A Square and A Cube Class 8 Notes Maths Chapter 1

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Class 8 Maths Chapter 1 Notes A Square and A Cube

Class 8 Maths Notes Chapter 1 – Class 8 A Square and A Cube Notes

We have been learning about numbers, their patterns and operations since long. Like numbers, number patterns too have a great influence on the way mathematics has evolved and is used to make sense of almost everything-from a tiny particle to the universe. It is exciting to know how number patterns form the foundation for deeper problem-solving.

Unlike arbitrary calculations, odd-even numbers, prime numbers, triangular numbers, factors, multiples, etc, reveal hidden patterns, make strong linkages to geometry-shapes, their areas and volumes, and even appear in real-world puzzles (like the famous locker problem).

In the journey ahead, this chapter facilitates the entry into the fascinating world of square and cube numbers – where mathematics meets mystery, patterns, and even a royal inheritance puzzle. Math isn’t just about calculations; it’s rather a path to logical thinking and discovering useful patterns.

→ A number multiplied by itself, e.g., a × a = a² is read as ‘ a squared’ and is always positive.

→ A natural number n is said to be a perfect square if there exists some natural number m such that n = m × m = m².

→ Properties of Perfect Squares

• A number having 2,3,7 or 8 at the unit place is never a perfect square number.
• The number of zeros at the end of a perfect square number is always even.
• Squares of even numbers are even and squares of odd numbers are odd.
• The number of non-perfect square numbers between the squares of two consecutive numbers n and (n+1) is 2 n, where n is a natural number.
• For any natural number n, the square of n, i.e., n² is equal to the sum of the first n odd natural numbers.
• The square of a natural number (other than 1) is either a multiple of 3 or exceeds a multiple of 3 by 1.
• The difference between the squares of two consecutive natural numbers is equal to their sum, i.e., for every natural number n, (n+1)² – n² = (n+1) + n.
• By adding two triangular numbers we get a square number.
• The square of any odd number, other than 1 can be expressed as the sum of two consecutive natural numbers.

→ The square root of a number n is that number which when multiplied by itself, gives the number n as the product. √ is the symbol for square root. The square root of n is denoted by √n.

→ A number when multiplied by itself thrice called its cube. The cube of x = x × x × x = x³.

→ A natural number ‘n’ is called a perfect cube if there exists a natural number ‘m’ such that n = m × m × m = m³.

→ Properties of Perfect Cubes

• The cubes of all odd numbers are odd, and the cubes of all even numbers are even.
• The number of zeros at the end of a perfect cube is always a multiple of 3.
• Cubes of positive integers are always positive, and cubes of negative integers are always negative.
• The cube of the numbers having digits 1,4,5,6 and 9 at their ones places ends with in the same digits respectively.
• The cube of a number ending with 2 ends with digit 8 at its ones place and vice versa.
• The cube of a number ending with 3 ends with digit 7 at its ones place and vice versa.