A Square and A Cube Class 8 Notes

A Square and A Cube Class 8 Notes

A Square and A Cube Class 8 Notes

Square Numbers

A square number is obtained by multiplying an integer by itself.

Example : (3 × 3 = 9), so 9 is a square number.

Perfect Squares : These are squares of natural numbers and have unique properties.

Properties of Perfect Squares

Units Digits :

Perfect squares can end in : 0, 1,4, 5, 6, or 9.

If a number ends in : 2, 3, 7, or 8, it is not a perfect square.

Specific Cases :

1. If a number ends in 1 or 9, its square ends in 1.
2. If a number ends in 2 or 8, its square ends in 4.
3. If a number ends in 3 or 7, its square ends in 9.
4. If a number ends in 4 or 6, its square ends in 6.
5. If a number ends in 5, its square ends in 5.
6. If a number ends in 0, its square ends in 0.

Perfect Squares and Odd Numbers

Relationship : The sum of the first n odd numbers equals (n²).

Example:

(1 = 1²)

(1 + 3 = 4 = 2²)

(1 + 3 + 5 = 9 = 3²)

Square Roots

Definition : The square root is the inverse operation of squaring.

Denoted as ([latex] \sqrt{{y}} [/latex]) where (y = x²).

Example : ([latex] \sqrt{{9}} [/latex] = 3) because (3 × 3 = 9).

Prime Factorisation and Perfect Squares

Method : To check if a number is a perfect square, use prime factorisation.

Example :

For (324): (324 = 2 × 2 × 3 × 3 × 3 × 3)

Grouping : (324 = (2 × 3 × 3) × (2 × 3 × 3)

= (2 × 3 × 3)²

Therefore, (324 = 18²)

Cubic Numbers

Definition : A number obtained by multiplying a number by itself three times.

Example: (1³ = 1), (2³ = 8), (3³ = 27), etc.

Perfect Cubes : Numbers that can be expressed as the cube of an integer.

Examples :

(1³ = 1) (2³ = 8)

(3³ = 27) (4³ = 64)

(5³ = 125)

Taxicab Numbers

Numbers that can be expressed as the sum of two cubes in more than one way.

Example : (1729 = 1³ + 12³ = 9³ + 10³).

Perfect Cubes and Consecutive Odd Numbers

Relationship : Each perfect cube can be expressed as the sum of consecutive odd numbers.

Example:

(1³ = 1) (sum of 1 odd number)

(2³ = 8) (sum of 2 odd numbers : (3 + 5))

(3³ = 27) (sum of 3 odd numbers : [7 + 9 + 11])

Cube Roots

Definition : The cube root of a number (y) is the value that; when multiplied by itself three times, gives (y).

Denoted as [latex]\sqrt[3]{y}[/latex].

Example: [latex]\sqrt[3]{8}[/latex] = 2 because(2³ = 8), [latex]\sqrt[3]{27}[/latex] = 3 because

3³ = 27, [latex]\sqrt[3]{64}[/latex] = 4, because 4³ = 64, etc.

Successive Differences

Concept: Finding the differences between consecutive perfect cubes reveals patterns.

Example: 1³ = 1, 2³ = 8

Differences: (8 - 1 = 7), (27 - 8 = 19), etc.

Further differences can show a constant pattern.

A Pinch of History

Historical Context : The study of perfect squares and cubes dates back to ancient civilizations.

• The Babylonians compiled lists of perfect squares and cubes as early as 1700 BCE.

• In ancient India, terms like ‘ varga’ (square) and ‘ghana’ (cube) were used in mathematical texts.

Square Numbers

A square number is a number that is obtained by multiplying an integer by itself. For example, if we take the integer 3 and multiply it by itself (3 x 3), we get 9. Therefore, 9 is a square number. The sequence of square numbers starts from 1 and continues as follows :

(1² = 1) (2² = 4) (3² = 9)

(4² = 16) (5² = 25) (6² = 36)

(7² = 49) (8² = 64) (9² = 81)

(10² = 100)

These numbers are also referred to as perfect squares.

Perfect Squares

Perfect squares are the squares of natural numbers. They have unique properties that can help us identify them. For instance, perfect squares end with specific digits.

Patterns and Properties of Perfect Squares

1. Units Digits:

If a number ends in 0, 1, 4, 5, 6, or 9, it could potentially be a perfect square.

However, if a number ends in 2, 3, 7, or 8, we can definitively say that it is not a perfect square. For example :

(16) (which is (42)) ends in (6), and so does (36) (which is (62)).

But (26) does not have a perfect square root, even though it ends in (6).

2. Specific Cases:

1. If a number has (1) or (9) in the units place, then its square will end in (1).
2. If a number has (2) or (8) in the units place, then its square will end in (4).
3. If a number has (3) or (7) in the units place, then its square will end in (9).
4. If a number has (4) or (6) in the units place, then its square will end in (6).
5. If a number has (5) in the units place, then its square will end in (5).
6. If a number has 0 in the units place, then its square will end in 0.

Perfect Squares and Odd Numbers

There is a fascinating relationship between perfect squares and odd numbers. The nth odd number can be expressed as :

Odd number = 2n - 1.

