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
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.