Class 8 Exam > Class 8 Notes > Mathematics (Maths) Class 8 > Points to Remember: Squares & Square Roots
Points to Remember: Squares & Square Roots | Mathematics (Maths) Class 8 PDF Download
| Table of contents |
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| Facts that Matter |
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| We know that |
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| Remember |
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| Properties of Square Number |
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| Some Interesting Patterns |
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| Solved Examples |
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Facts that Matter
- A natural number ‘n’ is a perfect square, if m2 = n for a natural number m.
- A number ending in 2, 3, 7, or 8 is never a perfect square.
- The squares of even numbers are even.
- The squares of odd numbers are odd.
- A number ending in an odd number of zeros is never a perfect square.
- There are 2n non-perfect square numbers between the squares of the numbers n and n + 1.
- For any natural number ‘n’ greater than 1, 2n, (n2 – 1), and (n2 + 1) form a Pythagorean triplet.
- Finding a square root is the inverse operation of squaring a number.
- If ‘n’ be the number of digits of a square number then the number of digits in its square root is given by n/2 (for ‘n’ is even) and (n+1) /2 (for ‘n’ is odd).
Question for Points to Remember: Squares & Square RootsTry yourself:Which of the following is a perfect square?
View Solution
We know that
- If a whole number is multiplied by itself, the product is called the square of that number.
Example: 3 * 3 = 9 = 32
i.e. the square of 3 is 9.
Example: 5 * 5 = 25 = 52
i.e. the square of 5 is 25.
- A natural number is called a perfect square or a square number if it is the square of some natural number.
Example: 16 is the square of 4, therefore, 16 is a perfect square.
Remember
- All-natural numbers are not perfect squares or square numbers, 32 is not a square number. In general, if a natural number ‘m’ can be expressed as n2, where n is also a natural number, then ‘m’ is the perfect square. The numbers like 1, 4, 9, 16, 25, and 36 are called square numbers.
Table: Square of numbers from 1 and 10.
Properties of Square Number
Table: Let us consider the square of all natural numbers from 1 to 20.
From the table, we conclude that:
Property 1: “The ending digits (the digits in the one’s place) of a square number is 0, 1, 4, 5, 6 or 9 only.”
Question for Points to Remember: Squares & Square RootsTry yourself: A perfect square that lies between 40 and 50 is:
View Solution
Some Interesting Patterns
- Triangular numbers are: 1, 3, 6, 10, 15, 21, etc. If we combine two consecutive triangular numbers, we get a square number.
1 + 3 = 4, ‘4’ is a square number
3 + 6 = 9, ‘9’ is a square number
6 + 10 = 16, ‘16’ is a square number
and so on.
- 12 =1
112 = 121
1112 = 12321
11112 = 1234321 - 72 = 49
672 = 4489
6672 = 444889
66672 = 44448889 and so on.
Solved Examples
Problem: What will be the unit’s digit in the square of the following numbers?1. 12487
2. 1324
3. 91478
4. 1251
Solution: The unit’s digit in the square of the following is:
1. 12487 is 9 (as 72 = 49, 9 in the unit’s place).
2. 1324 is 6 (as 42 = 16, 6 in the unit’s place).
3. 91478 is 4 (as 82 = 64, 4 in the unit’s place).
4. 1251 is 1 (as 12 = 1, 1 in the unit’s place).
Question for Points to Remember: Squares & Square RootsTry yourself:If 5278 is squared, then what will be at the unit place?
View Solution
Problem: Comment on the square of an even number and of an odd number.
Solution: The square of an even number is always an even number and the square of an odd number is always an odd number. The square of an even number will always have 4, 6, or even zeros in its unit’s place. And the square of an odd number will always have 1, 5, or 9 in its unit’s place.
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FAQs on Points to Remember: Squares & Square Roots - Mathematics (Maths) Class 8
| 1. What are some properties of square numbers? |  |
Ans. Square numbers are numbers that are obtained by multiplying a number by itself. Some properties of square numbers include:
- All square numbers are non-negative, meaning they are greater than or equal to zero.
- The square of any positive integer is always positive.
- The square of any negative integer is always positive.
- The square of an even number is always an even number.
- The square of an odd number is always an odd number.
| 2. What are some interesting patterns related to square numbers? | |
Ans. There are several interesting patterns related to square numbers, such as:
- The sum of the first n odd numbers is always equal to the square of n.
- The difference between two consecutive square numbers is always equal to the sum of their square roots.
- The sum of the first n square numbers is equal to n multiplied by (n + 1) multiplied by (2n + 1), divided by 6.
- The units digit of a square number can only be 0, 1, 4, 5, 6, or 9.
| 3. How can square numbers be used to solve problems? | |
Ans. Square numbers can be used to solve a variety of problems, including:
- Calculating areas of squares and rectangles: The area of a square or rectangle can be found by squaring the length of one side.
- Determining if a number is a perfect square: If a number is a perfect square, it can be expressed as the square of an integer.
- Finding the square root of a number: The square root of a number is the value that, when multiplied by itself, equals the original number.
- Analyzing patterns and relationships: Square numbers often exhibit interesting patterns and relationships that can be explored and analyzed in problem-solving scenarios.
| 4. How do square numbers relate to square roots? | |
Ans. Square numbers and square roots are closely related. A square root of a number is a value that, when multiplied by itself, equals the original number. The square root of a square number is always an integer. For example, the square root of 16 is 4, because 4 multiplied by itself equals 16. Similarly, the square root of 25 is 5, because 5 multiplied by itself equals 25. In mathematical notation, the square root of a number x is denoted as √x.
| 5. Can you provide some examples of problems involving square numbers? | |
Ans. Certainly! Here are a few examples of problems that involve square numbers:
1. Find the area of a square with a side length of 5 units.
- Solution: The area of a square is given by the formula A = s^2, where s is the length of a side. In this case, the side length is 5 units, so the area is 5^2 = 25 square units.
2. Determine if 64 is a perfect square.
- Solution: To determine if a number is a perfect square, we can take its square root. The square root of 64 is 8, and 8 multiplied by itself equals 64. Therefore, 64 is a perfect square.
3. Solve the equation x^2 = 81.
- Solution: To solve this equation, we need to find the value of x that, when squared, equals 81. Taking the square root of both sides, we have x = ±√81. The square root of 81 is 9, so the solutions are x = ±9.
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