What are Perfect Squares?

# What are Perfect Squares?

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Vikram Singh
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Updated on Oct 25, 2023 10:23 IST

In this article, we will discuss perfect squares, and different methods to check the perfect square and finally, we will use python to determine the number of perfect squares in the given range, and how to check a number is perfect square or not using python.

In mathematics, when we multiply two integers (identical integers), then we get the square, but when we take the square root of any integer and get the integer value, then the square values are called the perfect squares.

Confused!! Don’t worry, in this blog, we will cover everything from what is perfect square, list of perfect squares, methods to check perfect squares and finally we will discuss how to find the perfect squares in python.

## What is Perfect Square?

A perfect square is a number that is created by multiplying two identical integers.

Example:

1. 4 x 4 = 16 = -4 x -4
2. 7 x 7 = 49 = -7 x -7
3. 11 x 11 = 121 = -11 x -11

Here, 16, 49, and 121 are perfect squares that are created by multiplying 4 (-4), 7(-7), and 11(-11) by themselves respectively.

Note: Generally, perfect square are expressed as x2, where x belongs to an integer.

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## List of Perfect Squares

Here is the list of all perfect squares from 1 to 100.

## Polynomial as a Perfect Square

Perfect squares are not only limited to numerals but also exist in algebraic identities and these are identified using factorization techniques.

Algebraic Identities as a Perfect Square:

1. (x + y)2 = x2 + 2xy + y2
2. (x – y)2 = x2 – 2xy + y2

An algebraic expression that is obtained by squaring binomial expression (a binomial is a polynomial of two terms) is known as Perfect Square Trinomial.

1. (x + 3)2 = x2 + 6x + 9
2. (x – 5)2 = x210x + 25

Here, (x + 3)2 and (x – 5)2 are binomial while, x2 + 6x + 9 and x210x + 25 are known as Perfect Square Trinomials.

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## Methods to Check Perfect Square

### Square Root

If a square root of an integer is an integer, then the given number is a perfect square.

In the above three examples, 81, and 144 are perfect squares while 181 is not a perfect square since 13.4536 is not an integer.

### All perfect square ends with 00, 1, 4, 5, 6, or 9

This is a kind of elimination method to determine whether a given number is a perfect square or not if you don’t have a calculator.

If any given number doesn’t end with the above number then that number is not a perfect square.

There are some conditions attached to these numbers such as

1. To be a perfect square, if the number ends with 1, 4, or 9 then the tens digit must be an even (0, 2, 4, 6, 8) number
2. if the number ends with 5 then the tens digit must be 2
3. To be a perfect square, if the number ends with 6 then the tens digit must be an odd (1, 3, 5, 7, 9) number

### What areDigital Roots?

Let’s first discuss how to calculate digital roots.

To calculate the digital roots, just sum all the digits in the number.

If the obtained sum is two-digit, then sum them as well.

Example: Find the digital root of 729.

1. 729 -> 7 + 2 + 9 = 18 -> 1 + 8 = 9

Hence, the digital root of 729 is 9.

1. 64 -> 6 + 4 = 10 -> 1 + 0 = 1
2. 343 -> 3 + 4 + 3 = 10 -> 1 + 0 = 1

To be a perfect square, the digital root of that number must be 0, 1, 4, or 7 otherwise it’s not a perfect square

Note: If the digital root of any number is not equal to 0, 1, 4, or 7 then the number must be a non-perfect square number but if the digital roots of any number are one of them (0, 1, 4, or 7) then the number is probably a perfect square but we are not certain about it.

Let’s understand with the help of the above-discussed example

1. The digital sum of 729 is 9 which is not equal to any of 0, 1, 4, or 7. Hence, 729 is not a perfect square
1. The digital sum of 64 and 343 is 1 but 64 ( 8 x 8) is a perfect square while 343 ( 7 x 7 x 7) is not a perfect square.

