In square root there is a term called a perfect square a perfect square is a number that can be multiped by two of the same number with no decimals. I realized on my own after looking at a multiplication chart that each perfect square is usually highlighted and the gap between each perfect square is exactly the length of the previous gap plus 2. Using this we can calculate perfect squares without using multiplication functions and it gives us more information. Some information examples are how many perfect squares we can calculate as well as how big the gaps can get.
For those of you who are mathematically inclined: Provide a proof for RobbyZero’s statement, “The gap between each perfect square is exactly the length of the previous gap plus 2.”
Not good at math but hopefully this informal proof counts lol
“The gap between each perfect square is exactly the length of the previous gap plus 2.” here’s an example: 25 - 16 = 16 - 9 + 2 which is 5^2 - 4^2 = 4^2 - 3^2 + 2 which we can turn into variable form: x^2 - (x - 1)^2 = (x - 1)^2 - (x - 2)^2 + 2 which is always true.
I did the algebra out but I’m too lazy to write it up with LaTeX (mods enable the latex integration maybe please?!?!? ) or post a neat picture so here’s my Proof by Wolfram Alpha .
The +2 thing also works with a different set of numbers on a multiplication table like 2, 6, 12, 20, 30 or 3, 8, 15, 24, 35 as long your going diagonally but to get it to work I’m betting that its going to be messy
More Math Warning
There’s also a way to do it with x^3, x^4, and up but the math gets messier with the higher power number because you have to do another step for one power up you do but the + number is whatever step your one but with ! (5!=5x4x3x2x1=120 or 3!=3x2x1=6 not sure about x^0 though, since 0! = 1)
) or post a neat picture so here’s my 