Calculating squares and higher powers using division and addition

The other day I had to calculate 1.26^3 and I was thinking about the quickest way to mentally do this.

I realized that the following method is rare and needs to be talked about. It is extremely fast.

Here is how I did this.
First step. Think of 126, divide by 4 and add this to 126:

126 = 120 + 6, so division by 4 is easy: 30 + 1.5 = 31.5

Add 31.5 to 126 and get 157.50.

Second step. Add 1% of 126 = 1.26.

157.50 + 1.26 = 158.76.
Except for the decimal point, this is 126 squared!

Now multiply 1.5876 by 1.26. Same routine:

Divide 158.76 by 4:
158 = 160 - 2, so division by 4 gives 40 - 0.5 = 39.50

Add this to 158.76 gives 198.26.
Quickest way is to first add 40 to 158.76 and then subtract 0 5.

Now divide 0.76 by 4.
76 = 80 - 4
(80 - 4)/4 = 20 - 1 = 19

So add 19 cents and get 198.45

Now add 1.5876 (1% of 158.76):
The complement of 198.45 is 1.55.
Add that first.
198.45 + 1.5876 =198.45 + 1.55 +0.0376 = 200.0376

So 1.26^3 = 2.000376.

The whole idea is that adding a quarter is the same as multiplying by 1.25.
Since we are multiplying by 1.26 instead of 1.25, we need to also add a percent.

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I can follow your steps mathematically, but for the love of god I need a pen and paper to actually do it. Can you please give us some insight as to how you manage to do all this in your head and not get completely lost (like me)?

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Ok, let’s elaborate some more.

First step. I think of 126 and contemplate on dividing by 4.
I see 120 + 6 and and divide it by 4: 30 + 1.5.

In my mind I work with 2 numbers: 126 and the division of 126 by 4.
So the very moment I work out ‘30’, I add this to 126 and get 156. I repeat 156 a couple of times until it sticks and then go back to working out 6/4 = 1.5.

I now add 1.5 to 156 and get 157.5.

Next step 1% of 126 = 1.26.
Again I see 2 numbers: 157.5 from the previous step and 1.26 from this step.
Split 1.26 into 2 parts: 1 | 26.
I first add 1 to 157 and then 26 to 50 and get 158.76.
Again, let that number circle around in your mind until it sticks.

In mental calculation take small steps to working things out, otherwise you lose track.

Now for the next step I contemplate how to divide 158.76 by 4.
If you read my old posts on this forum I work with complements a lot. So I see ‘159 | -24’. However; 158 | 76 is already easily divisible by 4 if we see it as ‘160 - 2 | 80 - 4’.

If you took the complement ‘159 | -24’ to divide by 4 , you would have gotten (160 - 1 | -24) / 4 = 40 | -25 - 6 = 40 | -31. 158 | 76 + 40 | -31 = 198 | 45.
This works.
But for me, I see 158 | 76 = 160 - 2 | 80 -4 first and realise this is easy to divide by 4.

While circling 198.45 in my mind I realise it is 200 - 1.55 too.
This comes in handy when adding 1.5876.
So in order to add 1.5876 to 198.45 I first add 1.55 to 198.45. When seeing the complement previously, I already knew that this is 200, so now I only need to subtract 155 from 158.26 = 3.76.

Btw, as you can see the decimal point goes where it is most convenient.

So now I need to concatenate ‘376’ to 200.
While moving the decimal point I need to work out where the 376 goes. Is it 200.376 or 200.0376 or 203.76?

Since 198.45 + 1.55 = 200.00, 198.45 + 1.5876 becomes 200.0376. I don’t see how you can put the decimal point somewhere else.

Last step is realising where the decimal point goes in the final answer:

1.26^2 cannot be 158.26, so needs to be 1.5876.
and 1.5876 * 1.26 cannot be 200.0376.
So 200.0376. needs to be 2.000376.

Moving the decimal point just makes it easy for me to calculate. I don’t know why this is. Maybe because I calculate with money a lot?

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Damn, that seems so much work lol. How long did it take for you to have the answer?
Also, wouldn’t it be easier to do 126³ and then place the decimal point at the end? That’s what I did and it took me only a minute or so in my head.

But you are a genius!
What intermediate steps did you take?

(I needed a lot more than 1 minute. Maybe 3 or so? I am definitely not the fastest calculator in the world.)

thanks a lot, this is already easier

sorry, wrong link, I meant to thank Kinma.

Not that your post wasn’t good.

Indeed that is what I do.
Kind of…

I actually go from 126 to 158.76 to 200.0376.
And finally move it back to 2.000376.

I did:
100×126+20×126+6×126 = 15,876.
I then continued:
100x15,876+20×15,876+6×15,876 = 2,000,376.

The decimal point goes behind the 2 and done; 1.26³ = 2.000376.

When I did a step like 20×15,876, I would add the inbetween answers one by one. So first 20×15,000= 300,000 and I would add this to the answer and then continue with 20×800= 16,000 and add this to the answer as well. I point this out because I don’t want you to think that I immediately calculate 20x15,876 in my head without intermediate steps because that is impossible and that would be guessing, lol.

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Fantastic! I cannot do what you do.
I need smaller/different steps to work this out.

But this is the whole point of mental calculation; finding out what steps your brain can handle best.

This is the same as what I do:

This is an important observation btw.
Every intermediate step gets added to the subtotal.

It makes one remember one big number less and this is why most people don’t dear to work out such a calculation and we can.

By showing in great detail how I do this I hope more people take up the ancient and noble art of mental calculation.

The thing is; it really isn’t. It is just a lot of work to write down.
Let’s compare.
What you do is:

What I do is:

126 + 126/4 + 126/100 = 158.76.
Then:
158.76 + 158.76/4 + 158.76/100 = 200.0376.

You take the method that I think most people learn in school. And this works all the time.

The whole point of this post is that in these specific calculations, dividing by 4, add, and then adding the number itself (shifted two places) is easier than multiplying with 20, add, then multiplying with 6 and add again.

Another advantage is in rounding.
Sometimes I am fine with 3-4 decimals precicion.
After the first step you have multiplied by 1.20 and I have multiplied by 1.25.
I know that adding 1% adds one and small change, so if that is enough I am done.

But this thread is not about what is the best way.
It show a different method.

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Your method is faster and you don’t have to remember any unnecessary zero’s in intermediate steps where as I do because I do 126³ instead of 1,26³. When I do 20×15,000=300,000 I have to remember that it is 300 thousand and thus also remember the zero’s. It’s not very difficult because I can see the numbers but still something.