Perfect Square
Perfect Square is a number obtained by multiplying a whole number by itself, like 4 which is obtained when 2 is multiplied by itself, i.e. 2 × 2 = 4, thus 4 is a perfect square. In mathematical terms, the perfect square is expressed as a2.
In this article, we have covered the meaning and definition of perfect squares, methods of finding perfect squares, and a list of perfect squares and applications.
Table of Content
- What is Perfect Square?
- Perfect Square Definition
- How to Identify Perfect Square Numbers?
- Perfect Square Formula
- Perfect Squares Numbers from 1 to 100
- List of Perfect Squares from 1 to 100
- Properties of Perfect Square
- Perfect Square Chart
- Perfect Square – Tips and Tricks
- How many Perfect Squares are between 1 and 100?
- How many Perfect Squares are between 1 and 1000?
- Perfect Square Examples
- Practice Questions on Perfect Square
What is Perfect Square?
Perfect squares are numbers that you get when you multiply a whole number by itself. For instance, 4 is a perfect square because it’s 2 times 2. Another example is 9, which is 3 times 3. These numbers have a special property, being the result of multiplying a whole number by itself. Examples of perfect squares include 1, 4, 9, 16, and so on.
Perfect Square Definition
Perfect square is a number achieved by multiplying a whole number by itself. For example, 4 is a perfect square since it is the product of 2 multiplied by 2.
How to Identify Perfect Square Numbers?
To find a perfect square number, take a whole number and multiply it by itself. For example, let’s consider the number 16. If we take the whole number 4 and multiply it by itself (4 × 4), the result is 16.
Since the outcome is a whole number, 16 is a perfect square. In general, this method helps determine if a number is a perfect square by checking if it can be expressed as the product of a whole number multiplied by itself.
Perfect Square Formula
The formula for a perfect square is expressed as n2, where ‘n‘ is a whole number. In this formula, n is multiplied by itself, resulting in a perfect square. For example, if n is 3, the perfect square is 32, which equals 9.
Other formulas used for perfect square are,
- n2 − (n − 1)2 = 2n − 1
- n2 = (n − 1)2 + (n − 1) + n
Algebraic Identities as perfect squares:
- a2 + 2ab + b2 = (a + b)2
- a2 – 2ab + b2 = (a – b)2
Perfect Squares Numbers from 1 to 100
List of perfect squares from 1 to 100 is added in the table below,
Perfect Square Numbers From 1 to 100 | ||||
---|---|---|---|---|
1 | = | 1 × 1 | = | 12 |
4 | = | 2 × 2 | = | 22 |
9 | = | 3 × 3 | = | 32 |
16 | = | 4 × 4 | = | 42 |
25 | = | 5 × 5 | = | 52 |
36 | = | 6 × 6 | = | 62 |
49 | = | 7 × 7 | = | 72 |
64 | = | 8 × 8 | = | 82 |
81 | = | 9 × 9 | = | 92 |
100 | = | 10 × 10 | = | 102 |
List of Perfect Squares from 1 to 100
List of perfect squares between 1 to 100 are shown in the table below:
12 = 1 |
112 = 121 |
212 = 441 |
312 = 961 |
412 = 1681 |
512 = 2601 |
612 = 3721 |
712 = 5041 |
812 = 6561 |
912 = 8281 |
---|---|---|---|---|---|---|---|---|---|
22 = 4 |
122 = 144 |
222 = 482 |
322 = 1024 |
422 = 1764 |
522 = 2704 |
622 = 3844 |
722 = 5184 |
822 = 6724 |
922 = 8464 |
32 = 9 |
132 = 169 |
232 = 529 |
332 = 1089 |
432 = 1849 |
532 = 2809 |
632 = 3969 |
732 = 5329 |
832 = 6889 |
932 = 8649 |
44 = 16 |
142 = 196 |
242 = 576 |
342 = 1156 |
442 = 1936 |
542 = 2916 |
642 = 4096 |
742 = 5476 |
842 = 7056 |
942 = 8836 |
52 = 25 |
152 = 225 |
252 = 625 |
352 = 1225 |
452 = 2025 |
552 = 3025 |
652 = 4225 |
752 = 5625 |
852 = 7225 |
952 = 9025 |
62 = 36 |
162 = 256 |
262 = 676 |
362 = 1296 |
462 = 2116 |
562 = 3136 |
662 = 4356 |
762 = 5776 |
862 = 7396 |
962 = 9216 |
72 = 49 |
172 = 289 |
272 = 729 |
372 = 1369 |
472 = 2209 |
572 = 3249 |
672 = 4489 |
772 = 5929 |
872 = 7569 |
972 = 9409 |
82 = 64 |
182 = 324 |
282 = 784 |
382 = 1444 |
482 = 2304 |
582 = 3364 |
682 =4624 |
782 = 6084 |
882 = 7744 |
982 = 9604 |
92 = 81 |
192 = 361 |
292 = 841 |
392 = 1521 |
492 = 2401 |
592 =3481 |
692 =4761 |
792 = 6241 |
892 = 7921 |
992 = 9801 |
102 = 100 |
202 = 400 |
302 = 900 |
402 = 1600 |
502 = 2500 |
602 =3600 |
702 =4900 |
802 = 6400 |
902 = 8100 |
1002 = 10000 |
Properties of Perfect Square
Some important properties of perfect square are,
Result of Squaring an Integer | Perfect square is result of multiplying an integer by itself. |
---|---|
Negative Numbers Can Form Perfect Squares | Negative integers can form perfect square, e.g., (−4)2 = 16 |
Unique Square for Each Integer | Each integer has not a unique square. Two integers have one square, i.e. ‘a’ and ‘-a’ have same square. |
Zero is a Perfect Square | Zero is considered a perfect square because 02 = 0 |
Sum of Consecutive Odd Numbers | A perfect square is sum of consecutive odd numbers. |
Geometric Representation | Perfect square represents area of any figure. |
Perfect Square Chart
Chart for Perfect Square is added below as,
Perfect Square – Tips and Tricks
Some Tricks and Tips for Perfect Squares are given below.
