General discussion of mental maths

[Webmaster’s note: the original post was removed, but the mental math conversation continues below.]

I first want to say that I am a total amateur when it comes to mental math, but I have at least read the book. And from what I understand the procedures/methods are really quite simple and wouldn’t require any mnemonics to remember.

Perhaps, day of the week calculation can benefit from mnemonics, but only in the form of actually remembering numbers so it isn’t anything directly related to the actual method.


I am working through the trachtenberg system at the moment and I am quite impressed with the speed at which I can multiply numbers. I have only got to the point of multiplying by 11, 12 and 6 but it is lightening fast.

I use the major system to remember my result as the answer is calculated backwards, then I just work through the images in reverse to say the answer,

for example is someone asked me to multiply 453764523 X 11 (they would need to write it down of course) I could give them the answer in about 40 seconds, the answer actually does not fit on the readout of my calculator. If I am ‘allowed’ to write the answer down beneath the sum I can give the answer almost immediately(in the time it takes to write it)

Once I complete learning the system I am hoping that I can ask someone for a long number, say a 5 digit number, then ask somone else for another 5 digit number. Then ask them to open their phone and calculate the answer, in the meantime I can write the answer on a piece of paper before they even have it calculated (it would take them a while to open up their caclulator on their phone etc)



Arthur Benjamin uses the Anchor Method most:

Just practise this a bit. You will quickly get the hang of it.
The book discribes the techniques very well, so just practise after each chapter.

For practice I use an old programmable calculator emulator that I use on my phone:
This let’s me quickly program a training routine.

The book is a great way of getting used to these techniques. I highly recommend it.

Logarithms change multiplication into addition, powers into multiplication and roots into division. I guess you can imagine how useful that is in mental calculation.

The courses I don’t know how good they are. Try the book first. While waiting for the book, watch YouTube movies about him. He usually explains what he does, so it is easy to follow.

Don’t be put off by the link btw. That text describes most of the techniques the pros use. Take one at a time and practice. Imho the anchor method is best for multiplying numbers. My advice would be to stay with that. Arthur’s book uses it a lot, so if you receive it, you will use it soon anyway.

The book doesn’t address logarithms. If you understand powers the concept of logarithms is close.

Let’s see if I can explain the concept with a quick example.

8 times 4 makes 32.

8 is 2 times 2 times 2 or 2 to the power of 3. We write 2^3.

4 is 2 times 2 or 2 to the power of 2. We write 2^2.

32 is 2 to the power of 5. We write 2^5.

I hope you already see where this is going. The multiplication 8 X 4 can be written as two powers of 2.
When we multiply the numbers we might just as well add the powers 2 + 3 = 5.

This is what we mean when we say ‘logarithms turn multiplication into addition’.

Rewriting a number as a power of something is called taking the logarithm.
The 2-logarithm of 8 = 3
The 2-logarithm of 4 = 2
The 2-logarithm of 32 = 5

We usually work with 10-logarithms, because our number system is ten based.
The principles stay the same though.

I hope you now at least understand the concept.

You are very welcome.
If you like to learn more about logarithms, I have written a couple of posts about calculating them mentally.

I suggest starting with Arthur Benjamin’s book first though.

For the people who do not have the book, here is a spoiler.

Let’s say you need to calculate 43 X 77.
This can be rewritten as 40 X 80 + 3 X 37.
40 X 80 is easy: 3200.
3 X 37 is 111.
3200 + 111 =

Another one:
41 X 61 =
Rewrite to 40 X 62 + 1 X 21 =
2480 + 21 =

49^2 =
50 X 48 + 1 =
2400 + 1 =

51^2 =
50 X 52 + 1 =
2600 + 1 =

It’s the rewrite that makes the calculation easier.

Sam, I hope you are enjoying the book.

I would love a general discussion of these techniques, however there are usually not a lot of people on this part of the forum chiming in.

What I suggest as a general rule is trying to mentally calculate as much as possible.
Then every time you come across a difficult problem try to find a solution or try to find out how to make it easier.

