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 is a square number and how is it defined? |  |
Ans.A square number is a number that can be expressed as the product of an integer multiplied by itself. For example, \(4\) is a square number because it can be written as \(2 \times 2\). The general form for a square number is \(n^2\) where \(n\) is any whole number.
| 2. How can I find the square root of a number? | |
Ans.To find the square root of a number, you need to determine which number multiplied by itself gives you the original number. For instance, the square root of \(25\) is \(5\) because \(5 \times 5 = 25\). You can also use a calculator or look up a square root table for quick reference.
| 3. What are some properties of square numbers? | |
Ans.Square numbers have several interesting properties:
1. The square of any integer is non-negative.
2. The difference between consecutive square numbers is an odd number (e.g., \(1^2 - 0^2 = 1\), \(2^2 - 1^2 = 3\), \(3^2 - 2^2 = 5\)).
3. The square of an even number is even; the square of an odd number is odd.
| 4. Are there any interesting patterns related to square numbers? | |
Ans.Yes, square numbers display various patterns. For example, the sum of the first \(n\) odd numbers is always a perfect square (\(1 + 3 + 5 + ... + (2n-1) = n^2\)). Additionally, if you list square numbers, you'll notice that they are spaced further apart as numbers increase.
| 5. Can you provide an example of solving a problem involving square roots? | |
Ans.Certainly! If you need to find the square root of \(144\), you can determine that \(12 \times 12 = 144\), so the square root of \(144\) is \(12\). Alternatively, you can check values systematically or use a calculator for larger numbers.
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