Square and Square root
Square: - The product of a number by itself is known as the square of the number or complete square or square.
Number × Number = (Number)2
Learn the square of one to thirty.
वर्ग:- किसी संख्या की उसी संख्या में गुणा करने पर प्राप्त संख्या उस संख्या की वर्ग कहलाती है।संख्या × संख्या = (संख्या)²
एक से तीस तक के वर्ग याद करो।
Perfect Square Numbers: Numbers whose square roots can be determined are called perfect square numbers. For example, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, etc., are perfect square numbers because their square roots can be calculated, which are 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10, respectively.
Learn the square of these numbers:
निम्न संख्याऐं एवं वर्ग याद करो
संख्याऐं | वर्ग | संख्याऐं | वर्ग |
Number | Square | Number | Square |
1² | 1 | 16² | 256 |
2² | 4 | 17² | 289 |
3² | 9 | 18² | 324 |
42 | 16 | 19² | 361 |
5² | 25 | 20² | 400 |
6² | 36 | 21² | 441 |
7² | 49 | 22² | 484 |
8² | 64 | 23² | 529 |
9² | 81 | 24² | 576 |
10² | 100 | 25² | 625 |
11² | 121 | 26² | 676 |
12² | 144 | 27² | 729 |
13² | 169 | 28² | 784 |
14² | 196 | 29² | 841 |
15² | 225 | 30² | 900 |
Perfect Square –> Natural number n is called a perfect square or square number. if it is the square of some natural number such that n = m²
n = m × m = m²
If any number is written in the form of a square (m²) or a pair (m × m) is known as square number or perfect square.
25 = 5 × 5 = 5²
36 = (2 × 3) × (2 × 3) = 6 × 6 = 6²
180 = (2 × 2) × (3 × 3) × 5
Here 5 is not in the form of a square (5²) or a pair (5 × 5) so that 180 is not a perfect square.
1. Show that the following numbers are perfect square or not.
A.
(i) 1 (ii) 9 (ii) 8 (iv) 6 (v) 4
(vi) 7 (vii) 5 (viii) 2 (ix) 3
B.
(i) 16 (ii) 39 (ii) 40 (iv) 25 (v) 54
(vi) 36 (vii) 52 (viii) 49 (ix) 64 (x) 81
C.
(i) 169 (ii) 393 (ii) 400 (iv) 225 (v) 545
(vi) 316 (vii) 576 (viii) 484 (ix) 676 (x) 841
D.
(i) 1169 (ii) 1800 (ii) 1600 (iv) 4761
(v) 2025 (vi) 9408 (vii) 5762 (viii) 9484
(ix) 6760 (x) 8415
2. Show that each of the following numbers is a perfect square or not. Also, find the number whose square is the given number in each case:
(i) 1169 (ii) 1800 (ii) 1600 (iv) 4761
(v) 2025 (vi) 9408 (vii) 5762 (viii) 9484
(ix) 6760 (x) 8415
(1) 23805
(ii) 12150
(1) 14283
(1) 11025
(1) 25453
3. Find the smallest number by which the given number must be multiplied so that the product is a perfect square: Find the square root of the product.
(i) 1169 (ii) 1800 (ii) 1600 (iv) 4761
(v) 2025 (vi) 9408 (vii) 5762 (viii) 9484
(ix) 6760 (x) 8415
(1) 23805
(ii) 12150
(1) 14283
(1) 11025
(1) 25453
4. Find the smallest number by which the given number must be divided so that the resulting num-ber is a perfect square:
(i) 1169 (ii) 1800 (ii) 1600 (iv) 4761
(v) 2025 (vi) 9408 (vii) 5762 (viii) 9484
(ix) 6760 (x) 8415
(1) 23805
(ii) 12150
(1) 14283
(1) 11025
(1) 25453
5. Which of the following numbers are perfect squares?
8, 9, 11, 21, 16, 32, 25, 49, 36, 50, 64, 79, 81, 111, 121, 150, 169, 225
6. Using prime factorization method, find which of the following numbers are perfect squares? 189, 225, 2048, 343, 441, 2916, 11025, 3549
7. By what number should each of the following numbers be multiplied to get a perfect square in each case? Also, find the number whose square is the new number.
(6) 8820
(i) 3675
(iii) 605
(iv) 2880
(v) 4056
(vi) 3468
(vii) 7776
8. By what numbers should each of the following be divided to get a perfect square in each case? Also, find the number whose square is the new number.
(1) 16562
(ii) 3698
(iii) 5103
(iv) 3174
(v) 1575
MATHEMATICS VIII
9. Find the greatest number of two digits which is a perfect square.
10. Find the least number of three digits which is perfect square.
A SQUARE
1.5
11. Find the smallest number by which 4851 must be multiplied so that the product becomes a perfect square.
12. Find the smallest number by which 28812 must be divided so that the quotient becomes a perfect square.
13. Find the smallest number by which 1152 must be divided so that it becomes a perfect square. Also, find the number whose square is the resulting number.
