Every square from 1² to 25², what makes them useful, and the root of each one
A square number is what you get when a number is multiplied by itself. The first twenty-five are 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. They are the most valuable facts in mental arithmetic, because almost every other multiplication can be reached in one step from a nearby square.
1² = 1; 2² = 4; 3² = 9; 4² = 16; 5² = 25; 6² = 36; 7² = 49; 8² = 64; 9² = 81; 10² = 100; 11² = 121; 12² = 144; 13² = 169; 14² = 196; 15² = 225; 16² = 256; 17² = 289; 18² = 324; 19² = 361; 20² = 400; 21² = 441; 22² = 484; 23² = 529; 24² = 576; 25² = 625.
Read the other way, each of those is a square root: √625 = 25, √576 = 24, √529 = 23, and so on back to √1 = 1. Squares and roots are one fact read in two directions, so learning the list once covers both.
Every multiplication fact sits next to a square. If 8 × 8 = 64 is instant, then 8 × 9 is 64 + 8 = 72 and 8 × 7 is 64 − 8 = 56. One solid fact gives you both neighbours for the cost of an addition.
The difference between consecutive squares is also predictable: it goes up by two each time. From 1 to 4 is 3, from 4 to 9 is 5, from 9 to 16 is 7, from 16 to 25 is 9. That is a useful check — consecutive squares always differ by an odd number, so an even gap means one of them is wrong.
Knowing the squares by heart is what makes estimating a square root fast. Bracket the number between the two squares either side, then judge how far along the gap it sits. √200 sits between 196 and 225, much closer to 196, so the root is a little over 14.
That estimate is usually right to within a tenth, which is more than enough to sanity-check a calculator or to keep a longer calculation moving.
Areas, diagonals, distances on a grid, standard deviations, compound growth — all of them are built on squaring or un-squaring. The squares are not an isolated party trick; they are the lookup table that everything geometric quietly runs on.
They also make certain multiplications collapse. 19 × 21 looks awkward until you notice it is 20² − 1 = 399, because a pair either side of a square always comes to the square minus one.
A number multiplied by itself. 7 × 7 = 49, so 49 is a square number and 7 is its square root.
They are 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100 — the squares of 1 through 10.
Because moving from n² to (n+1)² adds 2n + 1. Each step adds two more than the last, which is why the differences run 3, 5, 7, 9, 11 and so on.
Up to 25² = 625 covers almost every practical estimate. Up to 12² is the minimum worth having, because that is what makes every neighbouring multiplication fact one addition away.
Each square root below has a page of its own with the working and a way to estimate it by hand.