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## Perfect Square in Python

### Print all the Perfect Square in a given range

` `
`# Print all the perfect square in a given rangeimport mathdef perfect_square_range (a,b): for i in range (a, b + 1): if (math.sqrt(i) == int(math.sqrt(i))): #here we want to check whether the sqrt of that number is an integer or not print(i, end = " ")Copy code`

### Number of Perfect Square in a given range

` `
```#Print the number of Perfect Square #here, we will use the formula (if the defined range is (a,b)): #floor(sqrt(b)) - ceil(sqrt(a)) + 1 #ceil: smallest integer greater or equal to x. It will be used to get the number after a.#Example: ceil(5.9) = 6, ceil(-5.9) = -5 #floor: largest integer less than or equal to x. It will be used to get the number before b.#Example: floor (5.9) = 5, floor (-5.9) = -6 import math def number_of_perfect_square(a,b): return (math.floor(math.sqrt(b)) - math.ceil(math.sqrt(a)) + 1)Copy code```

### Using Square Root Function

` `
```#Perfect Square# Python Program to identify whether the given number is a perfect square or not #using Square Root import math i = input("enter the number") #taking the inputsqrt = math.sqrt(i) #finding the square root of the given number if i == int(sqrt)**2 : print (i, " is a perfect square")else: print(i, " is not a perfect square")Copy code```

### Using Ceil and Floor Function

` `
```#Perfect Square#Using ceil and floor function#here we will use the concept that if the ceil and floor value of square root of any number are equal then the number is a perfect square else not import math i = int(input("enter the number")) #taking the input if (math.ceil(math.sqrt(i)) == math.floor(math.sqrt(i))): print (i, " is a perfect square")else: print(i, " is not a perfect square")Copy code```
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## Conclusion

In this article, we have discussed perfect squares, different methods to check the perfect square and finally we used python to determine the number of perfect square in the given range, and how to check a number is perfect square or not using python.

## FAQs

What is a Perfect Square?

A perfect square is a number that is created by multiplying two identical integers. Example: 16 is a perfect square that is a multiplication of 4 and 4.

How can you recognize a perfect square?

It is very easy to recognize the perfect square as every perfect square end with 00, 1, 4, 5, 6, or 9. Another method to check is find the square root of the number and if the square root is an integer, then the number is a perfect square.

What are some examples of perfect square?

Examples of Perfect Squares are: 4, 9, 16, 25, 36, 49, 64, 81, 100, etc.

What are the first few Perfect Squares?

The first few perfect squares are 1 (1u00d71), 4 (2u00d72), 9 (3u00d73), 16 (4u00d74), 25 (5u00d75), and so on.

Is 0 considered a Perfect Square?

Yes, 0 is considered a perfect square because 0 u00d7 0 = 0.

How do I find the next Perfect Square after a given number?

To find the next perfect square after a given number, first find the square root of the given number, round up to the nearest whole number, and then square that number.

Can negative numbers be Perfect Squares?

No, negative numbers cannot be perfect squares. Perfect squares are always non-negative because when you square a real number, the result is always non-negative.

What is the importance of Perfect Squares in mathematics?

Perfect squares are important in mathematics for various reasons, including simplifying radical expressions, solving quadratic equations, and understanding geometric concepts like area.

How can I calculate the square root of a Perfect Square without a calculator?

For smaller perfect squares, you can memorize them or use the method of prime factorization. For larger numbers, estimation or algorithms like the long division method can be used.

Are all even numbers Perfect Squares?

No, not all even numbers are perfect squares. A number being even or odd does not necessarily determine if it is a perfect square. For example, 8 is even but not a perfect square.

How are Perfect Squares used in real life?

Perfect squares are used in real life in areas such as architecture, computer graphics, and cryptography, where calculations involving areas and secure algorithms are crucial.

Can fractions or decimals be Perfect Squares?

Yes, fractions or decimals can be perfect squares if their square roots are rational numbers. For example, 0.25 is a perfect square because it is 0.5 u00d7 0.5.