# Squaring numbers mentally up to 125

I dont know if you guys already know about this, but i have a trick for squaring numbers from 0-125 but i have 2 different strategies,some work better than others sometimes.(4 if you didnt memorize squares of #s up to 25).

Note:the trick works for all real numbers but it is easily mentally computed over the range of integers from 0 to 125

Trick #1 -

for numbers more than 25 but less than 75.(this is because if the number is more than 75 you will have to do more calculations);

let the number be n ,the steps are as follows

-calculate n-25

-multiply it by 100

-calculate 50-n

-square it then add it to n-25

ex:$${39}^{2}$$

$$\boxed{1}$$ (39-25)*100=1400

$$\boxed{2}$$ $${11}^{2}$$=121 so 1400+121=1521

Trick #2-

if the number is more than 75 but less than 125 we have this rule:

-calculate 100-n

-subtract your result from the original number then multiply it by 100

-square (100-n) then add it to the previous result

ex:$${109}^{2}$$

100-109=-9

100(109+9)=11800

adding $${9}^{2}$$ we get 11881

These methods require you to know squares of numbers up to 25,

for 2 digit numbers starting with 1:

-add the ones digit to the number

-multiply it by 10 ,then add the square of the ones digit

for 2 digit #s starting with 2:

-add the ones digit to the number

-multiply it by 20

-add the square of the ones digit

$$ex:15\\ \\ 15+5=20\\ 20*10=200\quad \quad {5}^ {2}=25\quad \quad \quad \quad 200+25=225\\ \\ ex:23\\ 23+3=26\Rightarrow 2*260\Rightarrow 520+{3}^ {2}=529\\ \\ \\ \\ \\$$

Note by Hummus A
2 years, 5 months ago

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let me "formalize" the tricks.

1.$$n^2=(50-n)^2+100(n-25)$$

2.$$n^2=(n-(100-n))100+(100-n)^2=100(100-2n)+(100-n)^2$$

Both are true.

- 2 years, 5 months ago

Oooooh, could you add this to the mental math tricks wiki? We have a wiki collaboration coming up this weekend, and you should come join us!

Staff - 2 years, 5 months ago