Let’s focus on calculating squares.
They can be used to speed up regular multiplication, like we discuss in the difference of squares thread.
The tens are easy, 10 squared = 100, 20 squared = 400, 30 squared = 900, etc.
The fives are easy:
35 squared is 3 times 4 times 100 and add 25 = 1225.
45 squared is 4 times 5 times 100 and add 25 = 2025.
55 squared is 5 times 6 times 100 and add 25 = 3025.
In other words take the first digit, multiply this by one more and concatenate ‘25’ to the result.
We see that we can easily do the tens and the fives, so all that is left to do are the numbers in between.
This is easier than it might sound. Let’s start with an example:
31 squared. If we start by calculating 30 squared = 900, then to get to 31, all we need to do is add 30 and then 31.
Alternatively we can double the number we just squared (2X30=60), add it to our result (add to 900 = 960) and add one (961).
The latter is what I do most of the time. I hope you see how easy this is.
Since these are the very numbers you already have in your mind when you do this calculation, this keeps the amount of digits to keep in your mind to a bare minimum and makes it a great technique for mental calculation.
Another example 36^2.
Start with 35^2=1225. Then double the 35 to get 70. This makes 1295, add one to get 1296.
Done.
We can go back.
34^2. Start with 35^2=1225, again double the 35 to get 70. Now instead of adding 70, we now subtract 70. 1225 - 70 = 1155, again add one to get 1156.
This is based on (a + b)^2 = a^2 + 2ab +b^2
We used b=1 in the previus examples and b=-1 for the example of going back.
If b = -1, when we go back one step we get a^a - 2a + 1
This is the reason we subtract 70 and add 1 in the previous example.
Tip: use small steps. For 36 squared we could start with 30 and use a step of 6 to get the following
intermediate results:
30^2 = 900
3X6X2 = 360. Add to 900 = 1260
6^2=36. Add to 1260 to get 1296.
Same result. However, I do 1225 + 70 + 1 a lot quicker than 900 + 360 + 36.
For 37 we can evaluate 35, then calculate 36, then move to 37.
This is great training the squares. Just start at a random number, square it and do the next, etc.
For 37, instead of first caclculating 36 and then 37, we can do 2 steps at a time.
We can use (a + b)^2 = a^2 + 2ab +b^2 with 35 for a and 2 for b.
37^2 =
(35 + 2)^2 =
35^2 + 2X2X35 +2^2 =
1225 + 4X35 +4 =
1225 +140 + 4 =
1365 + 4 = 1369.
In my head I start with 1225, then do 2 times 35 = 70, then double it to get 140, then add to 1225, then
add 4.
If you cannot work with the formula, try this:
For one step forward add the number itself, then add the following number.
So for 36^2, if we know 35^2 we add first 35, then 36. So we now have added 35 twice and then 1.
For two steps forward, see this example.
For 37^2 if we know 35^2, we add first 35, then 36 (to go from 35 to 36), then 36, then 37 (to go from 36 to 37). So we add, 35, 36, 36, and 37.
If we add 35 four times, then there is 4 left to add.
It is a lot of text to write down, but once you get the hang of it, it goes really quick.
