Wow! Just wow! Well done. And in a short amount of time.
Wow! Just wow! Well done. And in a short amount of time.
You really can’t do better than get detailed advice from Kinma. IMO those posts are worth money!
Methods are part of a toolkit. With experience you learn to spot which problems are easiest done this way or that. Every method is useful though you will favor some. Each one takes time to learn.
I say, learnt the squares up to 100. Turns out it’s not that much work and it’s a real boon.
Somewhere in here, it was suggested it’s okay to bump an old thread. Man, am I putting that to the test, here.
I just discovered mental math, because of this exact thread. I was researching something else and stumbled upon it. Not only was I intrigued by the concept of “Mental Math,” especially since I was researching something similar, the idea of getting to choose a method appeals to people like me. Or people like us, right?
As an added bonus, I hadn’t participated in this forum for too long.
I learn through reading, so I went to Amazon, and discovered a book called “Mental Math: Tricks To Become A Human Calculator .” It’s not only the top rated in its category, it’s also ranked pretty high overall. Seemed a good place to start.
it uses something the author calls Ofpad, which I haven’t seen referenced here. It’s solid, but googling it seems to send me to mostly self-referential information.
And it is. I’m up to multiplication and progressing well, as long as I can see the original problem. Holding the numbers mentally is somewhat problematic. I can do it with effort, for the easier problems. I use a variation of the Dominic system for numbers, with nearly half of the two-digit combos being a number association link instead. GREAT for memorizing cards or long strings of numbers. But when I visualize Neil Patrick Harris shaving a dog, it may be easy to know that is 989165. It doesn’t help me to do calculations with the numbers.
Wondering if I should switch to a shape number system for mental math. It might help me visualize the actual numbers. I also noted in this thread people using the Major system. I’m familiar with it, but not practiced at making words from the sounds. I preferred Dominic with PAO.
Any thoughts on the best way to hold the numbers in memory? Does anybody use ofpad at all?
It’s always okay (and encouraged) to bump old threads.
We’ve just been saying how we want to go through the old threads and collect all the useful stuff.
I use the Major System. I never saw it as having any real advantage for computation. I can’t think of any reason why another well tested system should make a difference. If you are already part way down one track, I wouldn’t switch.
Keeping track of powers of ten is tricky for most. One approach is to break free of columns and use a | symbol so that you don’t have finalize each intermediate computation.
eg 522 = 25 | 200 | 4 = 2704
This allows overflows and even negative numbers. 9 can be written as 10-1.
There is a fair bit of discussion about this in some other thread which I can’t find right now.
Not using memory techniques, what is the largest amount of digits you can do? Let’s say for a multiplication?
Hope I’m understanding the question right.
I’m up to multiplication in the learning. (My progress slowed down due to work and writing.) I’ve got my tables down up to 12x12.
I’m doing 2 digits times 2 digits without aid, but I’m not always getting the right answer when not looking at the question.
The method is one of two, depending on the problem. In some cases I’m breaking up the multiplier.
53 x 24 =
(53x20) + (53x4) =
1060 + 212 = 1272
It’s when I begin adding the last step that I’m messing up unless I use a memory technique. And sometimes even with that, as I’m just beginning this step and still practicing.
The other way is rounding up.
(87x100) -(87x1) =
8700 - 87 =
I’m finding rounding up much easier.
I’m pretty good at single digit multiplication now, up to 5 digits times 1 digit.
Have a look here if you want to: How can I start mental calculating
…or just half of 2400 and then adding 3 x 24
…using 100 as a base:
87 -13 99 -1
either across 87 - 1 = 86 or 99 - 13 = 86 or as 100 - 13 - 1 = 86 is your left and side and 1 * 13 is your right hand side.
I would advise you to play around with the criss cross method. As it is easier on the brain (less stuff to memorise).
In the case of:
You start with the numbers that result in the hundreds. So multiply 50 X 20 = 1,000.
Then the digits that result in the tens or 3 X 20 + 4 X 50 = 260.