In general people can add easily.
Multiplication of single digit numbers is easy, 2 digit numbers for most people become a bit of a challenge.
Square roots where the answer is a whole number (this is the perfect power you where talking about) for numbers up to 100 is easy for most people. After that it quickly becomes more difficult.
Square numbers, start with 1, 2, 3, etc and see where your limit is.

Try these out. If you hit your limit, put it on the forum and i’ll try to find a way, system or method to make these easier.
I see a lot of new members on the forum interested in mental calculation. Let us know what you are struggling with and we will help you overcome it.

After a while I enjoy reading a bit here again. Doing a 5*5 and beating modern devices of technique may be impossible but you`ll still impress people enough :-).

If it’s squaring that’s a particular interest, Arthur Benjamin has an online video where he focuses on just that:

Here is a YouTube playlist consisting of 18 videos that teach various aspects of fast mental math for free:

It covers addition, subtraction, multiplication, division, squaring, and much more. Many of the techniques used by Arthur Benjamin are illustrated in these videos.

Kinma, here is one of my favorite videos that is perfect for introducing the concept of logarithms:

The following is also a great post for really building your intuition about logarithms:

Hi Greymatters,

Great to have you back.
The mental calculation forum is quiet.
Shall we make it a bit more active?

Sounds like a great idea!

Some faster speed you will naturally develop, just by memorizing answers. For example, Dr. Arthur Benjamin does 2-digits squares the fastest I’ve ever seen. He doesn’t even calculate them anymore because he’s done them so much, he just knows the answer. That only comes with regular practice and performance.

Glad you enjoyed the logarithm videos!

I use a programmable calculator for this.
I have the calculator take 2 random numbers and mentally add them or multiply them.

This one:

I would say that I can try 3 digit numbers, but that this is way more difficult.

A difficult one is 731 squared:
731^2 = 700X762 + 31^2
700 X 762 = 490,000 + 42,000 + 1,400 = 533,400 (basically this is a single digit number times a 3 digit number and 2 zero’s added.)
31^2 = 30X32 +1 = 961
Total is 534,361

You could be lucky and get 115 squared:
115^2 = 100X130 + 15^2 = 13,000 + 225 = 13,225

The difficulty lies in keeping all those numbers in your head at the same time.

For squaring 3 digit numbers, you could also use cross multiplication:

731^2 =
ten thousands: 7 X 7 = 49 X 10,000 = 490,000
thousands: 7 X 3 X 2 = 42 thousand. Add to previous = 532,000
hundreds: 7 X 1 X 2 + 3 X 3 = 23 X 100 = 2300. Add to previous = 534300
tens: 3 X 1 X 2 = 6 X 10 = 60 -> 534360
ones: 1 X 1 = 1 -> 534361

Alternatively you could start with squaring 73.
Add two zeros, so you squared 730.
Then move to 731 by adding 730 and 731.

These calculations need you to be able to hold more digits in memory.
So a lot of practice is needed.

or using the trick for squaring any number ending in 5
115^2 = 11x12 hundred + 25 = 13,225
This is fun :slight_smile:

Maybe this thread is dead but I think this is the best place for my question. I do know the techniques of mental math. And I can work through them pretty fast, but I want to be the fastest I can be. I am willing to practice the same thing a thousand times, but I want it to be effective and really effective and most of all push me. How should I do that? What practice principles do you use? I do know programming so I can create a program to give me the problems so that’s not the issue.

Not anymore.

Start with finding out where you slow down.
If you do let’s say a 2x2 multiplication, which steps take the most time?
Is it:

  • finding the right conversion?
  • adding the numbers?
  • juggling zero’s?
  • etc.

Let’s take an extreme example:

Using standard ‘criss-cross’, you would get:
70X60 = 4200
(7X9+6X9)X10 = (13X9)x10 = (90+27)X10 = 1170
9x9 = 81
Add them up to get: 5451

You could also do:
(80-1) X (70-1) =
5600 -80-70 + 1 = 5451

For me, the second version is much quicker.

When training this, try to do the same calculation in multiple ways.
Also use casting out nines and elevens to find out whether your answer is right.