14. Find the smallest square number that is divisible by 144, 180 and 192
पूर्ण वर्ग संख्याएँ:- वे संख्याऐं जिनके वर्गमूल निकाले जा सकते हैं पूर्ण वर्ग संख्याऐं कहलाती हैं। जैसे 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 आदि पूर्ण वर्ग संख्याऐं है क्योंकि इनके वर्गमूल निकाले जा सकते हैं जो क्रमशः 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 हैं।
Even Numbers: - The number that is divided by two is known as even number. It unit are 0, 2, 4, 6, 8.
Mind it that the square of even number will be even. These are divided by four.
सम संख्या:- वह संख्या जो 2 से विभाजित होता है। सम संख्या कहलाती है। इसके इकाई के अंक 0, 2, 4, 6, 8 आदि होते हैं।
याद रखें कि सम संख्या का वर्ग सम संख्या ही होता है। ये चार से विभाजित हो जाते हैं।
Odd Numbers: - The number not divided by two is known as odd number. It unit are 1, 3, 5, 7, 9.
Mind it that the square of odd number will be odd.
विषम संख्या:- वह संख्या जो 2 से विभाजित नहीं होती है। विशम संख्या कहलाती है। इसके इकाई के अंक 1, 3, 5, 7, 9 आदि होते हैं।
याद रखें कि सम संख्या का वर्ग सम संख्या ही होता है।
1. Find the squares of the followings: निम्न संख्याओं का वर्ग ज्ञात कीजिए। | ||
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What could be the possible 'one's digits of the square root of each of the following
numbers?
(1) 9801
(ii) 99856
(iv) 657666025
2. Without doing any calculation, find the numbers which are surely not perfect squares.
(1) 153
(iii) 998001
(ii) 257
(iii) 408
(iv) 441
3. Find the square roots of 100 and 169 by the method of repeated subtraction.
4. Find the square roots of the following numbers by the Prime Factorisation Method.
(1) 729
(ii) 400
(iii) 1764
(iv) 4096
(v) 7744
(ix) 529
(vi) 9604
(x) 8100
(vii) 5929
(viii) 9216
5. For each of the following numbers, find the smallest whole number by which it should be multiplied so as to get a perfect square number. Also find the square root of the square number so obtained.
(1) 252
(v) 1458
(ii) 180
(vi) 768
(iii) 1008
(iv) 2028
6. For each of the following numbers, find the smallest whole number by which it should be divided so as to get a perfect square. Also find the square root of the square number so obtained.
(1) 252
(ii) 2925
(iii) 396
(iv) 2645
(v) 2800
(vi) 1620
7. The students of Class VIII of a school donated Rs 2401 in all, for Prime Minister's National Relief Fund. Each student donated as many rupees as the number of students in the class. Find the number of students in the class.
8. 2025 plants are to be planted in a garden in such a way that each row contains as many plants as the number of rows. Find the number of rows and the number of plants in each row.
9. Find the smallest square number that is divisible by each of the numbers 4, 9 and 10.
10. Find the smallest square number that is divisible by each of the numbers 8, 15 and 20.
Square Root :–> Square root is the inverse operation of square. There are two integral square roots of a perfect square number. Positive square root of a number is denoted by the symbol √
For example, √9 = gives + 3, – 3
It may be Negative or Positive
√9 = gives + 3, – 3
02. Find the square root by factorization method: अभाज्य गुणनखंड विधि से निम्न संख्याओं का वर्गमूल ज्ञात कीजिए: | ||
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Find the square root of 6400
हल: Sol.
√6400
= √ 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5
= √ 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5
= 2 × 2 × 2 × 2 × 5
= 80
√6400 = 80
Find the square roots of the following numbers using the division method:
Find the square roots of the following numbers using the prime factorization method:
भाग विधि से निम्न संख्याओं का वर्गमूल ज्ञात कीजिए: | ||
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Find the square roots of the following numbers using the prime factorization method:
अभाज्य गुणनखंड विधि से निम्न संख्याओं का वर्गमूल ज्ञात कीजिए: | ||
01. 1.44 |
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6400 का वर्गमूल ज्ञात कीजिए?
Find the square root of 6400
हल: Sol.
√6400
Properties of Square
★ A natural Number having 2, 3, 7 or 8 at the ones place is never a perfect square.
★ A natural Number having 0, 4, 6 or 9 at the ones place may or may not be a perfect square.
★ The number of zeros at the end of a perfect square is always even.
★ A number ending with odd number of Zero is never a perfect square.
★ The square of even number is always even number.
★ The square of odd number is always odd number.
★ Factor of a perfect square can be groped in pairs of equal numbers.