For example :

The 1st odd number is (2(1) -1 = 1)

The 2nd odd number is (2(2) -1 = 3)

The 3rd odd number is (2(3) -1 = 5)

The sum of the first n odd numbers is equal to (n²2). For example :

(1 = 1²)

(1 + 3 = 4 = 2²)

(1 + 3 + 5 = 9 = 3²)

Perfect Squares and Triangular Numbers

Perfect squares are also related to triangular numbers, which are formed by the sum of the first n natural numbers. The nth triangular number can be expressed as :

T_n = [latex]\frac{n(n+1)}{2}[/latex]

Square Roots

Square root is the inverse operation of square. Every perfect square has two integral square roots. The positive square root of a number is. denoted by the symbol [latex]\sqrt{ } .[/latex] For example, [latex]\sqrt{9}[/latex] = 3.

The square root of a number (y) is a value (x) such that: y = x²

This means that if you multiply (x) by itself, you will get (y). For example, the square root of [latex]\sqrt{64}[/latex] is 8 because (8 × 8 = 64).

Prime Factorisation and Perfect Squares

To determine if a number is a perfect square, we can use prime factorisation. If we can divide the prime factors of a number into two equal groups, then the product of the prime factors in either group will give us the square root.

For example :

For (324):

Prime factorisation : (324 = 2 × 2 × 3 × 3 × 3 × 3)

Grouping : (324 = (2 × 3 × 3) × (2 × 3 × 3)

= (2 × 3 × 3)²)

Therefore, (324 = 18²), and the square root is (18).

Conversely, if we take (156):

1. Prime factorisation : (156 = 2 × 2 × 3 × 13)
2. We cannot pair the factors completely, which indicates that (156) is not a perfect square.

Cubic Numbers

A number obtained by multiplying a number by itself three times is called a cube. For example 1, 8, 27, ... , etc., are cubes.

Perfect Cubes : A perfect cube is a number that can be expressed as the cube of an integer. In simpler terms, if you multiply a whole number by itself three times, you get a perfect cube. We denote the cube of a number (n) as (n³).

For example :

(1³ = 1 × 1 × 1 = 1)

(2³ = 2 × 2 × 2 = 8)

(3³ = 3 × 3 × 3 = 27)

(4³ = 4 × 4 × 4 = 64)

(5³ = 5 × 5 × 5 = 125)

So, the perfect cubes we see are :

(1, 8, 27, 64, 125, ...)

Taxicab Numbers : A taxicab number is a special number that can be expressed as the sum of two cubes in more than one way. The most famous taxicab number is 1729, which can be expressed as :

(1³ + 12³ = 1 + 1728 = 1729 )

(9³ + 10³ = 729 + 1000 = 1729 )

This number was famously discussed by mathematicians G.H. Hardy and Srinivasa Ramanujan.

Perfect Cubes and Consecutive Odd Numbers

There is an interesting relationship between perfect cubes and consecutive odd numbers. Each perfect cube can be expressed as the sum of a certain number of consecutive odd numbers. Here’s how it works :

(1³ = 1) (sum of 1 odd number : (1))

(2³ = 8) (sum of 2 odd numbers : (3 + 5 = 8))

(3³ = 27) (sum of 3 odd numbers :

(7 + 9 + 11 = 27))

(4³ = 64) (sum of 4 odd numbers :

(13 + 15 + 17 + 19 = 64))

(5³ = 125) (sum of 5 odd numbers :

(21 + 23 + 25 + 27 + 29 = 125))

This pattern continues, showing that each perfect cube can be formed by adding consecutive odd numbers.

Cube Roots : The cube root of a number is the value that, when multiplied by itself three times, gives the original number. We denote the cube root of a number y as [latex]\sqrt[3]{y}[/latex].

For example :

([latex]\sqrt[3]{8}[/latex] = 2) because (2³ = 8)

([latex]\sqrt[3]{27}[/latex] = 3) because (3³ = 27)

([latex]\sqrt[3]{64}[/latex] = 4) because (4³ = 64)

In general, if (y = n³), then (n = [latex]\sqrt[3]{y}[/latex]).

Successive differences: Successive differences means we can find the differences between conse¬cutive perfect cubes and see what patterns emerge. For example, the first few perfect cubes are :

(1³ = 1) (2³ = 8) (3³ = 27)

(4³ = 64) (5³ = 125)

Now, let’s find the differences :

(8 - 1 = 7) (27 - 8 = 19)

(64 - 27 = 37) (125 - 64 = 61)

Now, let’s find the differences of these differences:

(19 - 7 = 12) (37 - 19 = 18)

(61 - 37 = 24)

And again :

(18 - 12 = 6) (24 - 18 = 6)

Notice that after a few levels, the differences become constant (in this case, 6). This shows a pattern similar to that of perfect squares but with a different constant value.

A Pinch of History

The study of perfect squares and cubes dates back to ancient civilizations. The Babylonians were among the first to compile lists of perfect squares and cubes as early as 1700 BCE. They used these lists for practical applications such as land measurement and architectural design.

In ancient India, the terms varga (for square) and ghana (for cube) were used in mathematical texts. The mathematician Aryabhata, who lived around 499 CE, discussed these concepts and their applications in geometry.