Squares and Square Roots
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Hello friends, how are you doing? I hope you are doing great! Friends, I am Dheeraj Sahni and welcome to my website - www.mathshindi.com. On this website, you learn complex topics related to maths in an easy and interesting way.
Similarly,
Square of 1 is 1 x 1, which means 1.
Square of 3 is 3 x 3, which means 9.
Square of 4 is 4 x 4, which means 16.
Square of 5 is 5 x 5, which means 25.
Square of 6 is 6 x 6, which means 36.
Square of 7 is 7 x 7, which means 49.
Square of 8 is 8 x 8, which means 64.
Square of 9 is 9 x 9, which means 81.
Square of 10 is 10 x 10, which means 100, and so on.
In the same way, you can find the square of any Natural number!
And not only of Natural numbers; we can find the square of any Whole number, Integer, Rational number, Irrational number, Real number or Complex number.
For example,
Square of 0 is 0 x 0, which means 0.
Square of -5 is (-5) x (-5), which means 25.
Square of 1/2 is (1/2) x (1/2), which means 1/4.
Square of √3 is √3 x √3, which means 3.
Square of 0.25 is 0.25 x 0.25, which means 0.0625.
Square of (2 + 3i) is (2 + 3i) x (2 + 3i), which means (4 - 9 + 12i), which means (-5 + 12i).
In the same way, you can find the square of any number either it is a Real number or a Complex number!
"Symbolically we represent 'square' by '2' writing it at the top-right side of the number whose square is to be found in a comparatively smaller size."
In this way, we represent the sentence 'Square of 1 or 1 squared' by '12'.
Now the above examples can be written as follow:
1 squared = 12 = 1 x 1 = 1
3 squared = 32 = 3 x 3 = 9
4 squared = 42 = 4 x 4 = 16
5 squared = 52 = 5 x 5 = 25
6 squared = 62 = 6 x 6 = 36
7 squared = 72 = 7 x 7 = 49
8 squared = 82 = 8 x 8 = 64
9 squared = 92 = 9 x 9 = 81
10 squared = 102 = 10 x 10 = 100
And also,
0 squared = 02 = 0 x 0 = 0
(-5) squared = (-5)2 = (-5) x (-5) = 25
(1/2) squared = (1/2)2 = (1/2) x (1/2) = 1/4
√3 squared = (√3)2 = √3 x √3 = 3
0.25 squared = (0.25)2 = 0.25 x 0.25 = 0.0625
(2 + 3i) squared = (2 + 3i)2 = (2 + 3i) x (2 + 3i) = (4 - 9 + 12i) = (-5 + 12i)
I hope, now you have a better understanding of - What the square is.
Now, let's move on to our main topic - What the square root is.
Similarly,
Square root of 0 is 0 because 0 x 0 = 0.
Square root of 1 is 1 because 1 x 1 = 1.
Square root of 4 is 2 because 2 x 2 = 4.
Square root of 16 is 4 because 4 x 4 = 16.
Square root of 25 is 5 because 5 x 5 = 25.
Square root of 36 is 6 because 6 x 6 = 36.
Square root of 49 is 7 because 7 x 7 = 49.
Square root of 64 is 8 because 8 x 8 = 64.
Square root of 81 is 9 because 9 x 9 = 81.
Square root of 100 is 10 because 10 x 10 = 100, etc.
"Symbolically we represent 'square root' by '√' writing it just before the number whose square root is to be found."
In this way, we represent the sentence 'Square root of 1 or square root 1' by '√1'.
Now the above examples can be written as follow:
√0 = 0 because 0 x 0 = 0
√1 = 1 because 1 x 1 = 1
√4 = 2 because 2 x 2 = 4
√25 = 5 because 5 x 5 = 25
√36 = 6 because 6 x 6 = 36
√49 = 7 because 7 x 7 = 49
√64 = 8 because 8 x 8 = 64
√81 = 9 because 9 x 9 = 81
√100 = 10 because 10 x 10 = 100
I hope, now you have a better understanding of - What the square root is.
Now I will ask a question and you have to give the answer, Ok! So, here is the question - What is the square root of 10? (Think about it!)
Can you able to find the answer? Can you able to find a number which when multiplied by itself, gives 10? I think you will not have found that number. Am I right? If I am right, now you will be thinking that - 'We cannot find the square root of 10. The square root of 10 does not exist.'
Well, I want to tell you, this is not so! We can find the square root of any Natural number. And not only of Natural numbers; we can find the square root of any Whole number, Non-negative Integer, Non-negative Rational number, Non-negative Irrational number, Non-negative Real number.
Note: The square root of any negative number whether it is negative Integer or negative Rational number or negative Irrational number or negative Real number, does not exist. They are Imaginary numbers (or Complex numbers).
Now, 10 is a Natural number (It is also a Whole number, a non-negative Integer, a non-negative Rational number, a non-negative real number.) So we can find out the square root of 10. Now you may be thinking that how can we find the square root of 10. How can we find the number which when multiplied by itself, gives 10? (You may be excited to know that. I think!)
There are methods and some short tricks for finding out the square root of imperfect square numbers or non-perfect square numbers like 10, 7, 11, 21, etc. A perfect square number is that whose square root is a Whole number. If I say in simple words, the squares of all Whole numbers are the perfect square numbers. For example: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, etc. are perfect squares numbers or just perfect squares.
Actually, there are three main methods to find out the square root of the given number:
1. Square root by Prime Factorisation Method
2. Square root by Estimation Method
3. Square root by Long Division Method
In spite of the above three main methods, there are some more methods that are used for finding the square root of a number, such as the Square root by Vedic Method etc.
Friends, today I am going to explain to you - "Square and Square Root"
So without wasting time, Let's get started...
Before we understand - What the square root is. We have to understand - What the square is.
