Square numbers
A square number is the result of multiplying a whole number by itself, such as 4 (2×2) or 25 (5×5).
Square numbers are a purely mathematical grouping — being n² for some whole number n. Several squares in this catalogue also carry separate cultural or everyday significance (16 and "sweet sixteen", 25 and a silver anniversary, 100 and the idea of a century), but that significance is unrelated to their square-number status and is covered on each number's own page.
Square numbers as arrays
A square number is what you get multiplying a whole number by itself — a number of dots that arranges perfectly into a square, with no leftovers and no gaps. 1, 4, 9, and 16 are the first four square numbers, from 1×1, 2×2, 3×3, and 4×4.
Worked example: is 36 square? Is 40?
36's square root is exactly 6— 6 × 6 = 36 with nothing left over, so 36 is a perfect square. 40 has no whole-number square root: 6 × 6 = 36 (4 short of 40), and 7 × 7 = 49 (9 too many) — there's no whole number that lands exactly on 40, so 40 is not a perfect square.
How to tell if a number is a perfect square
Take the square root and check whether it lands exactly on a whole number. 49's square root is 7 exactly, so 49 is a perfect square; 50's square root is approximately 7.07, so 50 is not. This is a purely computational check — no lookup table involved.
Squares, cubes and other powers
A square is a number raised to the power of 2 (n²). Some numbers are also perfect cubes (n³) — 1 and 64 are both a square and a cube, since 64 = 8² and 4³. Being a square says nothing about whether a number is also prime, triangular, or any other class; each property is checked independently.
Square numbers in the catalogue
1
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
841
900
961
1,024
1,089
1156
1225
1296
1369
1444
1521
1600
1681
1764
1849
1936
2025
2500
3,600
4,096
4900
6400
10,000
14,400
16,384
44,100
46,656
65,536
384,400
1,000,000
1,048,576
16,777,216
1,073,741,824
4,294,967,296
1,000,000,000,000
Common questions
What makes a number a perfect square?
A perfect square is the result of multiplying a whole number by itself — 4 is 2×2, 25 is 5×5, and so on. Every number in this catalogue is checked against this definition directly, not against a list.
How can I tell if a number is a perfect square?
Take its square root and check whether it lands exactly on a whole number. 36's square root is exactly 6, so 36 is a perfect square; 40's square root is approximately 6.32, so 40 is not.
Is every square number also culturally significant?
No. Being a square is a purely mathematical property. A few squares in this catalogue, like 16 or 25, also happen to carry separate cultural or everyday significance — but that's covered on their own profile page, not implied by being a square.