Add to 1,000 gives 1,260.
Last step the 3 X 4 = 12.
Add to 1260 and get 1,272.
Always do either (or both) the 9 proof or 11 proof. I’ll start with 11:
53 mod 11 = 9 (subtract 44 from 53)
24 mod 11 = 2 (subtract 22 from 24)
9 X 2 = 18 and 18 mod 11 = 7
So the original numbers you started out with mod 11 or 9 and 2 multiplied together and. then also mod 11 ends up is 7.
Now the answer needs to also end up in 7 and it does:
1272 mod 11 = 172 mod 11 (subtract 1100)
172 mod 11 = 62 mod 11 (subtract 110)
62 mod 11 = 7 (subtract 55)
The 9 proof is easier to do, but I personally like the 11 proof because of the continued subtraction:
53 mog 9 = 8 (add the digits 5 and 3)
24 mod 9 = 6 (add the digits)
8 x 6 = 48 and 48 mod 9 = 3 (add digits: 4+8=12. Again add digits 1 and 2 = 3)
Now the answer mod 9 also needs to end up in 3 and it does.
1272 mod 9 = 3 (1+2+7+2 = 12 and 1+2 = 3)
Choose either the 9 or 11 proof or - and this is excellent training - do both.
As always a lot to type out, but with some proficiency it goes quick!
This is how you check your answer without a calculator.
Comparing the 2 methods.
Both work and both are excellent.
For you, I advice the criss-cross method. The reason is that you can do it digit by digit.
You first work out the hundreds, then the tens and last the ones.
In your method the first result is 1060, which means you need to need to keep both a thousand and a ten in your short term memory.
In the (criss) cross multiplication method you r first result is 1,000.
The second result is 26 (tens) or 260 which can be effortlessly added to 1,000.
And finally 12 can also easily be added to 1,260.
If you do this for a while you quickly start to look ahead.
Since your first result is 10 (hundreds) and your next step will be evaluating the tens there is zero change there will be a carry from the tens into the thousands. There will be a carry from the tens into the hundreds but not into the thousands.
Because there is no change of a carry into the thousands you know for sure that the first digit is 1.
So call out ‘one thousand and …’ and continue calculating.
Since you called out one thousand, there is no need to keep this in short term memory.
Just drop it! You will find that with practice your brain will still hold it.
If you ask yourself ‘what didd I just say?’, it will come back.
Next result is 260 and you look ahead at the 3x4. There is a carry into the tens, but not into the hundreds. The hundreds are certain. Now call out “… two hundred and …”.
Now drop the 200 from memory and think of the 60.
Last result is 12. Add to 60 and call out ‘seventy two!’.
Now, from the result the most amount of digits you are working with is 2!
You dropped the 1,000 and the 200 and you only needed to keep 60 in short term memory in order to add to the 12.
So the most taxing thing is working out the cross or the tens (3 X 2 (tens) + 4 X 5 (tens) = 26 tens or 260). Form the 260 you only need to keep 60 in your brain while working out 3x4.
So with looking ahead you can call out the numbers that will not change in the calculation while working out the rest.
This alleviates the need to keep the whole answer in memory.
No need for any memory system until maybe you do 4 digit multiplication.
I don’t know if this is clear for everybody.
Check out Vedic Mathematics with a base.
Basically this is advanced cross multiplication.
in the previous example of:
we split 53 into 50 and 3 and 24 into 20 and 4.
Mentally try to see this as:
50 | 3
20 | 4 X
in your example of:
Try to see this as:
100 | -13
100 | -1 X
And start with 100 X 100 = 10,000.
Next step -13 x 100 + -1 X 100 = (-13-1) X 100 = -1,400. 10,000 - 1,400 = 8,600.
Last: -13 X -1 = 13
In the Vedic method you postpone the first multiplication with 100.
So you do: 100 - (13-1) = 86.
Then multiply with 100 or add 2 zeros: 8600.
So you look at the numbers from a “base 100”, meaning you look at 87 from 100 or -13.