What is Square?
"A square is a number which is obtained by multiplying the given number by itself, for example, square of 2 is 2 x 2, means 4."
Similarly,
Square of 1 is 1 x 1, which means 1.
Square of 3 is 3 x 3, which means 9.
Square of 4 is 4 x 4, which means 16.
Square of 5 is 5 x 5, which means 25.
Square of 6 is 6 x 6, which means 36.
Square of 7 is 7 x 7, which means 49.
Square of 8 is 8 x 8, which means 64.
Square of 9 is 9 x 9, which means 81.
Square of 10 is 10 x 10, which means 100, and so on.
In the same way, you can find the square of any Natural number!
And not only of Natural numbers; we can find the square of any Whole number, Integer, Rational number, Irrational number, Real number or Complex number.
For example,
Square of 0 is 0 x 0, which means 0.
Square of -5 is (-5) x (-5), which means 25.
Square of 1/2 is (1/2) x (1/2), which means 1/4.
Square of √3 is √3 x √3, which means 3.
Square of 0.25 is 0.25 x 0.25, which means 0.0625.
Square of (2 + 3i) is (2 + 3i) x (2 + 3i), which means (4 - 9 + 12i), which means (-5 + 12i).
In the same way, you can find the square of any number either it is a Real number or a Complex number!
"Symbolically we represent 'square' by '2' writing it at the top-right side of the number whose square is to be found in a comparatively smaller size."
In this way, we represent the sentence 'Square of 1 or 1 squared' by '12'.
Now the above examples can be written as follow:
1 squared = 12 = 1 x 1 = 1
3 squared = 32 = 3 x 3 = 9
4 squared = 42 = 4 x 4 = 16
5 squared = 52 = 5 x 5 = 25
6 squared = 62 = 6 x 6 = 36
7 squared = 72 = 7 x 7 = 49
8 squared = 82 = 8 x 8 = 64
9 squared = 92 = 9 x 9 = 81
10 squared = 102 = 10 x 10 = 100
And also,
0 squared = 02 = 0 x 0 = 0
(-5) squared = (-5)2 = (-5) x (-5) = 25
(1/2) squared = (1/2)2 = (1/2) x (1/2) = 1/4
√3 squared = (√3)2 = √3 x √3 = 3
0.25 squared = (0.25)2 = 0.25 x 0.25 = 0.0625
(2 + 3i) squared = (2 + 3i)2 = (2 + 3i) x (2 + 3i) = (4 - 9 + 12i) = (-5 + 12i)
I hope, now you have a better understanding of - What the square is.
Now, let's move on to our main topic - What the square root is.
What is Square Root?
"The square root is just opposite of what square is. Finding the square root of a number is finding a number which when multiplied by itself, gives the given number. For example, square root 9 is 3 because when we multiply 3 by itself, i.e. 3 x 3, gives 9."
Similarly,
Square root of 0 is 0 because 0 x 0 = 0.
Square root of 1 is 1 because 1 x 1 = 1.
Square root of 4 is 2 because 2 x 2 = 4.
Square root of 16 is 4 because 4 x 4 = 16.
Square root of 25 is 5 because 5 x 5 = 25.
Square root of 36 is 6 because 6 x 6 = 36.
Square root of 49 is 7 because 7 x 7 = 49.
Square root of 64 is 8 because 8 x 8 = 64.
Square root of 81 is 9 because 9 x 9 = 81.
Square root of 100 is 10 because 10 x 10 = 100, etc.
"Symbolically we represent 'square root' by '√' writing it just before the number whose square root is to be found."
In this way, we represent the sentence 'Square root of 1 or square root 1' by '√1'.
Now the above examples can be written as follow:
√0 = 0 because 0 x 0 = 0
√1 = 1 because 1 x 1 = 1
√4 = 2 because 2 x 2 = 4
√25 = 5 because 5 x 5 = 25
√36 = 6 because 6 x 6 = 36
√49 = 7 because 7 x 7 = 49
√64 = 8 because 8 x 8 = 64
√81 = 9 because 9 x 9 = 81
√100 = 10 because 10 x 10 = 100
I hope, now you have a better understanding of - What the square root is.
Now I will ask a question and you have to give the answer, Ok! So, here is the question - What is the square root of 10? (Think about it!)
Can you able to find the answer? Can you able to find a number which when multiplied by itself, gives 10? I think you will not have found that number. Am I right? If I am right, now you will be thinking that - 'We cannot find the square root of 10. The square root of 10 does not exist.'
Well, I want to tell you, this is not so! We can find the square root of any Natural number. And not only of Natural numbers; we can find the square root of any Whole number, Non-negative Integer, Non-negative Rational number, Non-negative Irrational number, Non-negative Real number.
Note: The square root of any negative number whether it is negative Integer or negative Rational number or negative Irrational number or negative Real number, does not exist. They are Imaginary numbers (or Complex numbers).
Now, 10 is a Natural number (It is also a Whole number, a non-negative Integer, a non-negative Rational number, a non-negative real number.) So we can find out the square root of 10. Now you may be thinking that how can we find the square root of 10. How can we find the number which when multiplied by itself, gives 10? (You may be excited to know that. I think!)
There are methods and some short tricks for finding out the square root of imperfect square numbers or non-perfect square numbers like 10, 7, 11, 21, etc. A perfect square number is that whose square root is a Whole number. If I say in simple words, the squares of all Whole numbers are the perfect square numbers. For example: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, etc. are perfect squares numbers or just perfect squares.
Actually, there are three main methods to find out the square root of the given number:
1. Square root by Prime Factorisation Method
2. Square root by Estimation Method
3. Square root by Long Division Method
In spite of the above three main methods, there are some more methods that are used for finding the square root of a number, such as the Square root by Vedic Method etc.
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