Square of a Number Ending in 5: To find square of a number ending in 5, multiply the digit before 5 with next digit and append 25. For example, 752= 7×8(25) = 5625
Square of Numbers Close to 100: For numbers close to 100, express the square as (100 – x)2= 1002 – 200x + x2. This simplifies calculations, especially for mentally calculating squares.
Odd Number Squares: Square of any odd number is an odd number. If n is an odd number, then n2 is odd.
Even Number Squares: Square of any even number is an even number. If m is an even number, then m2 is even.
Difference of Squares: Use difference of squares formula, a2− b2= (a+b)(a−b). This can help in factoring or simplifying expressions.
Square of a Sum: (a+b)2 = a2 + 2ab + b2
Square of a Difference: (a−b)2 = a2 − 2ab + b2
Observations on Perfect Squares
Perfect numbers end with any one of these digits 0, 1, 4, 5, 6, or 9. Also some observations on perfect squares are,
- Numbers ending with 3 and 7 have 9 as units place digit in their square number.
- Numbers ending with 5 have 5as units place digit in their square number.
- Numbers ending with 4 and 6 will have 6 as units place digit in their square number.
- Numbers ending with 2 and 8 will have 4 as units place digit in their square number.
- Numbers ending with 1 and 9 will have 1 as units place digit in their square number.
How many Perfect Squares are between 1 and 100?
There are 8 perfect squares between 1 and 100 (excluding 1 and 100). They are,
4, 9, 16, 25, 36, 49, 64 and 81
How many Perfect Squares are between 1 and 1000?
There are 30 perfect squares between 1 and 1000. They are,
4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900 and 961
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Perfect Square Examples
Example 1: Identify the first two perfect squares.
Solution:
First two perfect squares are obtained by squaring the first two whole numbers:
- 12=1 (Square of 1 is 1)
- 22= 42 (Square of 2 is 4)
Therefore, first two perfect squares are 1 and 4.
Example 2: If a number is a perfect square and its square root is 9, what is the number?
Solution:
If a number is a perfect square and its square root is 9, we can find the number by squaring the square root:
92 = 81
So, required number is 81, as it is a perfect square, and its square root is 9.
Example 3: If a number is a perfect square and its square root is a prime number, find the number.
Take the prime number 5. The square of 5 is 25 (52=25). Here, 25 is a perfect square, and 5 is a prime number.
So, the number we’re looking for is 25, where the square root (5) is a prime number
Practice Questions on Perfect Square
Some questions on perfect square are,
Q1: Find the square of 5.
Q2: Is 36 a perfect square?
Q3:. Determine the square root of 49.
Q4: Write next two perfect squares after 16.
Q5: Identify the perfect square closest to 150.
FAQs on Perfect Square
How many Perfect Squares are between 1 and 100?
There are 10 perfect squares between 1 and 100. These are 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.
How many Perfect Squares are between 1 and 1000?
There are 31 perfect squares between 1 and 1000. These include numbers like 1, 4, 9, 16, 25, and so on, up to 961.
Is 216 a perfect square?
Yes, 216 is a perfect square. The square root of 216 is 14, because 14 multiplied by itself (14 × 14) equals 216.
What defines a perfect square?
A perfect square is a number that can be made by multiplying a whole number by itself. For instance, 9 is a perfect square because it’s 3 times 3.
How does one determine if a number qualifies as a perfect square?
To check if a number is a perfect square, you see if it can be expressed as the product of a whole number multiplied by itself. If yes, it’s a perfect square.
In mathematical terms, what characterizes a perfect square trinomial?
A perfect square trinomial in math is an expression that can be factored into two identical binomials. It has the form (a+b)2.
Which numerical values are considered perfect squares?
Numbers like 1, 4, 9, 16, and so on, are perfect squares. They result from multiplying a whole number by itself.
What is the process for factoring perfect squares?
To factor perfect squares, you write them as the square of a binomial. For example, 25=(5)2
What approach is used to identify perfect squares?
Identifying perfect squares involves finding if a number can be written as the product of a whole number multiplied by itself.
Does the number 7 qualify as a perfect square?
No, 7 is not a perfect square. You can’t get it by multiplying a whole number by itself.
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