square has an even number of trailing zeroes.

WORKED IDEA

Why zeroes come in pairs

(30)² = 900 = 9 × 10²

One trailing zero in 30 becomes two in its square. More generally, (m × 10ᵏ)² = m² × 10²ᵏ when m has no trailing zero.

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Question

Is 56 a perfect square just because it ends in 6? Explain.

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NCERT reference: chapter PDF pages 4, 5.

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Grow a square with an odd L-shaped border

To turn a 4 by 4 square into a 5 by 5 square, add a row and a column. Count their shared corner only once.

The border explains the difference

y (units)

x (units)

Highlight a line or construction

Old square · The blue 4 × 4 square has area 16.
Top row · Five new unit squares along the top.
Right column · Four further unit squares, excluding the top corner.

116 old tiles · (2, 2)
25 new tiles · (2.5, 4.5)
34 new tiles · (4.5, 2)

The orange top row and green column together contain 9 unit squares. Boundaries are drawn to scale. Equal units on both axes.

Read the plotted values

Old square

x	y
0	0
4	0
4	4
0	4
0	0

Top row

x	y
0	4
5	4
5	5
0	5
0	4

Right column

x	y
4	0
5	0
5	4
4	4
4	0

Consecutive squares: Squares of consecutive whole numbers, such as 16 and 25.
Odd number: An integer of the form 2k + 1.

The added border contains 4 + 4 + 1 = 9 tiles. This explains 5² − 4² = 9.

Starting with one tile and adding odd borders gives 1 + 3 + 5 + … + (2n − 1) = n². There are 2n integers strictly between n² and (n + 1)².

1
Count the original square: 4 × 4 = 16.
2
Add 5 tiles across the top and 4 more at the right.
3
The new total is 16 + 9 = 25 = 5².

(n + 1)² − n² = 2n + 1.

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Question

How many tiles change a 6 by 6 square into a 7 by 7 square?

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NCERT reference: chapter PDF pages 5, 6, 7.

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Two triangular dot groups make one square

Split a square of dots along its diagonal. One group has one more row than the other.

Count the blue and orange triangles

Every dot is counted exactly once. The diagonal belongs to blue, avoiding an extra count.

Triangular number: A total of the form 1 + 2 + … + n = n(n + 1)/2.

A 5 by 5 array has 25 dots. The blue triangle has 1 + 2 + 3 + 4 + 5 = 15 dots. The orange triangle has 1 + 2 + 3 + 4 = 10.

So neighbouring triangular numbers add to a square: Tₙ + Tₙ₋₁ = n². The diagonal dots belong to only one group.

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Question

Which two neighbouring triangular numbers add to 36?

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NCERT reference: chapter PDF page 7.

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A square root gives the side; prime factors show pairs

Ask which nonnegative number, multiplied by itself, gives the number under the root sign.

Pair the factors of 784

Each outlined group contains two equal prime factors. Keep one from each pair: 2 × 2 × 7 = 28.

Square root: √a is the nonnegative number whose square is a, for a ≥ 0.
Prime factorisation: Writing a positive integer as a product of prime numbers.

The square root sign names one value: √64 = 8. The equation x² = 64 has two solutions, x = 8 and x = −8.

In a perfect square, each prime factor can be put into pairs. Take one factor from each pair to find its positive square root.

1
Factor 784: 784 = 2⁴ × 7².
2
Group equal factors into pairs: (2 × 2)(2 × 2)(7 × 7).
3
Take one factor from each pair: √784 = 2 × 2 × 7 = 28.
4
Check by squaring: 28 × 28 = 784.

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Question

Why is √((−9)²) equal to 9 rather than −9?

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NCERT reference: chapter PDF pages 8, 9.

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Locate a root between squares before guessing

Start with nearby perfect squares. They give reliable bounds for the side length.

Bracket √130

Value (number)

Number-line marks · Only horizontal position represents the number.

111 · (11, 0)
211.5 · (11.5, 0)
312 · (12, 0)

√130 lies to the right of 11 and to the left of 11.5. The marked midpoint is a trial value, not the root itself.

Read the plotted values

Number-line marks

Value	Height
11	0
11.5	0
12	0

Bound: A value that the answer cannot go below or above.

Since 11² = 121 and 12² = 144, √130 lies between 11 and 12.

For a tighter bound, test 11.5² = 132.25. So √130 is between 11 and 11.5. Always square a trial value to check it.

WORKED IDEA

Test the midpoint

11² < 130 < 11.5²

Taking nonnegative square roots keeps this order, giving 11 < √130 < 11.5.

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Question

A square board must have a whole-number side length and area at most 185 cm². What is the largest allowed side?

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NCERT reference: chapter PDF pages 9, 10.

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A cube is equal layers in three directions

A cube with side 3 unit lengths has 3 layers. Each layer contains 3 × 3 small cubes.

Three layers, nine unit cubes per layer

The visible grid suggests unit cubes throughout the solid, including hidden cubes. The perspective picture is a model; depth is not a flat area.

Cube of a number: n³ = n × n × n.
Volume: How much three-dimensional space a solid occupies, measured in cubic units.

A 3 by 3 by 3 cube contains 27 unit cubes. The superscript 3 means three equal factors.

A negative number cubed stays negative: (−5)³ = −125. Squaring a negative number instead gives a positive result.

WORKED IDEA

Do not confuse multiplying by 3 with cubing

6³ = 216; 3 × 6 = 18

Six cubed means 6 × 6 × 6. Three times six is a different operation.