★ The square of a number ending with
(1) Square of 1 or 9 ends with 1
(2) Square of 2 or 8 ends with 4
(3) Square of 3 or 7 ends with 9
(4) Square of 4 or 6 ends with 6
(5) Square of 5 ends with 5
(6) Square of 0 ends with 0
★ The perfect squares 1, 4, 9, 16,... can be represented geometrically by dots forming a square as shown below
★ The square of an Natural number ‘n’ is equal the of first ‘n’ odd numbers
1² = 1 = 1
2² = 1 + 3 = 4
3² = 1 + 2 + 3 = 9
4² = 1 + 2 + 3 + 5 = 16
5² = 1 + 2 + 3 + 5 + 6 = 25★ A natural Number with n digits has either (2n – 1 ) or 2n digits in its square.
★ One digit numbers have either 1 or 2 digits in square.
★ Two digits number have either 3 or 4 digits in square.
★ Three digits number have either 5 or 6 digits in square.
★ The square of any odd natural number can be expressed as the sum of two consecutive positive numbers.
3² = 9 = 4 + 5
5² = 25 = 12 + 13
7² = 49 = 24 + 25
★ Square of a Natural Number is greater than 1 and can be written as
3² = 9 = 3 × 3 or 2 × 4 + 1
4² = 16 = 4 × 4 or 3 × 5 + 1
5² = 25 = 5 × 5 or 4 × 6 + 1
6² = 36 = 6 × 6 or 5 × 7 + 1
n² = 25 = n × n or (n – 1) × (n + 1) + 1
★ Between the square of any 2 concuitive number ‘n’ and (n +1) there are 2n Non-perfect square numbers.
Between the square of 1 and 2 = 2 × 1 = 2
Between the square of 2 and 3 = 2 × 2 = 4
Between the square of 3 and 4 = 2 × 3 = 6
Between the square of 4 and 5 = 2 × 4 = 8
Between the square of ‘n’ and (n+1) = 2 × n = ….
★ A perfect square has ‘n’ digits in the corresponding square root.
★ If ‘n’ is even number then number of digits will n/2
√2209 will be 4/2 = 2 digits in its square root.
★ If ‘n’ is odd number then number of digits will (n+1)/2.
√14641 will be 5+1/2 = 6/2 = 3 digits in its square root.
is true in general.
It follows from the above property that a number and its square have the same parity.
PROPERTY 4 The square of a natural number other than one is either a multiple of 3 or exceeds a multiple of 3 by 1.
In other words, a perfect square leaves remainder 0 or 1 on division by 3.
Verification: The above property can be easily verified from the following computations:
Square number Remainder when divided by 3
2² = 4 = 3 * 1 + 1 1
3² = 9 = 3 * 3 + 0 0
4² = 16 = 3 * 5 + 1 1
5² = 25 = 3 * 8 + 1 1
6² = 36 = 3 * 12 + 0 0
7² = 49 = 3 * 16 + 1 1
8² = 64 = 3 * 21 + 1 1
9² = 81 = 3 * 27 + 0 0
★ Note: If a number when divided by 3 leaves remainder 2, then it is not a perfect square.
For example 98, 122, 635 are not perfect squares as they leave remainder 2 when divided by 3.
REMARK If a number leaves remainder 0 or 1 when divided by 3, it is not necessarily a perfect square. However, when it leaves remainder other than 0 and 1, it is not a perfect square.
PROPERTY 5 The square of a natural number other than one is either a multiple of 4 or exceeds a multiple of 4 by 1.
In other words, a perfect square leaves remainder 0 or 1 on division by 4.
Pythagorean Triplets: - the triplet which follows the rule of Pythagoras is known as Pythagorean triplets.
It is shown by the rule of 2m, m² – 1, m² + 1, where ‘m’ is a positive integer.
पाइथागोरियन त्रिक:- वह त्रिक जो पाईथा गोरस के नियम का पालन करता है। पाइथागोरियन त्रिक कहलाता है।
ये निम्न सूत्र द्वारा ज्ञात किये जा सकते हैं। 2m, m² – 1, m² + 1 जहाँ m एक धनात्मक पूर्णाक है।
01. पाइथागोरियन त्रिक लिखिए जिसका एक सदस्य है,
8
6
12
14
16
18
01. एक पाइथागोरस त्रिक ज्ञात कीजिए जिसकी एक संख्या 8 है।
हलः- पाइथागोरस त्रिक पाने का सूत्र 2m, m² – 1, m² + 1 है। जहाँ m एक धनात्मक पूर्णाक है।
02. एक पाइथागोरस त्रिक ज्ञात कीजिए जिसकी एक संख्या 12 है।
वर्ग सर्ववमिका
वर्ग का क्षेत्र फल = (भुजा)²
वर्ग की भुजा = √ Area
ध्यान रखें कि
mind it
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