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Check if any number is a perfect square, find its square root if it is, list perfect squares up to a given number, and find the nearest perfect squares above and below.
Enter a number to check if it's a perfect square
Type any non-negative number in the input field. You can use whole numbers or decimals. The calculator accepts any value from 0 upwards.
The calculator instantly determines if your number is a perfect square. If yes, it shows the square root. If no, it shows the nearest perfect squares above and below.
Review the list of all perfect squares up to your number (up to 20 entries) to see the pattern and understand where your number falls in the sequence.
n² = n × n
A perfect square is a number that equals some integer multiplied by itself.
√x ∈ ℤ ⟹ x is perfect square
If the square root is an integer, the number is a perfect square.
1+3+5+...+(2n-1) = n²
Sum of first n odd numbers equals n². Example: 1+3+5 = 9 = 3²
(n+1)² - n² = 2n + 1
Consecutive perfect squares differ by consecutive odd numbers.
a² × b² = (ab)²
Product of perfect squares is a perfect square. Example: 4 × 9 = 36
Last digit: 0,1,4,5,6,9 only
Perfect squares can only end in 0, 1, 4, 5, 6, or 9.
| Number (n) | Perfect Square (n²) | Calculation |
|---|---|---|
| 1 | 1 | 1 × 1 |
| 2 | 4 | 2 × 2 |
| 3 | 9 | 3 × 3 |
| 4 | 16 | 4 × 4 |
| 5 | 25 | 5 × 5 |
| 6 | 36 | 6 × 6 |
| 7 | 49 | 7 × 7 |
| 8 | 64 | 8 × 8 |
| 9 | 81 | 9 × 9 |
| 10 | 100 | 10 × 10 |
| 11 | 121 | 11 × 11 |
| 12 | 144 | 12 × 12 |
| 13 | 169 | 13 × 13 |
| 14 | 196 | 14 × 14 |
| 15 | 225 | 15 × 15 |
| 16 | 256 | 16 × 16 |
| 17 | 289 | 17 × 17 |
| 18 | 324 | 18 × 18 |
| 19 | 361 | 19 × 19 |
| 20 | 400 | 20 × 20 |
A perfect square is a number that can be expressed as the product of an integer with itself. For example, 16 is a perfect square because 16 = 4 × 4 = 4². Perfect squares are also called square numbers. The sequence starts: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100...
Take the square root of the number. If the result is a whole number (integer), then it's a perfect square. For example, √49 = 7 (integer), so 49 is a perfect square. But √50 ≈ 7.07 (not integer), so 50 is not a perfect square.
Perfect squares have several properties: (1) They can only end in 0, 1, 4, 5, 6, or 9. (2) The difference between consecutive perfect squares is always an odd number. (3) Every perfect square is the sum of consecutive odd numbers starting from 1. (4) The product of two perfect squares is always a perfect square.
No, perfect squares are always non-negative (0 or positive). When you square any real number (positive or negative), you always get a non-negative result. For example, both 5² = 25 and (-5)² = 25, so 25 is a perfect square, but -25 is not.
A perfect square is the result of squaring an integer (like 25 = 5²), while a square root is the operation that reverses squaring (√25 = 5). Not all numbers are perfect squares, but every non-negative number has a square root. Only perfect squares have integer square roots.
Perfect squares appear in many areas: (1) Geometry - areas of squares with integer sides. (2) Pythagorean theorem - 3² + 4² = 5² gives us right triangles. (3) Statistics - variance and standard deviation calculations. (4) Computer science - algorithms and data structures. (5) Physics - inverse square laws in gravity and electromagnetism.
Perfect squares have fascinated mathematicians for thousands of years. Ancient Greek mathematicians studied them extensively, and they form the basis of many important mathematical concepts. The name "square" comes from geometry - if you arrange n² objects in a square grid, you get a perfect square arrangement.
One beautiful property is that every perfect square is the sum of consecutive odd numbers starting from 1. For example, 16 = 1 + 3 + 5 + 7. This connects perfect squares to arithmetic sequences and provides a way to calculate them without multiplication.