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Question

Why does a cube of side 4 cm have volume 64 cm³, not 12 cm?

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NCERT reference: chapter PDF pages 11, 12, 13.

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A cube root collects prime factors in threes

Three identical prime factors make one factor of a cube root.

Complete the missing triple

The orange dot is the extra factor 2. Each group of three equal factors contributes one factor to the cube root.

Cube root: ∛a is the real number whose cube is a.
Perfect cube: An integer that is the cube of an integer.

A positive integer is a perfect cube when every exponent in its prime factorisation is a multiple of 3.

For 108 = 2² × 3³, the two factors of 2 need one more 2 to complete a triple. Multiplying by 2 gives 216 = 6³.

WORKED IDEA

A negative cube root

∛(−343) = −7

Because (−7) × (−7) × (−7) = −343. A real cube root has one real value, not a ± pair.

Go deeper: Last digits and zeroes

A cube may end in any digit from 0 to 9. For a nonzero integer cube, the number of trailing zeroes is a multiple of 3. Its parity is the same as the parity of the original integer.

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Question

What is the smallest positive integer that makes 250 a cube when multiplied by it?

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NCERT reference: chapter PDF pages 14, 15.

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A cube can be a group of consecutive odd numbers

Choose n odd numbers centred around n². Their total is n × n² = n³.

Five odd numbers balance around 25

Value (number)

Number-line marks · Only horizontal position represents the number.

121 · (21, 0)
223 · (23, 0)
325 · (25, 0)
427 · (27, 0)
529 · (29, 0)

Numbers equally far from 25 pair to give 50: 21 + 29 and 23 + 27. The central 25 remains.

Read the plotted values

Number-line marks

Value	Height
21	0
23	0
25	0
27	0
29	0

Average: The sum of values divided by how many values there are.

For 5³, take five consecutive odd numbers: 21, 23, 25, 27 and 29. Their average is 25, so their sum is 5 × 25 = 125.

The first odd number in the nth group is n² − n + 1, and the last is n² + n − 1. These groups continue without gaps.

1
Pair the outside values: 21 + 29 = 50.
2
Pair the next values: 23 + 27 = 50.
3
Add the middle: 50 + 50 + 25 = 125.

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Question

Which three consecutive odd numbers make 3³?

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NCERT reference: chapter PDF pages 14, 15.

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Difference tables reveal different growth patterns

Equal steps in n do not give equal steps in n² or n³. Repeated differences reveal the pattern.

Squares and cubes grow at different rates

Value (number)

n (integer)

Highlight a line or construction

n² · Values at the marked integers; joining lines help compare growth.
n³ · Cubing gives much larger values once n exceeds 1.

This plot shows integer samples. It is not a claim that the straight segments are the complete curves.

Read the plotted values

n²

n	Value
0	0
1	1
2	4
3	9
4	16

n³

n	Value
0	0
1	1
2	8
3	27
4	64

First difference: The change from one value to the next in a sequence.
Second difference: The change between consecutive first differences.

n	n³	First difference	Second difference	Third difference
0	0	—	—	—
1	1	1	—	—
2	8	7	6	—
3	27	19	12	6
4	64	37	18	6

For squares of consecutive integers, first differences are odd numbers and second differences are 2.

For cubes of consecutive integers, third differences are 6. This rule assumes a step of 1 in the input.

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Question

The squares are 1, 4, 9, 16, 25. Find the first and second differences.

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NCERT reference: chapter PDF pages 15, 16.

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One number can have two different cube decompositions

Ramanujan noticed a special property of 1729: it is the smallest positive integer expressible as a sum of two positive integer cubes in two different ways.

Two routes to the same total

The arrows connect two calculated sums to the same total. They do not show cube size or distance.

Taxicab number: In this famous example, the smallest number with two different decompositions as a sum of two positive integer cubes.

1729 = 1³ + 12³ = 9³ + 10³. Check each cube before adding. Changing the requirement from positive integers changes the problem.

Words such as varga (square), ghana (cube) and mūla (root) connect these ideas with the history of mathematics. Learn the meaning, rather than memorising a story alone.

WORKED IDEA

Verify both decompositions

1 + 1728 = 1729; 729 + 1000 = 1729

The decompositions use different unordered pairs, {1,12} and {9,10}.

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Question

Check that 9³ + 10³ really equals 1729.

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NCERT reference: chapter PDF pages 13, 15, 16.

A Square and a Cube Class 8 Notes

Chapter notes

A Square and a Cube

A square number is an area you can count

Count rows, then multiply

A decimal and a fraction

Use digit clues, but do not mistake a clue for proof

The ten possible last digits

Why zeroes come in pairs

Grow a square with an odd L-shaped border

The border explains the difference

Two triangular dot groups make one square

Count the blue and orange triangles

A square root gives the side; prime factors show pairs

Pair the factors of 784

Locate a root between squares before guessing

Bracket √130

Test the midpoint

A cube is equal layers in three directions

Three layers, nine unit cubes per layer

Do not confuse multiplying by 3 with cubing

A cube root collects prime factors in threes

Complete the missing triple

A negative cube root

A cube can be a group of consecutive odd numbers

Five odd numbers balance around 25

Difference tables reveal different growth patterns

Squares and cubes grow at different rates

One number can have two different cube decompositions

Two routes to the same total

Verify both decompositions

Now practise A Square and a Cube, free